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Search: a051873 -id:a051873
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A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards. +10
49
0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.
The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.
Index to sequences related to polygonal numbers
FORMULA
T(n,k) = n*(k-1)*k/2+k.
T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016
From Stefano Spezia, Apr 12 2024: (Start)
G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)
EXAMPLE
The square array of nonnegatives together with polygonal numbers begins:
=========================================================
....................... A A . . A A A A
....................... 0 0 . . 0 0 1 1
....................... 0 0 . . 1 1 3 3
....................... 0 0 . . 6 7 9 9
....................... 0 0 . . 9 3 6 6
....................... 0 1 . . 5 2 0 0
....................... 4 2 . . 7 9 6 7
=========================================================
Nonnegatives . A001477: 0, 1, 2, 3, 4, 5, 6, 7, ...
Triangulars .. A000217: 0, 1, 3, 6, 10, 15, 21, 28, ...
Squares ...... A000290: 0, 1, 4, 9, 16, 25, 36, 49, ...
Pentagonals .. A000326: 0, 1, 5, 12, 22, 35, 51, 70, ...
Hexagonals ... A000384: 0, 1, 6, 15, 28, 45, 66, 91, ...
Heptagonals .. A000566: 0, 1, 7, 18, 34, 55, 81, 112, ...
Octagonals ... A000567: 0, 1, 8, 21, 40, 65, 96, 133, ...
9-gonals ..... A001106: 0, 1, 9, 24, 46, 75, 111, 154, ...
10-gonals .... A001107: 0, 1, 10, 27, 52, 85, 126, 175, ...
11-gonals .... A051682: 0, 1, 11, 30, 58, 95, 141, 196, ...
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217, ...
...
=========================================================
The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.
MAPLE
T:= (n, k)-> n*(k-1)*k/2+k:
seq(seq(T(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, Oct 14 2018
MATHEMATICA
T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)
PROG
(Python)
def A139600Row(n):
x, y = 1, 1
yield 0
while True:
yield x
x, y = x + y + n, y + n
for n in range(8):
R = A139600Row(n)
print([next(R) for _ in range(11)]) # Peter Luschny, Aug 04 2019
(Magma)
T:= func< n, k | k*(n*(k-1)+2)/2 >;
A139600:= func< n, k | T(n-k, k) >;
[A139600(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024
(SageMath)
def T(n, k): return k*(n*(k-1)+2)/2
def A139600(n, k): return T(n-k, k)
flatten([[A139600(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024
CROSSREFS
Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.
A formal extension negative n is in A326728.
Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011
Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
KEYWORD
nonn,tabl,easy
AUTHOR
Omar E. Pol, Apr 27 2008
EXTENSIONS
Edited by Omar E. Pol, Jan 05 2009
STATUS
approved
A303298 Generalized 21-gonal (or icosihenagonal) numbers: m*(19*m - 17)/2 with m = 0, +1, -1, +2, -2, +3, -3, ... +10
30
0, 1, 18, 21, 55, 60, 111, 118, 186, 195, 280, 291, 393, 406, 525, 540, 676, 693, 846, 865, 1035, 1056, 1243, 1266, 1470, 1495, 1716, 1743, 1981, 2010, 2265, 2296, 2568, 2601, 2890, 2925, 3231, 3268, 3591, 3630, 3970, 4011, 4368, 4411, 4785, 4830, 5221, 5268, 5676, 5725, 6150, 6201, 6643, 6696, 7155, 7210
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Numbers k for which 152*k + 289 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317317. - Omar E. Pol, Jul 28 2018
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
G.f.: -(x^2+17*x+1)*x/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 23 2018
From Colin Barker, Jun 24 2018: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = (19*n^2 + 34*n) / 8 for n even.
a(n) = (19*n^2 + 4*n - 15) / 8 for n odd.
(End)
Sum_{n>=1} 1/a(n) = 38/289 + 2*Pi*cot(2*Pi/19)/17. - Amiram Eldar, Feb 28 2022
MAPLE
a:= n-> (m-> m*(19*m-17)/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60); # Alois P. Heinz, Jun 23 2018
MATHEMATICA
CoefficientList[Series[-(x^2 + 17 x + 1) x/((x + 1)^2*(x - 1)^3), {x, 0, 55}], x] (* or *)
Array[PolygonalNumber[21, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 56, 0] (* Michael De Vlieger, Jul 10 2018 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 18, 21, 55}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
PROG
(PARI) concat(0, Vec(x*(1 + 17*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jun 24 2018
(GAP) a:=[0, 1, 18, 21, 55];; for n in [6..60] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Jul 10 2018
CROSSREFS
Cf. A051873, A316672, A317317.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), this sequence (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Jun 23 2018
STATUS
approved
A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k. +10
16
0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - Omar E. Pol, Dec 21 2008
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.
FORMULA
T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)
EXAMPLE
The square array of polygonal numbers begins:
========================================================
Triangulars .. A000217: 0, 1, 3, 6, 10, 15, 21, 28,
Squares ...... A000290: 0, 1, 4, 9, 16, 25, 36, 49,
Pentagonals .. A000326: 0, 1, 5, 12, 22, 35, 51, 70,
Hexagonals ... A000384: 0, 1, 6, 15, 28, 45, 66, 91,
Heptagonals .. A000566: 0, 1, 7, 18, 34, 55, 81, 112,
Octagonals ... A000567: 0, 1, 8, 21, 40, 65, 96, 133,
9-gonals ..... A001106: 0, 1, 9, 24, 46, 75, 111, 154,
10-gonals .... A001107: 0, 1, 10, 27, 52, 85, 126, 175,
11-gonals .... A051682: 0, 1, 11, 30, 58, 95, 141, 196,
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
And so on ..............................................
========================================================
MATHEMATICA
T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)
PROG
(Magma)
T:= func< n, k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n, k | T(n-k, k) >;
[A139601(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024
(SageMath)
def T(n, k): return k*((n+1)*(k-1)+2)/2
def A139601(n, k): return T(n-k, k)
flatten([[A139601(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024
CROSSREFS
Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.
KEYWORD
nonn,tabl,easy
AUTHOR
Omar E. Pol, Apr 27 2008
STATUS
approved
A255184 25-gonal numbers: a(n) = n*(23*n-21)/2. +10
12
0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, 8722, 9367, 10035, 10726, 11440, 12177, 12937, 13720, 14526, 15355, 16207, 17082, 17980
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac).
Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples:
for k=4, A000290(n) = A000217(n) + A000217(n-1);
for k=5, A000326(n) = A000290(n) + A000217(n-1);
for k=6, A000384(n) = A000326(n) + A000217(n-1), etc.
This is the case k=25.
REFERENCES
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).
E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.
LINKS
Luciano Ancora, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: x*(-1 - 22*x)/(-1 + x)^3.
a(n) = A000217(n) + 22*A000217(n-1) = A051876(n) + A000217(n-1), see comments.
Product_{n>=2} (1 - 1/a(n)) = 23/25. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 23*x^2/2). - Nikolaos Pantelidis, Feb 05 2023
MATHEMATICA
Table[n (23 n - 21)/2, {n, 40}]
PROG
(Magma) k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015
(PARI) a(n)=n*(23*n-21)/2 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22), A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Apr 03 2015
STATUS
approved
A195527 Integers n that are k-gonal for precisely 3 distinct values of k, where k >= 3. +10
7
15, 21, 28, 51, 55, 64, 70, 75, 78, 91, 96, 100, 111, 112, 117, 126, 135, 136, 141, 144, 145, 148, 154, 156, 165, 175, 176, 186, 189, 195, 201, 204, 216, 232, 235, 238, 246, 255, 256, 285, 286, 288, 291, 297, 300, 306, 315, 316, 321, 322, 324, 330, 333, 336
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A177025 for number of ways a number can be represented as a polygonal number.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000
EXAMPLE
21 is in the sequence because it is a triangular number (A000217), an octagonal number (A000567) and an icosihenagonal number (A051873).
MATHEMATICA
data1=Reduce[1/2 n (n(k-2)+4-k)== # && k>=3 && n>0, {k, n}, Integers]&/@Range[336]; data2=If[Head[#]===And, 1, Length[#]] &/@data1; data3=DeleteCases[Table[If[data2[[k]]==3, k], {k, 1, Length[data2]}], Null]
PROG
(Python)
A195527_list = []
for m in range(1, 10**4):
n, c = 3, 0
while n*(n+1) <= 2*m:
if not 2*(n*(n-2) + m) % (n*(n - 1)):
c += 1
if c > 2:
break
n += 1
if c == 2:
A195527_list.append(m) # Chai Wah Wu, Jul 28 2016
CROSSREFS
Cf. A000217, A000567, A051873, A177025, A177029.
KEYWORD
nonn
AUTHOR
Ant King, Sep 21 2011
STATUS
approved
A317302 Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards. +10
5
0, 0, 1, 0, 1, 0, 0, 1, 1, -3, 0, 1, 2, 0, -8, 0, 1, 3, 3, -2, -15, 0, 1, 4, 6, 4, -5, -24, 0, 1, 5, 9, 10, 5, -9, -35, 0, 1, 6, 12, 16, 15, 6, -14, -48, 0, 1, 7, 15, 22, 25, 21, 7, -20, -63, 0, 1, 8, 18, 28, 35, 36, 28, 8, -27, -80, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, -35, -99, 0, 1, 10, 24, 40, 55, 66
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,10
COMMENTS
Note that the formula gives several kinds of numbers, for example:
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the nonnegative numbers.
For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).
LINKS
Table of n, a(n) for n=0..84.
Omar E. Pol, Polygonal numbers.
The OEIS, Polygonal numbers.
University of Surrey, Dept. of Mathematics, Polygonal Numbers - or Numbers as Shapes.
Eric Weisstein's World of Mathematics, Figurate Number.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers
FORMULA
T(n,k) = A139600(n-2,k) if n >= 2.
T(n,k) = A139601(n-3,k) if n >= 3.
EXAMPLE
Array begins:
------------------------------------------------------------------------
n\k Numbers Seq. No. 0 1 2 3 4 5 6 7 8
------------------------------------------------------------------------
0 ............ (A258837): 0, 1, 0, -3, -8, -15, -24, -35, -48, ...
1 ............ (A080956): 0, 1, 1, 0, -2, -5, -9, -14, -20, ...
2 Nonnegatives A001477: 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
3 Triangulars A000217: 0, 1, 3, 6, 10, 15, 21, 28, 36, ...
4 Squares A000290: 0, 1, 4, 9, 16, 25, 36, 49, 64, ...
5 Pentagonals A000326: 0, 1, 5, 12, 22, 35, 51, 70, 92, ...
6 Hexagonals A000384: 0, 1, 6, 15, 28, 45, 66, 91, 120, ...
7 Heptagonals A000566: 0, 1, 7, 18, 34, 55, 81, 112, 148, ...
8 Octagonals A000567: 0, 1, 8, 21, 40, 65, 96, 133, 176, ...
9 9-gonals A001106: 0, 1, 9, 24, 46, 75, 111, 154, 204, ...
10 10-gonals A001107: 0, 1, 10, 27, 52, 85, 126, 175, 232, ...
11 11-gonals A051682: 0, 1, 11, 30, 58, 95, 141, 196, 260, ...
12 12-gonals A051624: 0, 1, 12, 33, 64, 105, 156, 217, 288, ...
13 13-gonals A051865: 0, 1, 13, 36, 70, 115, 171, 238, 316, ...
14 14-gonals A051866: 0, 1, 14, 39, 76, 125, 186, 259, 344, ...
15 15-gonals A051867: 0, 1, 15, 42, 82, 135, 201, 280, 372, ...
...
CROSSREFS
Column 0 gives A000004.
Column 1 gives A000012.
Column 2 gives A001477, which coincides with the row numbers.
Main diagonal gives A060354.
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the same as column 2.
For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30).
Cf. A139600, A139601.
Cf. A303301 (similar table but with generalized polygonal numbers).
KEYWORD
sign,tabl,easy
AUTHOR
Omar E. Pol, Aug 09 2018
STATUS
approved
A098923 33-gonal numbers: n(31n-29)/2. +10
2
0, 1, 33, 96, 190, 315, 471, 658, 876, 1125, 1405, 1716, 2058, 2431, 2835, 3270, 3736, 4233, 4761, 5320, 5910, 6531, 7183, 7866, 8580, 9325, 10101, 10908, 11746, 12615, 13515, 14446, 15408, 16401, 17425, 18480, 19566, 20683, 21831, 23010
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Similar to 21-gonal and 15-gonal numbers (A051873, A051867).
LINKS
Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = n*(31*n-29)/2.
G.f.: x*(1+30*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(n) = 31*n + a(n-1) - 30 (with a(0)=0). - Vincenzo Librandi, Nov 16 2010
E.g.f.: exp(x)*(x + 31*x^2/2). - Nikolaos Pantelidis, Feb 10 2023
MATHEMATICA
Table[n(31n - 29)/2, {n, 0, 40}] (* Stefan Steinerberger, Feb 15 2006 *)
PolygonalNumber[33, Range[0, 40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 02 2018 *)
PROG
(PARI) a(n)=n*(31*n-29)/2 \\ Charles R Greathouse IV, Oct 16 2015
CROSSREFS
Cf. A051867, A051873.
KEYWORD
nonn,easy
AUTHOR
Parthasarathy Nambi, Oct 18 2004
EXTENSIONS
More terms from Stefan Steinerberger, Feb 15 2006
STATUS
approved
A215144 a(n) = 19*n + 1. +10
2
1, 20, 39, 58, 77, 96, 115, 134, 153, 172, 191, 210, 229, 248, 267, 286, 305, 324, 343, 362, 381, 400, 419, 438, 457, 476, 495, 514, 533, 552, 571, 590, 609, 628, 647, 666, 685, 704, 723, 742, 761, 780, 799, 818, 837, 856, 875, 894, 913, 932, 951, 970, 989
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Jeremy Gardiner, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
From G. C. Greubel, Apr 19 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (1+18*x)/(1-x)^2.
E.g.f.: (1+19*x)*exp(x). (End)
MATHEMATICA
Range[1, 1000, 19]
19*Range[0, 60]+1 (* Harvey P. Dale, Nov 14 2014 *)
LinearRecurrence[{2, -1}, {1, 20}, 50] (* G. C. Greubel, Apr 19 2018 *)
PROG
(PARI) for(n=0, 50, print1(19*n + 1, ", ")) \\ G. C. Greubel, Apr 19 2018
(Magma) I:=[1, 20]; [n le 2 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Apr 19 2018
CROSSREFS
Cf. A141868, A161705.
First differences of A051873.
KEYWORD
nonn,easy
AUTHOR
Jeremy Gardiner, Aug 04 2012
STATUS
approved
A237618 a(n) = n*(n + 1)*(19*n - 16)/6. +10
2
0, 1, 22, 82, 200, 395, 686, 1092, 1632, 2325, 3190, 4246, 5512, 7007, 8750, 10760, 13056, 15657, 18582, 21850, 25480, 29491, 33902, 38732, 44000, 49725, 55926, 62622, 69832, 77575, 85870, 94736, 104192, 114257, 124950, 136290, 148296, 160987, 174382
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Also 21-gonal (or icosihenagonal) pyramidal numbers.
REFERENCES
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (nineteenth row of the table).
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pyramidal Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: x*(1 + 18*x) / (1 - x)^4.
a(n) = (1/2)*( n*A226490(n) - Sum_{j=0..n-1} A226490(j) ).
a(n) = Sum_{i=0..n-1} (n-i)*(19*i+1), for n>0; see the generalization in A237616 (Formula field).
From G. C. Greubel, May 27 2022: (Start)
a(n) = binomial(n+2, 3) + 18*binomial(n+1, 3).
E.g.f.: (1/6)*x*(6 + 60*x + 19*x^2)*exp(x). (End)
EXAMPLE
After 0, the sequence is provided by the row sums of the triangle:
1;
2, 20;
3, 40, 39;
4, 60, 78, 58;
5, 80, 117, 116, 77;
6, 100, 156, 174, 154, 96;
7, 120, 195, 232, 231, 192, 115;
8, 140, 234, 290, 308, 288, 230, 134;
9, 160, 273, 348, 385, 384, 345, 268, 153;
10, 180, 312, 406, 462, 480, 460, 402, 306, 172; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 19*r-18 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.
MATHEMATICA
Table[n(n+1)(19n-16)/6, {n, 0, 40}]
CoefficientList[Series[x(1+18x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
PROG
(Magma) [n*(n+1)*(19*n-16)/6: n in [0..40]];
(Magma) I:=[0, 1, 22, 82]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014
(SageMath) b=binomial; [b(n+2, 3) +18*b(n+1, 3) for n in (0..50)] # G. C. Greubel, May 27 2022
CROSSREFS
Cf. A051873, A226490.
Cf. similar sequences listed in A237616.
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Feb 11 2014
STATUS
approved
A098230 75-gonal numbers: a(n) = n*(73*n-71)/2. +10
1
0, 1, 75, 222, 442, 735, 1101, 1540, 2052, 2637, 3295, 4026, 4830, 5707, 6657, 7680, 8776, 9945, 11187, 12502, 13890, 15351, 16885, 18492, 20172, 21925, 23751, 25650, 27622, 29667, 31785, 33976, 36240, 38577, 40987, 43470, 46026, 48655, 51357, 54132, 56980, 59901, 62895, 65962, 69102, 72315, 75601, 78960, 82392, 85897, 89475
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: -x*(1+72*x) / (x-1)^3. - R. J. Mathar, Feb 05 2011
a(n) = n*(73*n - 71)/2.
E.g.f.: exp(x)*(x + 73*x^2/2). - Nikolaos Pantelidis, Feb 10 2023
MAPLE
A098230 := proc(n) n*(73*n-71)/2 ; end proc:
seq(A098230(n), n=0..20) ; # R. J. Mathar, Feb 04 2011
PROG
(Magma) [ n*(73*n - 71)/2: n in [0..50] ]; // Vincenzo Librandi, Feb 04 2011
(PARI) a(n)=n*(73*n-71)/2 \\ Charles R Greathouse IV, Oct 16 2015
CROSSREFS
Cf. A051867, A051873.
KEYWORD
nonn,easy
AUTHOR
Parthasarathy Nambi, Oct 25 2004
STATUS
approved
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Last modified September 19 08:53 EDT 2024. Contains 376007 sequences. (Running on oeis4.)