A051624 -id:A051624 - cp's OEIS frontend

A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

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

Offset: 0

Omar E. Pol, Aug 09 2018

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).

			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, ...
...

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

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).

T(n,k) = A139600(n-2,k) if n >= 2.

T(n,k) = A139601(n-3,k) if n >= 3.

A342709 12-gonal (dodecagonal) square numbers.

Original entry on oeis.org

1, 64, 3025, 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, 7442093853169599697984, 349619996931001511354641, 16424697761903901433970161

Offset: 1

Bernard Schott, Mar 19 2021

The 12-gonal square numbers k correspond to the nonnegative integer solutions of the Diophantine equation k = d*(5*d-4) = c^2, equivalent to (5*d-2)^2 - 5*c^2 = 4. Substituting x = 5*d-2 and y = c gives the Pell-Fermat's equation x^2 - 5*y^2 = 4.
The solutions x are in A342710, while corresponding solutions y that are also the indices c of the squares which are 12-gonal are in A033890.
The indices d of the corresponding 12-gonal which are squares are in A081068.

			142129 = 169*(5*169-4) = 377^2, so 142129 is the 169th 12-gonal number and the 377th square, hence 142129 is a term.

Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Intersection of A000290 (squares) and A051624 (12-gonal numbers).

Similar for n-gonal squares: A001110 (triangular), A036353 (pentagonal), A046177 (hexagonal), A036354 (heptagonal), A036428 (octagonal), A036411 (9-gonal), A188896 (there are no 10-gonal squares > 1), A333641 (11-gonal), this sequence (12-gonal).

with(combinat):
seq(fibonacci(4*n-2)^2, n=1..16);

Table[Fibonacci[4*n - 2]^2, {n, 1, 16}] (* Amiram Eldar, Mar 19 2021 *)

a(n) = fibonacci(4*n-2)^2; \\ Michel Marcus, Mar 21 2021

G.f.: x*(1 + 16*x + x^2)/((1 - x)*(1 - 47*x + x^2)). - Stefano Spezia, Mar 20 2021
a(n) = 48*a(n-1) - 48*a(n-2) + a(n-3). - Kevin Ryde, Mar 20 2021
a(n) = 9*A161582(n) + 1. - Hugo Pfoertner, Mar 19 2021
a(n) = A033890(n-1)^2.

A051799 Partial sums of A007587.

Original entry on oeis.org

1, 14, 60, 170, 385, 756, 1344, 2220, 3465, 5170, 7436, 10374, 14105, 18760, 24480, 31416, 39729, 49590, 61180, 74690, 90321, 108284, 128800, 152100, 178425, 208026, 241164, 278110, 319145, 364560, 414656, 469744, 530145, 596190

Offset: 0

Barry E. Williams, Dec 11 1999

4-dimensional pyramidal number, composed of consecutive 3-dimensional slices; each of which is a 3-dimensional 12-gonal (or dodecagonal) pyramidal number; which in turn is composed of consecutive 2-dimensional slices 12-gonal numbers. - Jonathan Vos Post, Mar 17 2006

Convolution of A000027 with A051624 (excluding 0). - Bruno Berselli, Dec 07 2012

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A007587, A001622, A051624.
Cf. A093645 ((10, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

/* A000027 convolved with A051624 (excluding 0): */ A051624:=func; [&+[(n-i+1)*A051624(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Accumulate[Table[n(n+1)(10n-7)/6,{n,0,50}]] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.
G.f.: (1+9*x)/(1-x)^5.
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = (125*log(5) + 10*sqrt(5*(5-2*sqrt(5)))*Pi - 50*sqrt(5)*log(phi) - 84)/104, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = (50*sqrt(5)*log(phi) + 5*sqrt(50-10*sqrt(5))*Pi - 256*log(2) + 90)/52. (End)

A172117 a(n) = n(n+1)(20*n-17)/6.

Original entry on oeis.org

0, 1, 23, 86, 210, 415, 721, 1148, 1716, 2445, 3355, 4466, 5798, 7371, 9205, 11320, 13736, 16473, 19551, 22990, 26810, 31031, 35673, 40756, 46300, 52325, 58851, 65898, 73486, 81635, 90365, 99696, 109648, 120241, 131495, 143430, 156066

Offset: 0

Vincenzo Librandi, Jan 26 2010

Generated by the formula n*(n+1)*(2*d*n-2*d+3)/6 for d=10.
This sequence is related to A051624 by a(n) = n*A051624(n) - Sum_{i=0..n-1} A051624(i) = n*(n+1)*(20*n-17)/2; in fact, this is the case d=10 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Aug 26 2010
Also, a(n) = n*A190816(n) - Sum_{i=0..n-1} A190816(i) for n>0. - Bruno Berselli, Dec 18 2013
Starting with offset 1, the sequence is the binomial transform of (1, 22, 41, 20, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. [Bruno Berselli, Feb 13 2014]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051624.

Cf. similar sequences listed in A237616.

[n*(n+1)*(20*n-17)/6: n in [0..50]]; // Vincenzo Librandi, Aug 01 2015

Table[(20n^3+3n^2-17n)/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,23,86},40] (* Harvey P. Dale, May 15 2011 *)

a(n)=n*(20*n^2+3*n-17)/6 \\ Charles R Greathouse IV, Jan 11 2012

[sum( (-1)^j*(20-j)*binomial(n+2-j, 3-j) for j in (0..1)) for n in (0..50)] # G. C. Greubel, Apr 15 2022

G.f.: x*(1+19*x)/(1-x)^4. - Bruno Berselli, Aug 26 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Harvey P. Dale, May 15 2011
a(n) = Sum_{i=0..n-1} (n-i)*(20*i+1), with a(0)=0. - Bruno Berselli, Feb 11 2014
E.g.f.: (1/6)*x*(6 + 63*x + 20*x^2)*exp(x). - G. C. Greubel, Apr 15 2022

A226490 a(n) = n(19n-15)/2.

Original entry on oeis.org

0, 2, 23, 63, 122, 200, 297, 413, 548, 702, 875, 1067, 1278, 1508, 1757, 2025, 2312, 2618, 2943, 3287, 3650, 4032, 4433, 4853, 5292, 5750, 6227, 6723, 7238, 7772, 8325, 8897, 9488, 10098, 10727, 11375, 12042, 12728, 13433, 14157, 14900, 15662, 16443, 17243, 18062

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th hendecagonal number and n-th dodecagonal number.

Sum of reciprocals of a(n), for n > 0: 0.59314195720519963010713286193275...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A051682, A051873.

Cf. numbers of the form n*(n*k - k + 4)/2, this sequence is the case k=19: see list in A226488.

Magma
```
[n*(19*n-15)/2: n in [0..50]];
```

I:=[0,2,23]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (19 n - 15)/2, {n, 0, 50}]
CoefficientList[Series[x (2 + 17 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,2,23},50] (* Harvey P. Dale, Aug 17 2017 *)

a(n)=n*(19*n-15)/2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(2+17*x)/(1-x)^3.
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(4 + 19*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n + A051873(n). (End)

A226491 a(n) = n(21n-17)/2.

Original entry on oeis.org

0, 2, 25, 69, 134, 220, 327, 455, 604, 774, 965, 1177, 1410, 1664, 1939, 2235, 2552, 2890, 3249, 3629, 4030, 4452, 4895, 5359, 5844, 6350, 6877, 7425, 7994, 8584, 9195, 9827, 10480, 11154, 11849, 12565, 13302, 14060, 14839, 15639, 16460, 17302, 18165, 19049, 19954

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th dodecagonal number and n-th tridecagonal number.

Sum of reciprocals of a(n), for n > 0: 0.58517199913243139233033474262449...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A051865, A064762.

Cf. numbers of the form n*(n*k - k + 4)/2, this sequence is the case k=21: see list in A226488.

Magma
```
[n*(21*n-17)/2: n in [0..50]];
```

I:=[0,2,25]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (21 n - 17)/2, {n, 0, 50}]
CoefficientList[Series[x (2 + 19 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,2,25},50] (* Harvey P. Dale, Feb 01 2023 *)

a(n)=n*(21*n-17)/2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(2+19*x)/(1-x)^3.
a(n) + a(-n) = A064762(n).
From Elmo R. Oliveira, Jan 12 2025: (Start)
E.g.f.: exp(x)*x*(4 + 21*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A372025 Maximum second Zagreb index of maximal 3-degenerate graphs with n vertices.

Original entry on oeis.org

12, 54, 120, 210, 324, 462, 624, 810, 1020, 1254, 1512, 1794, 2100, 2430, 2784, 3162, 3564, 3990, 4440, 4914, 5412, 5934, 6480, 7050, 7644, 8262, 8904, 9570, 10260, 10974, 11712, 12474, 13260, 14070, 14904, 15762, 16644, 17550, 18480, 19434, 20412, 21414, 22440, 23490, 24564, 25662, 26784, 27930

Offset: 3

Allan Bickle, Apr 16 2024

The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.

A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are 3-stars, so the bound also applies to 3-trees.

			The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
Cf. A372027 (second Zagreb index of MOPs).

LinearRecurrence[{3, -3, 1}, {12, 54, 120}, 50] (* Paolo Xausa, Jan 22 2025 *)

a(n) = 3*(n-1)^2 + 9*(n-3)*(n-1).
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
G.f.: x^3*(6*x^2 - 18*x - 12)/(x - 1)^3. (End)
a(n) = 6*A014107(n-1). Sum_{n>=3} 1/a(n) = (1/2+log(2))/9 = 0.1325719... - R. J. Mathar, Apr 22 2024

A372026 Minimum second Zagreb index of maximal 2-degenerate graphs with n vertices.

Original entry on oeis.org

12, 33, 51, 86, 116, 147, 178, 210, 242, 274, 306, 338, 370, 402, 434, 466, 498, 530, 562, 594, 626, 658, 690, 722, 754, 786, 818, 850, 882, 914, 946, 978, 1010, 1042, 1074, 1106, 1138, 1170, 1202, 1234, 1266, 1298, 1330, 1362, 1394, 1426, 1458, 1490, 1522, 1554, 1586, 1618, 1650, 1682, 1714, 1746, 1778, 1810

Offset: 3

Allan Bickle, Apr 16 2024

nonn

The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.

A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices. The extremal graphs are described in (Bickle 2024).

			The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
Cf. A372027 (second Zagreb index of MOPs).

LinearRecurrence[{2, -1}, {12, 33, 51, 86, 116, 147, 178, 210}, 60] (* Paolo Xausa, Jan 22 2025 *)

a(n) = 32*n-110 for n>8.
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 10.
G.f.: x^3*(x^7 + x^5 - 5*x^4 + 17*x^3 - 3*x^2 + 9*x + 12)/(x - 1)^2. (End)

A372027 Maximum second Zagreb index of maximal outerplanar graphs with n vertices.

Original entry on oeis.org

12, 33, 61, 96, 135, 181, 233, 291, 355, 425, 501, 583, 671, 765, 865, 971, 1083, 1201, 1325, 1455, 1591, 1733, 1881, 2035, 2195, 2361, 2533, 2711, 2895, 3085, 3281, 3483, 3691, 3905, 4125, 4351, 4583, 4821, 5065, 5315, 5571, 5833, 6101, 6375, 6655, 6941, 7233, 7531

Offset: 3

Allan Bickle, Apr 16 2024

nonn

The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.

A maximal outerplanar graph has all vertices on the exterior region, and all other regions triangles. The extremal graphs are fans, except when n=6. Then the extremal graph is the triangular grid with degrees 4,4,4,2,2,2.

			The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
A. Hou, S. Li, L. Song, and B. Wei, Sharp bounds for Zagreb indices of maximal outerplanar graphs, J Comb Optim 22 (2011), 252-269.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).

Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).

LinearRecurrence[{3, -3, 1}, {12, 33, 61, 96, 135, 181, 233}, 50] (* Paolo Xausa, Jan 22 2025 *)

a(n) = 3*n^2 + n - 19 when n is not 3 or 6.
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 9.
G.f.: x^3*(x^6 - 3*x^5 + 3*x^4 + 2*x^2 + 3*x - 12)/(x - 1)^3. (End)

A193872 Even dodecagonal numbers: a(n) = 4n(5*n - 2).

Original entry on oeis.org

0, 12, 64, 156, 288, 460, 672, 924, 1216, 1548, 1920, 2332, 2784, 3276, 3808, 4380, 4992, 5644, 6336, 7068, 7840, 8652, 9504, 10396, 11328, 12300, 13312, 14364, 15456, 16588, 17760, 18972, 20224, 21516, 22848, 24220, 25632, 27084, 28576, 30108, 31680, 33292, 34944

Offset: 0

Omar E. Pol, Aug 19 2011

Even 12-gonal numbers. Bisection of A051624.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016885, A051624, A147874.

PolygonalNumber[12,2*Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 02 2017 *)

a(n)=4*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*A147874(n+1).
a(n) = 4*n*A016885(n-1), n >= 1.
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 4*x*(3 + 7*x)/(1 - x)^3.
E.g.f.: 4*x*(3 + 5*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

cp's OEIS Frontend

A317302 Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

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A342709 12-gonal (dodecagonal) square numbers.

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Maple

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A051799 Partial sums of A007587.

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Magma

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A172117 a(n) = n*(n+1)*(20*n-17)/6.

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Magma

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A226490 a(n) = n*(19*n-15)/2.

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Magma

Magma

Mathematica

PARI

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A226491 a(n) = n*(21*n-17)/2.

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A372025 Maximum second Zagreb index of maximal 3-degenerate graphs with n vertices.

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A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

A172117 a(n) = n(n+1)(20*n-17)/6.

A226490 a(n) = n(19n-15)/2.

A226491 a(n) = n(21n-17)/2.

A193872 Even dodecagonal numbers: a(n) = 4n(5*n - 2).