A051682 -id:A051682 - cp's OEIS frontend

A051879 Partial sums of A051798.

1, 14, 69, 224, 574, 1260, 2478, 4488, 7623, 12298, 19019, 28392, 41132, 58072, 80172, 108528, 144381, 189126, 244321, 311696, 393162, 490820, 606970, 744120, 904995, 1092546, 1309959, 1560664

Offset: 0

Barry E. Williams, Dec 14 1999

Convolution of triangular numbers (A000217) and 11-gonal numbers (A051682). [Bruno Berselli, Jul 21 2015]

A. 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. 1-16.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A051798; A000217, A051682.
Cf. A093644((9, 1) Pascal, column m=5).
Cf. A050405.

Accumulate[Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}]] (* Harvey P. Dale, Aug 19 2012 *)

a(n) = C(n+4, 4)*(9n+5)/5.

G.f.: (1+8*x)/(1-x)^6.

A261191 40-gonal numbers: a(n) = 38n(n-1)/2 + n.

Original entry on oeis.org

0, 1, 40, 117, 232, 385, 576, 805, 1072, 1377, 1720, 2101, 2520, 2977, 3472, 4005, 4576, 5185, 5832, 6517, 7240, 8001, 8800, 9637, 10512, 11425, 12376, 13365, 14392, 15457, 16560, 17701, 18880, 20097, 21352, 22645, 23976, 25345, 26752, 28197, 29680, 31201

Offset: 0

Sergey Pavlov, Aug 11 2015

According to the common formula for the polygonal numbers: (s-2)*n*(n-1)/2 + n (here s = 40).
The 4th number of the sequence, 117, is also the 10th pentagonal number (see A000326). The next number of the series, 232, is also the 9th decagonal number (see A001107), while 576 is the 25th square number (see A000290). The 12th number of the sequence, 2101, is the 23rd 11-gonal number (see A051682).
From Bruno Berselli, Aug 21 2015: (Start)
a(n) and a(n) - 2*n + 1 provide the numbers m such that 19*m + 81 is a square.
Partial sums of the numbers of the type 38*h + 1 (quadrisections of A113541 and A151979). (End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

JavaScript
```
function a(n){return 38*n*(n-1)/2+n}
```

[n*(19*n-18): n in [0..45]]; // Vincenzo Librandi, Aug 12 2015

A261191:=n->38*n*(n-1)/2+n: seq(A261191(n), n=0..50); # Wesley Ivan Hurt, Aug 15 2015

Table[n (19 n - 18), {n, 0, 45}] (* Bruno Berselli, Aug 21 2015 *)

concat(0, Vec(-x*(37*x+1)/(x-1)^3 + O(x^100))) \\ Colin Barker, Aug 11 2015

first(m)=my(v=vector(m,i,i--;38*i*(i-1)/2+i));v; \\ Anders Hellström, Aug 13 2015

a(n) = n*(19*n - 18).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. - Colin Barker, Aug 11 2015
G.f.: -x*(37*x+1) / (x-1)^3. - Colin Barker, Aug 11 2015
E.g.f.: exp(x)*(x + 19*x^2). - Nikolaos Pantelidis, Feb 10 2023

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

Original entry on oeis.org

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.

A152995 Twice 11-gonal numbers: a(n) = n(9n-7).

Original entry on oeis.org

0, 2, 22, 60, 116, 190, 282, 392, 520, 666, 830, 1012, 1212, 1430, 1666, 1920, 2192, 2482, 2790, 3116, 3460, 3822, 4202, 4600, 5016, 5450, 5902, 6372, 6860, 7366, 7890, 8432, 8992, 9570, 10166, 10780, 11412, 12062, 12730, 13416, 14120

Offset: 0

Omar E. Pol, Dec 22 2008

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

Cf. A051682 (11-gonal numbers).

Cf. A226488.

List([0..50], n-> n*(9*n-7)); # G. C. Greubel, Sep 01 2019

Magma
```
[n*(9*n-7): n in [0..50]];
```

seq(n*(9*n-7), n=0..50); # G. C. Greubel, Sep 01 2019

Table[n(9n-7),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,2,22},40] (* Harvey P. Dale, Nov 02 2011 *)
2*PolygonalNumber[11,Range[0,40]] (* Harvey P. Dale, May 31 2024 *)

a(n)=n*(9*n-7) \\ Charles R Greathouse IV, Jun 17 2017

[n*(9*n-7) for n in (0..50)] # G. C. Greubel, Sep 01 2019

a(n) = 9*n^2 - 7*n = A051682(n)*2.
a(n) = a(n-1) + 18*n - 16 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
a(0)=0, a(1)=2, a(2)=22, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Nov 02 2011
From G. C. Greubel, Sep 01 2019: (Start)
G.f.: 2*x*(1+8*x)/(1-x)^3.
E.g.f.: x*(2+9*x)*exp(x). (End)

A172082 a(n) = n(n+1)(6*n-5)/2.

Original entry on oeis.org

0, 1, 21, 78, 190, 375, 651, 1036, 1548, 2205, 3025, 4026, 5226, 6643, 8295, 10200, 12376, 14841, 17613, 20710, 24150, 27951, 32131, 36708, 41700, 47125, 53001, 59346, 66178, 73515, 81375, 89776, 98736, 108273, 118405, 129150, 140526

Offset: 0

Vincenzo Librandi, Jan 25 2010

Generated by formula: n*(n+1)*(2*d*n-2*d+3)/6 with d=9.

This sequence is related to A051682 by a(n) = n*A051682(n) - Sum_{i=0..n-1} A051682(i); in fact this is the case d=9 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, Apr 16 2012

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 to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051682.

Cf. similar sequences listed in A237616.

List([0..40], n-> n*(n+1)*(6*n-5)/2); # G. C. Greubel, Aug 30 2019

[(18*n^3+3*n^2-15*n)/6: n in [0..40]]; // Vincenzo Librandi, Jan 02 2014

seq(n*(n+1)*(6*n-5)/2, n=0..40); # G. C. Greubel, Aug 30 2019

Table[(18n^3+3n^2-15n)/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,21,78}, 40] (* Harvey P. Dale, Jun 29 2011 *)
CoefficientList[Series[x*(1+17*x)/(1-x)^4, {x,0,40}], x] (* Vincenzo Librandi, Jan 02 2014 *)

vector(40, n, n*(n-1)*(6*n-11)/2) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(6*n-5)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(0)=0, a(1)=1, a(2)=21, a(3)=78; for n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jun 29 2011
G.f.: x*(1+17*x)/(1-x)^4. - Harvey P. Dale, Jun 29 2011
a(n) = Sum_{i=0..n-1} (n-i)*(18*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(2 + 19*x + 6*x^2)*exp(x)/2. - G. C. Greubel, Aug 30 2019
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(3*sqrt(3)*Pi + 9*log(3) + 12*log(2) - 5)/55.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(6*Pi + 6*sqrt(3)*log(sqrt(3)+2) - 16*log(2) + 5)/55. (End)

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)

A131215 Numbers which are both 11-gonal and centered 11-gonal.

Original entry on oeis.org

1, 606, 241396, 96075211, 38237692791, 15218505655816, 6056927013322186, 2410641732796574421, 959429352726023297581, 381850471743224475863026, 151975528324450615370186976, 60485878422659601692858553631

Offset: 1

Richard Choulet, Sep 27 2007

nonn

A centered 11-gonal number is defined by (11*r^2 - 11*r + 2)/2 = A069125(r); a 11-gonal number by (9*p^2 - 7*p)/2 = A051682(p).
A number is both these numbers iff exist p and r such that (18*p - 7)^2 = 99*(2*r - 1) + 22.
The Diophantine equation X^2 = 99*Y^2 + 22 is such that : X is given by the sequence 11, 209, 4169, 83171,... in A131216; Y is given by the sequence 1, 21, 419, 8359,... in A083043.
The first equation is such that : p is given by 1, 12, 232, 4621,... which satisfies a(n+2) = 20*a(n+1) - a(n) - 7 and a(n+1) = 10*a(n) - 7/2 + sqrt(396*a(n)^2 - 308*a(n) + 33)/2 with g.f.  (1 -9*x +x^2)/( (1-x) * (1 -20*x + x^2) ); r is given by 1, 11, 210, 4180,... which satisfies a(n+2) = 20*a(n+1) - a(n) - 9 and a(n+1) = 10*a(n) - 9/2 + sqrt(396*a(n)^2 - 396*a(n) + 121)/2 with g.f. (1 - 10*x)/( (1-x)*(1 -20*x +x^2) ).

G. C. Greubel, Table of n, a(n) for n = 1..380
Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

Cf. A128922.

a:=[1,606,241396];; for n in [4..20] do a[n]:=399*a[n-1]-399*a[n-2] +a[n-3]; od; a; # G. C. Greubel, Dec 06 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) )); // G. C. Greubel, Dec 06 2019

seq(coeff(series(x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)), x, n+1), x, n), n = 1..20); # G. C. Greubel, Dec 06 2019

LinearRecurrence[{399,-399,1},{1,606,241396},20] (* Harvey P. Dale, Mar 04 2015 *)

my(x='x+O('x^20)); Vec(x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2))) \\ G. C. Greubel, Dec 06 2019

def A131215_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) ).list()
a=A131215_list(20); a[1:] # G. C. Greubel, Dec 06 2019

a(n+2) = 398*a(n+1) - a(n) + 209.
a(n+1) = 199*a(n) + 209/2 + (5/2)*sqrt(6336*a(n)^2 + 6688*a(n) + 1617).
G.f.: z*(1 +207*z +z^2)/((1-z)*(1-398*z+z^2)).
a(1)=1, a(2)=606, a(3)=241396, a(n) = 399*a(n-1) - 399*a(n-2) + a(n-3). - Harvey P. Dale, Mar 04 2015

More terms from Paolo P. Lava, Sep 26 2008

A139619 a(n) = 171*n + 19.

Original entry on oeis.org

19, 190, 361, 532, 703, 874, 1045, 1216, 1387, 1558, 1729, 1900, 2071, 2242, 2413, 2584, 2755, 2926, 3097, 3268, 3439, 3610, 3781, 3952, 4123, 4294, 4465, 4636, 4807, 4978, 5149, 5320, 5491, 5662, 5833, 6004, 6175, 6346, 6517, 6688

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 19th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 19th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016957, A017329, A057145, A139600, A139606-A139618, A139620.

Cf. A017173, A051682.

[19*(9*n + 1): n in [0..50]]; // Vincenzo Librandi, Jul 23 2011

Range[0, 100]*171 + 19 (* Paolo Xausa, Apr 17 2024 *)

a(n)=171*n+19 \\ Charles R Greathouse IV, Oct 05 2011

From Chai Wah Wu, Apr 14 2017: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (152*x + 19)/(x - 1)^2. (End)
From Elmo R. Oliveira, Apr 11 2024: (Start)
E.g.f.: 19*exp(x)*(1 + 9*x).
a(n) = 19*A017173(n) = 19*(A051682(n+1) - A051682(n)). (End)

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A227321 a(n) is the least r>=3 such that the difference between the nearest r-gonal number >= n and n is an r-gonal number.

Original entry on oeis.org

3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 5, 3, 8, 3, 3, 4, 5, 3, 11, 3, 3, 3, 5, 4, 3, 10, 3, 3, 11, 3, 17, 4, 3, 5, 3, 3, 7, 14, 3, 4, 15, 3, 23, 3, 3, 5, 11, 4, 3, 5, 5, 3, 19, 3, 3, 3, 8, 5, 21, 3, 32, 14, 3, 4, 3, 3, 15, 3, 5, 5, 25, 3, 38, 7, 3, 6, 3, 3, 13, 4, 3

Offset: 0

Vladimir Shevelev, Jul 30 2013

nonn

The n-th r-gonal numbers is n((n-1)r-2(n-2))/2, such that 3-gonal numbers are triangular numbers, 4-gonal numbers are squares, etc.

Peter J. C. Moses, Table of n, a(n) for n = 0..1999

Cf. A000217 (r=3), A000290 (r=4), A000326 (r=5), A000384 (r=6), A000566 (r=7), A000567 (r=8), A001106-7 (r=9,10), A051682 (r=11), A051624 (r=12), A051865-A051876 (r=13-24).

rGonalQ[r_,0]:=True; rGonalQ[r_,n_]:=IntegerQ[(Sqrt[((8r-16)n+(r-4)^2)]+r-4)/(2r-4)]; nthrGonal[r_,n_]:=(n (r-2)(n-1))/2+n; nextrGonal[r_,n_]:=nthrGonal[r,Ceiling[(Sqrt[((8r-16)n+(r-4)^2)]+r-4)/(2r-4)]]; (* next r-gonal number greater than or equal to n *) Table[NestWhile[#+1&,3,!rGonalQ[#,nextrGonal[#,n]-n]&],{n,0,99}] (* Peter J. C. Moses, Aug 03 2013 *)

If n is prime, then n == 1 or 2 mod (a(n)-2). If n >= 13 is the greater of a pair of twin primes (A006512), then a(n) = (n+3)/2. - Vladimir Shevelev, Aug 07 2013

More terms from Peter J. C. Moses, Jul 30 2013

cp's OEIS Frontend

A051879 Partial sums of A051798.

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A261191 40-gonal numbers: a(n) = 38*n*(n-1)/2 + n.

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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.

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A152995 Twice 11-gonal numbers: a(n) = n*(9*n-7).

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A172082 a(n) = n*(n+1)*(6*n-5)/2.

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

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A131215 Numbers which are both 11-gonal and centered 11-gonal.

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A261191 40-gonal numbers: a(n) = 38n(n-1)/2 + n.

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

A152995 Twice 11-gonal numbers: a(n) = n(9n-7).

A172082 a(n) = n(n+1)(6*n-5)/2.

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

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.