A051682 -id:A051682 - cp's OEIS frontend

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

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

Offset: 0

Luciano Ancora, Apr 03 2015

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.

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.

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

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).

k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015

Mathematica
```
Table[n (23 n - 21)/2, {n, 40}]
```

a(n)=n*(23*n-21)/2 \\ Charles R Greathouse IV, Oct 07 2015

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

A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n(n+1)(3*n-2)/2.

Original entry on oeis.org

0, 1, 12, 42, 100, 195, 336, 532, 792, 1125, 1540, 2046, 2652, 3367, 4200, 5160, 6256, 7497, 8892, 10450, 12180, 14091, 16192, 18492, 21000, 23725, 26676, 29862, 33292, 36975, 40920, 45136, 49632, 54417, 59500, 64890, 70596, 76627, 82992, 89700, 96760, 104181

Offset: 0

N. J. A. Sloane, R. K. Guy

Starting with 1 equals binomial transform of [1, 11, 19, 9, 0, 0, 0, ...]. - Gary W. Adamson, Nov 02 2007

			From _Vincenzo Librandi_, Feb 12 2014: (Start)
After 0, the sequence is provided by the row sums of the triangle (see above, third formula):
  1;
  2, 10;
  3, 20, 19;
  4, 30, 38, 28;
  5, 40, 57, 56, 37;
  6, 50, 76, 84, 74, 46; etc. (End)

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051682.
Cf. A093644 ((9,1) Pascal, column m=3).
Cf. similar sequences listed in A237616.

List([0..45], n-> n*(n+1)*(3*n-2)/2); # G. C. Greubel, Aug 30 2019

I:=[0,1,12,42]; [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

seq(n*(n+1)*(3*n-2)/2, n=0..45); # G. C. Greubel, Aug 30 2019

Table[n(n+1)(3n-2)/2,{n,0,45}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,12,42}, 45] (* Harvey P. Dale, Apr 09 2012 *)
CoefficientList[Series[x(1+8x)/(1-x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Feb 12 2014 *)

a(n)=n*(n+1)*(3*n-2)/2 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(3*n-2)/2 for n in (0..45)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1+8*x)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, a(0)=0, a(1)=1, a(2)=12, a(3)=42. - Harvey P. Dale, Apr 09 2012
a(n) = Sum_{i=0..n-1} (n-i)*(9*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
From Amiram Eldar, Jun 28 2020: (Start)
Sum_{n>=1} 1/a(n) = (9*log(3) + sqrt(3)*Pi - 4)/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)*Pi + 2 - 4*log(2))/5. (End)
E.g.f.: exp(x)*x*(2 + 10*x + 3*x^2)/2. - Elmo R. Oliveira, Aug 03 2025

More terms from Vincenzo Librandi, Feb 12 2014

A051874 22-gonal numbers: a(n) = n(10n-9).

Original entry on oeis.org

0, 1, 22, 63, 124, 205, 306, 427, 568, 729, 910, 1111, 1332, 1573, 1834, 2115, 2416, 2737, 3078, 3439, 3820, 4221, 4642, 5083, 5544, 6025, 6526, 7047, 7588, 8149, 8730, 9331, 9952, 10593, 11254, 11935, 12636, 13357, 14098, 14859

Offset: 0

N. J. A. Sloane, Dec 15 1999

Sequence found by reading the line from 0, in the direction 0, 22,... and the parallel line from 1, in the direction 1, 63,..., in the square spiral whose vertices are the generalized 22-gonal numbers. - Omar E. Pol, Jul 18 2012
Also sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 22,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 29 2012
This is also a star hendecagonal number: a(n) = A051682(n) + 11*A000217(n-1). - Luciano Ancora, Mar 30 2015

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Vincenzo Librandi, 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).

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+20 od: seq(a[n], n=0..39); # Zerinvary Lajos, Feb 18 2008

Table[n (10 n -9), {n, 0, 40}] (* Harvey P. Dale, Sep 19 2011 *)
CoefficientList[Series[x (1 + 19 x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)

a(n)=n*(10*n-9) \\ Charles R Greathouse IV, Jan 24 2014

a(n) = 2*a(n-1)-a(n-2)+20 with n>1, a(0)=0, a(1)=1. - Zerinvary Lajos, Feb 18 2008
a(n) = 20*n+a(n-1)-19 with n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
G.f.: x*(1+19*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(20*a(n)+191*n+1) = a(20*a(n)+191*n) + a(20*n+1). - Vladimir Shevelev, Jan 24 2014
Product_{n>=2} (1 - 1/a(n)) = 10/11. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 10*x^2). - Nikolaos Pantelidis, Feb 05 2023

A064225 a(n) = (9n^2 + 5n + 2)/2.

Original entry on oeis.org

1, 8, 24, 49, 83, 126, 178, 239, 309, 388, 476, 573, 679, 794, 918, 1051, 1193, 1344, 1504, 1673, 1851, 2038, 2234, 2439, 2653, 2876, 3108, 3349, 3599, 3858, 4126, 4403, 4689, 4984, 5288, 5601, 5923, 6254, 6594, 6943, 7301, 7668, 8044, 8429, 8823, 9226

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Michael Somos, Jul 22 2006
Ehrhart polynomial of closed quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006
In the natural number array A000027 this sequence is the first knight moves diagonal (A081267 is the second, A001844 is the main diagonal). It can be used to define this diagonal for any array: A007318(A064225-1)=A005809 (Subtraction by 1 because A007318 is defined with offset 0.) - Tilman Piesk, Mar 24 2012
Or positions of pentagonal numbers, such that p(a(n)) = p(a(n)-1) + p(3*n+1), where p=A000326. - Vladimir Shevelev, Jan 21 2014

			Illustration of initial terms:
.
.                                    o
.                                 o o
.                      o       o o o o
.                   o o     o o o o o
.           o    o o o o     o o o o o
.        o o      o o o     o o o o o
.   o     o o    o o o o     o o o o o
.        o o      o o o     o o o o o
.           o    o o o o     o o o o o
.                   o o     o o o o o
.                      o       o o o o
.                                 o o
.                                    o
.
.   1     8        24           49
- _Aaron David Fairbanks_, Feb 23 2025

Harry J. Smith, Table of n, a(n) for n = 0..1000
National Security Agency, Intrigued? (advertisement), Notices of the Amer. Math. Soc., vol. 49 (2002), p. 216.
J. A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869, Table 3, k=2 (different offset)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000326, A001844, A005809, A007318, A051682, A064226, A081267, A235332.

Column w=2 of A371967.

Table[(9n^2+5n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,8,24},51] (* Harvey P. Dale, Sep 13 2011 *)

{a(n) = 1 + n * (9*n + 5) / 2}; /* Michael Somos, Jul 22 2006 */

(define (A064225 n) (/ (+ (* 9 n n) (* 5 n) 2) 2))

a(n) = 9*n+a(n-1)-2, with n>0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(0)=1, a(1)=8, a(2)=24, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 13 2011
G.f.: (1+5*x+3*x^2)/(1-x)^3. - Colin Barker, Feb 23 2012
A064226(n) = a(-1-n). - Michael Somos, Jul 22 2006 (While the sequence itself is only one-way infinite, this identity works, as the defining formula (in the Name-field) produces integers also for the negative values of n, -1, -2, -3, etc.) - Antti Karttunen, Mar 24 2012
E.g.f.: exp(x)*(2 + 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A045946 Star of David matchstick numbers: a(n) = 6n(3*n+1).

Original entry on oeis.org

0, 24, 84, 180, 312, 480, 684, 924, 1200, 1512, 1860, 2244, 2664, 3120, 3612, 4140, 4704, 5304, 5940, 6612, 7320, 8064, 8844, 9660, 10512, 11400, 12324, 13284, 14280, 15312, 16380, 17484, 18624, 19800, 21012, 22260, 23544, 24864, 26220, 27612, 29040, 30504, 32004

Offset: 0

R. K. Guy

Vertical spoke of triangular spiral in A051682. - Paul Barry, Mar 15 2003

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

Cf. A005449, A033580, A049451, A045945, A051682, A081266.

Table[6n(3n+1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,24,84},40] (* Harvey P. Dale, Nov 23 2012 *)

a(n)=18*n^2+6*n \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 24*C(n,1) + 36*C(n,2); binomial transform of (0, 24, 36, 0, 0, 0, ...). - Paul Barry, Mar 15 2003
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=24, a(2)=84. - Harvey P. Dale, Nov 23 2012
G.f.: 12*x*(2+x)/(1-x)^3. - Ivan Panchenko, Nov 13 2013
a(n) = 2*A045945(n). - Michel Marcus, Nov 13 2013
a(n) = 12*A005449(n). - R. J. Mathar, Feb 08 2016
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=1} 1/a(n) = 1/2 - Pi/(12*sqrt(3)) - log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = -1/2 + Pi/(6*sqrt(3)) + log(2)/3. (End)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: 6*exp(x)*x*(4 + 3*x).
a(n) = 6*A049451(n) = 4*A081266(n) = 3*A033580(n). (End)

A064226 a(n) = (9n^2 + 13n + 6)/2.

Original entry on oeis.org

3, 14, 34, 63, 101, 148, 204, 269, 343, 426, 518, 619, 729, 848, 976, 1113, 1259, 1414, 1578, 1751, 1933, 2124, 2324, 2533, 2751, 2978, 3214, 3459, 3713, 3976, 4248, 4529, 4819, 5118, 5426, 5743, 6069, 6404, 6748, 7101, 7463, 7834, 8214, 8603, 9001, 9408, 9824

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003

Ehrhart polynomial of open quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006

Harry J. Smith, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
National Security Agency, Intrigued? (advertisement), Notices of the American Mathematical Society, Vol. 49 (2002), p. 216.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682, A064225, A081267, A081268, A235332.

I:=[3,14,34]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 19 2015

A064226:=n-> (9*n^2 + 13*n + 6) / 2; seq(A064226(n), n=0..50); # Wesley Ivan Hurt, May 08 2014

Table[(9 n^2 + 13 n + 6)/2, {n, 0, 50}] (* Wesley Ivan Hurt, May 08 2014 *)
LinearRecurrence[{3, -3, 1}, {3, 14, 34}, 50] (* Vincenzo Librandi, Jul 19 2015 *)

{a(n) = 3 + n * (9*n + 13) / 2}; /* Michael Somos, Jul 22 2006 */

From Paul Barry, Mar 15 2003: (Start)
a(n) = 3*C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (3, 11, 9, 0, 0, 0, ...).
G.f.: (3 + 5*x + x^2)/(1-x)^3.
a(n) = A081268(n) + 2. (End)
A064225(n) = a(-1-n). -  Michael Somos, Jul 22 2006
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Apr 16 2023
E.g.f.: (3 + 11*x + 9*x^2/2)*exp(x). - Elmo R. Oliveira, Oct 21 2024

A062725 Write 0, 1, 2, 3, 4, ... in a triangular spiral, then a(n) is the sequence found by reading the terms along the line from 0 in the direction 0, 7, ...

Original entry on oeis.org

0, 7, 23, 48, 82, 125, 177, 238, 308, 387, 475, 572, 678, 793, 917, 1050, 1192, 1343, 1503, 1672, 1850, 2037, 2233, 2438, 2652, 2875, 3107, 3348, 3598, 3857, 4125, 4402, 4688, 4983, 5287, 5600, 5922, 6253, 6593, 6942, 7300, 7667, 8043, 8428, 8822, 9225, 9637, 10058

Offset: 0

Floor van Lamoen, Jul 21 2001

Central terms of triangle A245300. - Reinhard Zumkeller, Jul 17 2014

Digital root of a(n) = A180597(n). - Gionata Neri, Apr 29 2015

			The spiral begins:
.
            15
            / \
          16  14
          /     \
        17   3  13
        /   / \   \
      18   4   2  12
      /   /     \   \
    19   5   0---1  11
    /   /             \
  20   6---7---8---9--10
.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017245, A051682, A180597, A218470, A245300.

a062725 n = n * (9 * n + 5) `div` 2 -- Reinhard Zumkeller, Jul 17 2014

s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
CoefficientList[Series[x (7 + 2 x)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jan 11 2020 *)

a(n) = n*(9*n+5)/2 \\ Charles R Greathouse IV, Apr 30 2015

a(n) = n*(9*n+5)/2.
a(n) = 9*n + a(n-1) - 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Jul 07 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(7+2*x)/(1-x)^3. (End)
a(n) = A218470(9*n+6). - Philippe Deléham, Mar 27 2013
a(n) = a(n-1) + A017245(n-1), a(0)=0. - Gionata Neri, Apr 30 2015
E.g.f.: exp(x)*x*(14 + 9*x)/2. - Elmo R. Oliveira, Dec 12 2024

Formula that confused indices corrected by R. J. Mathar, Jun 04 2010

A051798 a(n) = (n+1)(n+2)(n+3)*(9n+4)/24.

Original entry on oeis.org

1, 13, 55, 155, 350, 686, 1218, 2010, 3135, 4675, 6721, 9373, 12740, 16940, 22100, 28356, 35853, 44745, 55195, 67375, 81466, 97658, 116150, 137150, 160875, 187551, 217413, 250705, 287680, 328600, 373736, 423368, 477785, 537285

Offset: 0

Barry E. Williams, Dec 11 1999

Partial sums of A007586.

Convolution of A000027 with A051682 (excluding 0). - Bruno Berselli, Dec 07 2012

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

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

Cf. A007586, A051682.
Cf. A093644 ((9, 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 A051682 (excluding 0): */ A051682:=func; [&+[(n-i+1)*A051682(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,13,55,155,350},40] (* Harvey P. Dale, Aug 19 2012 *)

a(n)=(n+1)*(n+2)*(n+3)*(9*n+4)/24 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = C(n+3, 3)*(9*n+4)/4.
G.f.: (1+8*x)/(1-x)^5.
a(0)=1, a(1)=13, a(2)=55, a(3)=155, a(4)=350, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Aug 19 2012
a(n) = A080852(9,n). - R. J. Mathar, Jul 28 2016

A188892 Numbers n such that there is no triangular n-gonal number greater than 1.

Original entry on oeis.org

11, 18, 38, 102, 198, 326, 486, 678, 902, 1158, 1446, 1766, 2118, 2918, 3366, 3846, 4358, 4902, 5478, 6086, 6726, 7398, 8102, 8838, 9606, 10406, 11238, 12102, 12998, 13926, 14886, 15878, 16902, 17958, 19046, 20166, 21318, 22502, 24966, 26246

Offset: 1

T. D. Noe, Apr 13 2011

nonn

It is easy to find triangular numbers that are square, pentagonal, hexagonal, etc.  So it is somewhat surprising that there are no triangular 11-gonal numbers other than 0 and 1. For these n, the equation x^2 + x = (n-2)*y^2 - (n-4)*y has no integer solutions x>1 and y>1.
Chu shows how to transform the equation into a generalized Pell equation. When n has the form k^2+2 (A059100), then the Pell equation has only a finite number of solutions and it is simple to select the n that produce no integer solutions greater than 1.
The general case is in A188950.

Robert Israel, Table of n, a(n) for n = 1..10000
Wenchang Chu, Regular polygonal numbers and generalized pell equations, Int. Math. Forum 2 (2007), 781-802.

Cf. A051682 (11-gonal numbers), A051870 (18-gonal numbers), A188891, A188896.

filter:= n -> nops(select(t -> min(subs(t,[x,y]))>=2, [isolve(x^2 + x = (n-2)*y^2 - (n-4)*y)])) = 0:
select(filter, [seq(t^2+2,t=3..200)]); # Robert Israel, May 13 2018

A264804 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 11-gonal: (9n^2 - 7n)/2.

Original entry on oeis.org

1, 1, 526, 64095, 21420730041, 4528059468080555555556, 3834345160635370971474665069772601398563211, 100751687713984558500838936986634939491022212000570658953744730444103042117925197608458

Offset: 1

Anders Hellström, Nov 25 2015

Chai Wah Wu, Table of n, a(n) for n = 1..11
Wikipedia, Polygonal number

Cf. A051671, A051682 (11-gonal numbers), A061109, A061110, A261696, A264733, A264738, A264776, A264777, A264842, A264848, A264849.

hendecagonal(n)=ispolygonal(n,11)
first(m)=my(v=vector(m),s="");s="1";print1(1, ", ");for(i=2,m,n=1;while(!hendecagonal(eval(concat(s,Str(n)))),n++);print1(n, ", ");s=concat(s,Str(n)))

a(5)-a(8) from Chai Wah Wu, Mar 16 2018

cp's OEIS Frontend

A255184 25-gonal numbers: a(n) = n*(23*n-21)/2.

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A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n*(n+1)*(3*n-2)/2.

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A051874 22-gonal numbers: a(n) = n*(10*n-9).

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Maple

Mathematica

PARI

Formula

A064225 a(n) = (9*n^2 + 5*n + 2)/2.

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Scheme

Formula

A045946 Star of David matchstick numbers: a(n) = 6*n*(3*n+1).

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A064226 a(n) = (9*n^2 + 13*n + 6)/2.

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PARI

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n(n+1)(3*n-2)/2.

A051874 22-gonal numbers: a(n) = n(10n-9).

A064225 a(n) = (9n^2 + 5n + 2)/2.

A045946 Star of David matchstick numbers: a(n) = 6n(3*n+1).

A064226 a(n) = (9n^2 + 13n + 6)/2.

A051798 a(n) = (n+1)(n+2)(n+3)*(9n+4)/24.