A051866 -id:A051866 - cp's OEIS frontend

A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 25, 19, 1, 1, 25, 37, 37, 25, 1, 1, 31, 49, 55, 49, 31, 1, 1, 37, 61, 73, 73, 61, 37, 1, 1, 43, 73, 91, 97, 91, 73, 43, 1, 1, 49, 85, 109, 121, 121, 109, 85, 49, 1, 1, 55, 97, 127, 145, 151, 145, 127, 97, 55, 1, 1, 61, 109, 145, 169, 181, 181, 169, 145, 109, 61, 1

Offset: 0

Kolosov Petro, Aug 31 2017

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(1, n, k), i.e T(n, k) is partial case of L(m, n, k) for m = 1.
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
  ----------------------------------------
  k=    0   1   2   3   4   5   6   7   8
  ----------------------------------------
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  7,  1;
  n=3:  1, 13, 13,  1;
  n=4:  1, 19, 25, 19,  1;
  n=5:  1, 25, 37, 37, 25,  1;
  n=6:  1, 31, 49, 55, 49, 31,  1;
  n=7:  1, 37, 61, 73, 73, 61, 37,  1;
  n=8:  1, 43, 73, 91, 97, 91, 73, 43,  1;

Georg Fischer, Table of n, a(n) for n = 0..495 [rows 0..10 and 12..30 from Kolosov Petro]
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Columns k=0..6 give A000012, A016921, A017533, A161705, A103214, A128470, A158065.
Column sums k=0..4 give A000027, A000567, A051866, A051872, A255185.
Row sums give A001093.
Various cases of L(m, n, k): This sequence (m=1), A300656(m=2), A300785(m=3). See comments for L(m, n, k).
Differences of cubes n^3 are T(A000124(n), 1).
Cf. A000124, A000578, A003154, A003215, A007318, A008458, A016969.
Cf. A038593, A055012, A068601, A077028, A094053, A166873, A227776.
Cf. A294317, A302971, A304042.

Flat(List([0..11],n->List([0..n],k->6*k*(n-k)+1))); # Muniru A Asiru, Oct 09 2018

/* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 26 2018

T := (n, k) -> 6*k*(n-k) + 1:
seq(seq(T(n, k), k=0..n), n=0..11); # Muniru A Asiru, Oct 09 2018

T[n_, k_] := 6 k (n - k) + 1; Column[Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jun 02 2019 *)

t(n, k) = 6*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */
trianglerows(9) \\ Felix Fröhlich, Jan 09 2018

def A287326(n,k): return 6*k*(n-k) + 1
flatten([[A287326(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 25 2024

T(n, k) = 6*k*(n-k) + 1.
G.f. of column k: n^k*(1+(6*k-1)*n)/(1-n)^2.
G.f.: (1 - x - x*y + 7*x^2*y)/((1 - x)^2*(1 - x*y)^2). - Stefano Spezia, Oct 09 2018 [Adapted by Stefano Spezia, Sep 25 2024]
From Kolosov Petro, Jun 05 2019: (Start)
T(n, k) = 1/2 * T(A294317(n, k), k) + 1/2.
T(n+1, k) = 2*T(n, k) - T(n-1, k), for n >= k.
T(n, k) = 6*A077028(n, k) - 5.
T(2n, n) = A227776(n).
T(2n+1, n) = A003154(n+1).
T(2n+3, n) = A166873(n+1).
Sum_{k=0..n-1} T(n, k) = Sum_{k=1..n} T(n, k) = A000578(n).
Sum_{k=1..n-1} T(n, k) = A068601(n).
(n+1)^3 - n^3 = T(A000124(n), 1). (End)
Sum_{k=0..n} (-1)^k*T(n, k) = (-1/2)*(1 + (-1)^n)*A016969(floor(n/2) - 1). - G. C. Greubel, Sep 25 2024

A161935 28-gonal numbers: a(n) = n(13n - 12).

Original entry on oeis.org

0, 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728, 2041, 2380, 2745, 3136, 3553, 3996, 4465, 4960, 5481, 6028, 6601, 7200, 7825, 8476, 9153, 9856, 10585, 11340, 12121, 12928, 13761, 14620, 15505, 16416, 17353, 18316, 19305, 20320, 21361, 22428, 23521

Offset: 0

Pierre Gayet, Jun 22 2009

The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]

These are also the star 14-gonal numbers: a(n) = A051866(n) + 14*A000217(n-1). Luciano Ancora, Apr 04 2015

			G.f. = x + 28*x^2 + 81*x^3 + 160*x^4 + 265*x^5 + 396*x^6 + 553*x^7 + ...

Daniel Mondot, Table of n, a(n) for n = 0..1000
Pierre Gayet, Note et Compte rendu (gif version).
Pierre Gayet, Note et Compte Rendu (pdf version).
Pierre Gayet, 98 séquences générées ... par la formule générale indiquée.
Claude Monet, Nymphéas.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A051866, A161532, A161549, A161587, A161617, A162316.

Magma
```
[ (n+1)*(13*n+1): n in[0..50] ];
```

lst={}; Do[a=13*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
Table[n*(13*n - 12), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

{a(n) = n*(13*n - 12)}; /* Michael Somos, Dec 07 2016 */

a(n+1) = a(n) + 26*n + 1. - Vincenzo Librandi, Nov 30 2010
a(n) = A000217(n) + 25*A000217(n-1). - Luciano Ancora, Apr 04 2015
Product_{n>=2} (1 - 1/a(n)) = 13/14. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 13*x^2). - Nikolaos Pantelidis, Feb 05 2023
From Elmo R. Oliveira, Dec 14 2024: (Start)
G.f.: x*(1 + 25*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Edited by N. J. A. Sloane, Dec 07 2016 at the suggestion of Daniel Sterman.

Definition simplified by Omar E. Pol, Aug 10 2018

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

Original entry on oeis.org

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

A172073 a(n) = (4n^3 + n^2 - 3n)/2.

Original entry on oeis.org

0, 1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, 3510, 4459, 5565, 6840, 8296, 9945, 11799, 13870, 16170, 18711, 21505, 24564, 27900, 31525, 35451, 39690, 44254, 49155, 54405, 60016, 66000, 72369, 79135, 86310, 93906, 101935, 110409, 119340, 128740

Offset: 0

Vincenzo Librandi, Jan 25 2010

14-gonal (or tetradecagonal) pyramidal numbers generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=6.
In fact, the sequence is related to A000567 by a(n) = n*A000567(n) - Sum_{i=0..n-1} A000567(i) and this is the case d=6 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Nov 29 2010
Except for the initial 0, this is the principal diagonal of the convolution array A213761. - Clark Kimberling, Jul 04 2012
Starting (1, 15, 54, ...), this is the binomial transform of (1, 14, 25, 12, 0, 0, 0, ...). - Gary W. Adamson, Jul 29 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 to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A237616.

Cf. A000567, A051866, A194454.

List([0..40], n-> n*(n+1)*(4*n-3)/2); # G. C. Greubel, Aug 30 2019

[(4*n^3+n^2-3*n)/2: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

seq(n*(n+1)*(4*n-3)/2, n=0..40); # G. C. Greubel, Aug 30 2019

f[n_]:= n(n+1)(4n-3)/2; Array[f, 40, 0]
LinearRecurrence[{4,-6,4,-1},{0,1,15,54},40] (* Harvey P. Dale, Jan 29 2013 *)
CoefficientList[Series[x (1+11x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)

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

[n*(n+1)*(4*n-3)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(n) = n*(n+1)*(4*n-3)/2.
From Bruno Berselli, Dec 15 2010: (Start)
G.f.: x*(1+11*x)/(1-x)^4.
a(n) = Sum_{i=0..n} A051866(i). (End)
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)=15, a(3)=54. - Harvey P. Dale, Jan 29 2013
a(n) = Sum_{i=0..n-1} (n-i)*(12*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi/21 + 8*log(2)/7 - 2/7.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/21 + 8*sqrt(2)*log(sqrt(2)+2)/21 - (20 + 4*sqrt(2))*log(2)/21 + 2/7. (End)
E.g.f.: exp(x)*x*(2 + 13*x + 4*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

Edited by Bruno Berselli, Dec 14 2010

A211014 Second 14-gonal numbers: n(6n+5).

Original entry on oeis.org

0, 11, 34, 69, 116, 175, 246, 329, 424, 531, 650, 781, 924, 1079, 1246, 1425, 1616, 1819, 2034, 2261, 2500, 2751, 3014, 3289, 3576, 3875, 4186, 4509, 4844, 5191, 5550, 5921, 6304, 6699, 7106, 7525, 7956, 8399, 8854, 9321, 9800, 10291, 10794, 11309, 11836, 12375

Offset: 0

Omar E. Pol, Aug 04 2012

Sequence found by reading the line from 0, in the direction 0, 34, ... and the line from 11 in the direction 11, 69, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See 3rd formula in Proposition 3 p. 36 giving a(n+1).
Leo Tavares, Star illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195818.
Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, A211013, this sequence.
Cf. A051866.
Cf. A003154.

List([0..50], n-> n*(6*n+5)); # G. C. Greubel, Jul 04 2019

[n*(6*n+5): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[n*(6*n+5), {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)
LinearRecurrence[{3,-3,1},{0,11,34},50] (* Harvey P. Dale, Dec 12 2022 *)

a(n)=n*(6*n+5) \\ Charles R Greathouse IV, Jun 17 2017

[n*(6*n+5) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = -2*Sum_{k=0..n-1} binomial(6*n+5, 6*k+8)*Bernoulli(6*k+8). - Michel Marcus, Jan 11 2016
From G. C. Greubel, Jul 04 2019: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(11+x)/(1-x)^3.
E.g.f.: x*(11+6*x)*exp(x). (End)
From Amiram Eldar, Feb 28 2022: (Start)
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/10 + 6/25 - 3*log(3)/10 - 2*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/5 + log(2)/5 - 6/25 - sqrt(3)*log(sqrt(3)+2)/5. (End)
a(n) = A003154(n+1) - n - 1. - Leo Tavares, Jan 29 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.

A033594 a(n) = (n-1)(2n-1)(3n-1).

Original entry on oeis.org

-1, 0, 15, 80, 231, 504, 935, 1560, 2415, 3536, 4959, 6720, 8855, 11400, 14391, 17864, 21855, 26400, 31535, 37296, 43719, 50840, 58695, 67320, 76751, 87024, 98175, 110240, 123255, 137256, 152279, 168360

Offset: 0

N. J. A. Sloane

The sequence of n such that n is prime and (2*n+1) is prime is the sequence of Sophie Germain primes A005384 and the subsequence of those for which in addition (3*n+2) is prime is A067256. - Jonathan Vos Post, Dec 15 2004

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. A005384, A016921, A067256.

List([0..40], n-> (n-1)*(2*n-1)*(3*n-1) ); # G. C. Greubel, Mar 05 2020

[(n-1)*(2*n-1)*(3*n-1): n in [0..40]]; // Vincenzo Librandi, May 24 2011

A033594:=n->(n-1)*(2*n-1)*(3*n-1); seq(A033594(n), n=0..40); # Wesley Ivan Hurt, Feb 24 2014

Table[(n-1)*(2*n-1)*(3*n-1),{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{4,-6,4,-1},{-1,0,15,80},40] (* Harvey P. Dale, Aug 23 2012 *)

vector(41, n, my(m=n-1); (m-1)*(2*m-1)*(3*m-1) ) \\ G. C. Greubel, Mar 05 2020

[-1]+[n^3*rising_factorial((n-1)/n, 3) for n in (1..40)] # G. C. Greubel, Mar 05 2020

a(n)*A016921(n) + 1 = A051866(n)^2. - Bruno Berselli, May 23 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=-1, a(1)=0, a(2)=15, a(3)=80. - Harvey P. Dale, Aug 23 2012
G.f.: (-1 +4*x +9*x^2 +24*x^3)/(1-x)^4. - R. J. Mathar, Feb 06 2017
E.g.f.: (-1 + x + 7*x^2 + 6*x^3)*exp(x). - G. C. Greubel, Mar 05 2020
From Amiram Eldar, Jan 03 2021: (Start)
Sum_{n>=2} 1/a(n) = (7 - sqrt(3)*Pi - 16*log(2) + 9*log(3))/4.
Sum_{n>=2} (-1)^n/a(n) = Pi - 7/4 - sqrt(3)*Pi/2 + 2*log(2). (End)

A194454 a(n) = 12n^2 + 2n + 1.

Original entry on oeis.org

1, 15, 53, 115, 201, 311, 445, 603, 785, 991, 1221, 1475, 1753, 2055, 2381, 2731, 3105, 3503, 3925, 4371, 4841, 5335, 5853, 6395, 6961, 7551, 8165, 8803, 9465, 10151, 10861, 11595, 12353, 13135, 13941, 14771, 15625, 16503, 17405, 18331, 19281

Offset: 0

Bruno Berselli, Aug 24 2011

A142241 gives the first differences.
Inverse binomial transform of this sequence: 1, 14, 24, 0, 0 (0 continued).
a(n)*a(n-1)-11 is a square, precisely 4*A051866(n)^2.
Sequence found by reading the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

			Using these numbers we can write:
  1, 15, 53, 115, 201, 311, 445,  603,  785,  991, 1221, ...
  0,  0,  1,  15,  53, 115, 201,  311,  445,  603,  785, ...
  0,  0,  0,   0,   1,  15,  53,  115,  201,  311,  445, ...
  0,  0,  0,   0,   0,   0,   1,   15,   53,  115,  201, ...
  0,  0,  0,   0,   0,   0,   0,    0,    1,   15,   53, ...
  0,  0,  0,   0,   0,   0,   0,    0,    0,    0,    1, ...
  ======================================================
  The sums of the columns give the sequence A172073 (after 0):
  1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, ...

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

Cf. A154106, A172073, A049453.

Magma
```
[12*n^2+2*n+1: n in [0..40]];
```

Table[12 n^2 + 2 n + 1, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)

PARI
```
for(n=0, 40, print1(12*n^2+2*n+1", "));
```

G.f.: (1+x)*(1+11*x)/(1-x)^3.
a(n) = A154106(-n-1).
a(n) = 2*A049453(n) + 1.
a(n) = A051866(n) + A051866(n+1). - Charlie Marion, Nov 15 2019
E.g.f.: exp(x)*(1 + 14*x + 12*x^2). - Stefano Spezia, Nov 15 2019

A322123 Fermat pseudoprimes to base 2 that are tetradecagonal.

Original entry on oeis.org

31609, 60701, 458989, 513629, 679729, 729061, 745889, 1207361, 1994689, 2746589, 4361389, 4974971, 5173601, 5444489, 6749021, 9056501, 12659989, 13295281, 15525241, 15757741, 16070429, 16705021, 20770621, 21400481, 23822329, 23966011, 27492581, 34003061

Offset: 1

Amiram Eldar, Nov 27 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite.
Intersection of A001567 and A051866.
The corresponding indices of the tetradecagonal numbers are 73, 101, 277, 293, 337, 349, 353, 449, 577, 677, 853, 911, 929, 953, 1061, 1229, 1453, 1489, 1609, 1621, 1637, 1669, 1861, 1889, 1993, 1999, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A001567, A051866, A293623, A293624, A322124.

tetradec[n_] := n(6n-5); Select[tetradec[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]
Select[PolygonalNumber[14,Range[2400]],PowerMod[2,#-1,#]==1&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 11 2018 *)

isok(n) = ispolygonal(n, 14) && (Mod(2, n)^n==2) && !isprime(n) && (n>1); \\ Michel Marcus, Nov 28 2018

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

cp's OEIS Frontend

A287326 Triangle read by rows: T(n, k) = 6*k*(n-k) + 1; n >= 0, 0 <= k <= n.

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A161935 28-gonal numbers: a(n) = n*(13*n - 12).

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

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A172073 a(n) = (4*n^3 + n^2 - 3*n)/2.

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A211014 Second 14-gonal numbers: n*(6*n+5).

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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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A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.

A161935 28-gonal numbers: a(n) = n(13n - 12).

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

A172073 a(n) = (4n^3 + n^2 - 3n)/2.

A211014 Second 14-gonal numbers: n(6n+5).

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

A033594 a(n) = (n-1)(2n-1)(3n-1).

A194454 a(n) = 12n^2 + 2n + 1.