A005448 -id:A005448 - cp's OEIS frontend

A077588 Maximum number of regions into which the plane is divided by n triangles.

1, 2, 8, 20, 38, 62, 92, 128, 170, 218, 272, 332, 398, 470, 548, 632, 722, 818, 920, 1028, 1142, 1262, 1388, 1520, 1658, 1802, 1952, 2108, 2270, 2438, 2612, 2792, 2978, 3170, 3368, 3572, 3782, 3998, 4220, 4448, 4682, 4922, 5168, 5420, 5678, 5942, 6212, 6488

Offset: 0

Joshua Zucker and the Castilleja School MathCounts club, Nov 07 2002

			a(2) = 8 because a Star of David divides the plane into 8 regions: 6 triangles at the vertices, the interior hexagon, and the exterior.

Keyang Li, figures for n=1,2,3,4,5
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

Cf. A005448, A077591.
a(n) = A096777(3*n-1) for n > 0. - Reinhard Zumkeller, Dec 29 2007
For n > 0, a(n) = 2 * A005448(n). - Jon Perry, Apr 14 2013
a(n) = A242658(n) for n > 0. - Eric W. Weisstein, Nov 29 2017

CoefficientList[Series[(-z^3 - 5*z^2 + z - 1)/(z - 1)^3, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)

a(n) = 3n^2 - 3n + 2 for n > 0.
Proof (from Joshua Zucker and N. J. A. Sloane, Dec 01 2017)
Represent the configuration of n triangles by a planar graph with a node for each vertex of the triangles and for each intersection point. Let there be v_n nodes and e_n edges. By classical graph theory, a(n) = e_n - v_n + 2. When we go from n to n+1 triangles, each side of the new triangle can meet each side of the existing triangles at most twice, so Dv_n := v_{n+1}-v_n <= 6n.
Each of these intersection points increases the number of edges in the graph by 2, so De_n := e_{n+1}-e_n = 3 + 2*Dv_n, Da_n := a(n+1)-a(n) = 3 + Dv_n <= 3+6*n.
These upper bounds can be achieved by taking 3n points equally spaced around a circle and drawing n concentric overlapping equilateral triangles in the obvious way, and we achieve a(n) = 3n^2 - 3n + 2 (and v_n = 3n^2, e_n = 3n(2n-1)) for n>0. QED
a(n) is the nearest integer to (Sum_{k>=n} 1/k^2)/(Sum_{k>=n} 1/k^4). - Benoit Cloitre, Jun 12 2003
a(n) = a(n-1) + 6*n - 6 (with a(1) = 2). - Vincenzo Librandi, Dec 07 2010
For n > 0, a(n) = A002061(n-1) + A056220(n); and for n > 1, a(n) = A002061(n+1) + A056220(n-1). - Bruce J. Nicholson, Sep 22 2017

A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

Original entry on oeis.org

0, 1, 3, 6, 6, 6, 12, 18, 12, 6, 12, 24, 30, 24, 30, 42, 24, 6, 12, 24, 30, 30, 42, 66, 66, 36, 30, 60, 84, 72, 78, 96, 48, 6, 12, 24, 30, 30, 42, 66, 66, 42, 42, 78, 114, 114, 114, 150, 138, 60, 30, 60, 84, 90, 114, 174, 198, 132, 90, 144, 210, 192, 192, 210, 96, 6, 12, 24

Offset: 0

David Applegate and N. J. A. Sloane, Jun 15 2009

See the comments in A161644.

It appears that a(n) is also the number of V-toothpicks or Y-toothpicks added at the n-th stage in a toothpick structure on hexagonal net, starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2 (see A161206, A160120, A182633). - Omar E. Pol, Dec 07 2010

			From _Omar E. Pol_, Apr 08 2015: (Start)
The positive terms written as an irregular triangle in which the row lengths are the terms of A011782:
1;
3;
6,6;
6,12,18,12;
6,12,24,30,24,30,42,24;
6,12,24,30,30,42,66,66,36,30,60,84,72,78,96,48;
6,12,24,30,30,42,66,66,42,42,78,114,114,114,150,138,60,30,60,84,90,114,174,198,132,90,144,210,192,192,210,96;
...
It appears that the right border gives A003945.
(End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]

Rémy Sigrist, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A161644, A139251, A160161, A161207, A182633, A295559, A295560.

See A342271 for a(n)/3.

A069131 Centered 18-gonal numbers.

Original entry on oeis.org

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations.
Lamine Ngom, An origin of A069131 (illustration).
Leo Tavares, Illustration: Tri-Hexagons.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,19,55},50] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A309148 A(n,k) is (1/k) times the number of n-member subsets of [k*n] whose elements sum to a multiple of n; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 4, 0, 1, 3, 10, 9, 1, 1, 4, 19, 42, 26, 0, 1, 5, 31, 115, 201, 76, 1, 1, 6, 46, 244, 776, 1028, 246, 0, 1, 7, 64, 445, 2126, 5601, 5538, 809, 1, 1, 8, 85, 734, 4751, 19780, 42288, 30666, 2704, 0, 1, 9, 109, 1127, 9276, 54086, 192130, 328755, 173593, 9226, 1

Offset: 1

Alois P. Heinz, Jul 14 2019

For k > 1 also (1/(k-1)) times the number of n-member subsets of [k*n-1] whose elements sum to a multiple of n.

The sequence of row n satisfies a linear recurrence with constant coefficients of order n.

			Square array A(n,k) begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  1,   4,   10,    19,     31,     46,      64, ...
  0,   9,   42,   115,    244,    445,     734, ...
  1,  26,  201,   776,   2126,   4751,    9276, ...
  0,  76, 1028,  5601,  19780,  54086,  124872, ...
  1, 246, 5538, 42288, 192130, 642342, 1753074, ...

Alois P. Heinz, Rows n = 1..150, flattened

Columns k=1-10 give: A000035, A145855, A309182, A309183, A309184, A309185, A309186, A309187, A309188, A309189.
Rows n=1-3 give: A000012, A001477(k-1), A005448.
Main diagonal gives A308667.
Cf. A000010, A304482.

with(numtheory):
A:= (n, k)-> add(binomial(k*d, d)*(-1)^(n+d)*
             phi(n/d), d in divisors(n))/(n*k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

A[n_, k_] := 1/(n k) Sum[Binomial[k d, d] (-1)^(n+d) EulerPhi[n/d], {d, Divisors[n]}];
Table[A[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 04 2019 *)

A(n,k) = 1/(n*k) * Sum_{d|n} binomial(k*d,d)*(-1)^(n+d)*phi(n/d).

A(n,k) = (1/k) * A304482(n,k).

A080855 a(n) = (9n^2 - 3n + 2)/2.

Original entry on oeis.org

1, 4, 16, 37, 67, 106, 154, 211, 277, 352, 436, 529, 631, 742, 862, 991, 1129, 1276, 1432, 1597, 1771, 1954, 2146, 2347, 2557, 2776, 3004, 3241, 3487, 3742, 4006, 4279, 4561, 4852, 5152, 5461, 5779, 6106, 6442, 6787, 7141, 7504, 7876, 8257, 8647, 9046

Offset: 0

Paul Barry, Feb 23 2003

The old definition of this sequence was "Generalized polygonal numbers".
Row T(3,n) of A080853.
Equals binomial transform of [1, 3, 9, 0, 0, 0, ...] - Gary W. Adamson, Apr 30 2008
a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is congruent to 2 modulo 3. The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4, are 2+2, 5+5+2+2+2, 8+8+5+5+5+2+2+2, 11+11+8+8+8+5+5+5+2+2+2. - Augustine O. Munagi, Dec 18 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 3, a(n-1) = -coeff(charpoly(A,x), x^(n-2)). - Milan Janjic, Jan 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Leo Tavares, Illustration: Hexagonal Tri-Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A027468, A038764.
Cf. A283394 (see Crossrefs section).
Cf. A003215, A000217, A227776.

List([0..50],n->(9*n^2-3*n+2)/2); # Muniru A Asiru, Nov 02 2018

[(9*n^2 - 3*n +2)/2: n in [0..50]]; // G. C. Greubel, Nov 02 2018

seq((9*n^2-3*n+2)/2,n=0..50); # Muniru A Asiru, Nov 02 2018

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 500, 9}]; lst (* Zerinvary Lajos, Jul 11 2009 *)
Table[(9n^2-3n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1}, {1,4,16}, 50] (* Harvey P. Dale, Jul 24 2013 *)

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

G.f.: (1 + x + 7*x^2)/(1 - x)^3.
a(n) = 9*n + a(n-1) - 6 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = n*A005448(n+1) - (n-1)*A005448(n), with A005448(0)=1. - Bruno Berselli, Jan 15 2013
a(0)=1, a(1)=4, a(2)=16; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 24 2013
a(n) = A152947(3*n+1). - Franck Maminirina Ramaharo, Jan 10 2018
E.g.f.: (2 + 6*x + 9*x^2)*exp(x)/2. - G. C. Greubel, Nov 02 2018
From Leo Tavares, Feb 20 2022: (Start)
a(n) = A003215(n-1) + 3*A000217(n). See Hexagonal Tri-Rays illustration in links.
a(n) = A227776(n) - 3*A000217(n). (End)

Definition replaced with the closed form by Bruno Berselli, Jan 15 2013

A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 7, 10, 7, 1, 1, 1, 1, 11, 19, 19, 11, 1, 1, 1, 1, 16, 31, 38, 31, 16, 1, 1, 1, 1, 22, 46, 66, 66, 46, 22, 1, 1, 1, 1, 29, 64, 106, 126, 106, 64, 29, 1, 1, 1, 1, 37, 85, 162, 226, 226, 162, 85, 37, 1, 1

Offset: 0

Paul Barry, May 14 2006

Third column is essentially A000124. Fourth column is essentially A005448. Fifth column is A119327. Product of Pascal's triangle A007318 and A119328. Row sums are A038504. T(n,k) = T(n,n-k).

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 1,  1,  1;
  1, 1,  2,  1,  1;
  1, 1,  4,  4,  1,  1;
  1, 1,  7, 10,  7,  1, 1;
  1, 1, 11, 19, 19, 11, 1, 1;
  ...

Lukas Spiegelhofer and Jeffrey Shallit, Continuants, Run Lengths, and Barry's Modified Pascal Triangle, Volume 26(1) 2019, of The Electronic Journal of Combinatorics, #P1.31.

Seiichi Manyama, Rows n = 0..139, flattened
Jeffrey Shallit, Lukas Spiegelhofer, Continuants, run lengths, and Barry's modified Pascal triangle, arXiv:1710.06203 [math.CO], 2017.

Cf. A119358.

Column k has g.f.: (x^k/(1-x))* Sum{j=0..k} C(k,2j)*(x/(1-x))^(2j).

T(2n,n) = A119358(n). - Alois P. Heinz, Aug 31 2018

A129445 Numbers k > 0 such that k^2 is a centered triangular number.

Original entry on oeis.org

1, 2, 8, 19, 79, 188, 782, 1861, 7741, 18422, 76628, 182359, 758539, 1805168, 7508762, 17869321, 74329081, 176888042, 735782048, 1751011099, 7283491399, 17333222948, 72099131942, 171581218381, 713707828021, 1698478960862, 7064979148268, 16813208390239

Offset: 1

Alexander Adamchuk, Apr 15 2007, Apr 26 2007

Corresponding numbers n such that centered triangular number A005448(n) is a perfect square are listed in A129444(n).
Consider Diophantine equation 3*x*(x-1) + 2 - 2*y^2 = 0. Sequence gives solutions for y. - Zak Seidov, Jun 11 2013
Positive values of x (or y) satisfying x^2 - 10xy + y^2 + 15 = 0. - Colin Barker, Feb 09 2014
Nonnegative values of x of solutions (x, y) to the Diophantine equation 8*x^2 - 3*y^2 = 5. - Jon E. Schoenfield, Feb 02 2021

Alexander Adamchuk, Table of n, a(n) for n = 1..100
Tom Beldon and Tony Gardiner, Triangular Numbers and Perfect Squares, The Mathematical Gazette, Vol. 86, No. 507, (2002), pp. 423-431. - _Ant King_, Dec 07 2010
Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -1).

Cf. A005448, A129444.
Prime terms are listed in A129446.
Cf. A125602 (prime CTN), A184481 (semiprime CTN), A125603.
Cf. A000290, A249483.

Do[f = 3n(n-1)/2 + 1; If[IntegerQ[Sqrt[f]], Print[Sqrt[f]]], {n, 150000}]
LinearRecurrence[{0, 10, 0, -1}, {1, 2, 8, 19}, 30] (* T. D. Noe, Jun 13 2013 *)

a(n) = sqrt(3*A129444(n)*(A129444(n) - 1)/2 + 1).
G.f.: x*(1-x)*(1+3*x+x^2)/(1-10*x^2+x^4). - Colin Barker, Apr 11 2012
a(n) = 10*a(n-2) - a(n-4), a(1..4) =  1, 2, 8, 19. - Zak Seidov, Jun 11 2013

More terms from Alexander Adamchuk, Apr 26 2007

A255437 In positive integers: replace k^2 with the first k odd numbers.

Original entry on oeis.org

1, 2, 3, 1, 3, 5, 6, 7, 8, 1, 3, 5, 10, 11, 12, 13, 14, 15, 1, 3, 5, 7, 17, 18, 19, 20, 21, 22, 23, 24, 1, 3, 5, 7, 9, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 1, 3, 5, 7, 9, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 1, 3, 5, 7, 9, 11, 13, 50, 51

Offset: 1

Reinhard Zumkeller, Mar 23 2015

nonn

a(A005448(n)) = 1;
conjecture: a(A068722(n)) = (2*n+1)^2, i.e. A068722(n) = gives the position of the first occurrence of n-th odd square;
A164514(n) = a(A255527(n)) and a(m) < A164514(n) for m < A255527(n).

			.  A000290 | 1,    4,          9,                      16,         . . .
.  A000027 | _,2,3,___,5,6,7,8,_____,10,11,12,13,14,15,_______,17,18,...
.  A158405 | 1,    1,3,        1,3,5,                  1,3,5,7,
.  --------+-------------------------------------------------------------
.     a(n) | 1,2,3,1,3,5,6,7,8,1,3,5,10,11,12,13,14,15,1,3,5,7,17,18,19 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A256188, A000290, A000037, A158405, A016742, A164514, A255527, A005448, A255507 (first differences), A255508 (partial sums).

a255437 n = a255437_list !! (n-1)
a255437_list = f 0 [1..] a158405_tabl where
   f k xs (zs:zss) = us ++ zs ++ f (k + 2) vs zss
                     where (us, v:vs) = splitAt k xs

A288638 Number A(n,k) of n-digit biquanimous strings using digits {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 33, 16, 1, 1, 1, 6, 31, 92, 106, 32, 1, 1, 1, 7, 46, 201, 421, 333, 64, 1, 1, 1, 8, 64, 376, 1206, 1830, 1030, 128, 1, 1, 1, 9, 85, 633, 2841, 6751, 7687, 3153, 256, 1

Offset: 0

Alois P. Heinz, Jun 12 2017

A biquanimous string is a string whose digits can be split into two groups with equal sums.

			A(2,2) = 3: 00, 11, 22.
A(3,2) = 10: 000, 011, 022, 101, 110, 112, 121, 202, 211, 220.
A(3,3) = 19: 000, 011, 022, 033, 101, 110, 112, 121, 123, 132, 202, 211, 213, 220, 231, 303, 312, 321, 330.
A(4,1) = 8: 0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111.
Square array A(n,k) begins:
  1,  1,    1,    1,     1,      1,      1,      1, ...
  1,  1,    1,    1,     1,      1,      1,      1, ...
  1,  2,    3,    4,     5,      6,      7,      8, ...
  1,  4,   10,   19,    31,     46,     64,     85, ...
  1,  8,   33,   92,   201,    376,    633,    988, ...
  1, 16,  106,  421,  1206,   2841,   5801,  10696, ...
  1, 32,  333, 1830,  6751,  19718,  48245, 104676, ...
  1, 64, 1030, 7687, 36051, 128535, 372345, 939863, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Columns k=0-9 give: A000012, A011782, A053156, A288687, A288688, A288689, A288690, A288691, A288692, A065024.
Rows n=0+1,2-3 give: A000012, A000027(k+1), A005448(k+1).
Main diagonal gives A288693.
Cf. A064544, A064914.

b:= proc(n, k, s) option remember;
      `if`(n=0, `if`(s={}, 0, 1), add(b(n-1, k, select(y->
       y<=(n-1)*k, map(x-> [abs(x-i), x+i][], s))), i=0..k))
    end:
A:= (n, k)-> b(n, k, {0}):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, k_, s_] := b[n, k, s] = If[n == 0, If[s == {}, 0, 1], Sum[b[n-1, k, Select[Flatten[{Abs[#-i], #+i}& /@ s], # <= (n-1)*k&]], {i, 0, k}]];
A[n_, k_] := b[n, k, {0}];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 08 2018, from Maple *)

A069127 Centered 14-gonal numbers.

Original entry on oeis.org

1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201, 4551, 4915, 5293, 5685, 6091, 6511, 6945, 7393, 7855, 8331, 8821, 9325, 9843, 10375, 10921, 11481, 12055, 12643, 13245, 13861, 14491

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Binomial transform of [1, 14, 14, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 14, 0, 0, 0, ...]. - Gary W. Adamson, Jul 29 2011

Centered tetradecagonal numbers or centered tetrakaidecagonal numbers. - Omar E. Pol, Oct 03 2011

			a(5) = 141 because 7*5^2 - 7*5 + 1 = 175 - 35 + 1 = 141.
a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
From _Bruno Berselli_, Oct 27 2017: (Start)
1   =         -(1) + (2).
15  =       -(1+2) + (3+4+5+6).
43  =     -(1+2+3) + (4+5+6+7+8+9+10).
85  =   -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14).
141 = -(1+2+3+4+5) + (6+7+8+9+10+11+12+13+14+15+16+17+18). (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Heptagonal Stars.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005448, A001844, A005891, A003215, A069099.

Cf. A163756, A193053.

FoldList[#1 + #2 &, 1, 14 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[14*Range[0,50]]+1 (* Harvey P. Dale, Apr 09 2012 *)

a(n)=7*n^2-7*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 7*n^2 - 7*n + 1.
a(n) = 14*n+a(n-1)-14 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+12*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
a(n) = A163756(n-1) + 1. - Omar E. Pol, Oct 03 2011
a(n) = a(-n+1) = A193053(2n-2) + A193053(2n-3). - Bruno Berselli, Oct 21 2011
Sum_{n>=1} 1/a(n) = Pi * tan(sqrt(3/7)*Pi/2) / sqrt(21). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 8*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 8/e - 1. (End)
a(n) = A069099(n) + 7*A000217(n-1). - Leo Tavares, Jul 09 2021
E.g.f.: exp(x)*(1 + 7*x^2) - 1. - Stefano Spezia, Aug 01 2024

cp's OEIS Frontend

A077588 Maximum number of regions into which the plane is divided by n triangles.

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A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

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A069131 Centered 18-gonal numbers.

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A309148 A(n,k) is (1/k) times the number of n-member subsets of [k*n] whose elements sum to a multiple of n; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A080855 a(n) = (9*n^2 - 3*n + 2)/2.

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A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j).

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A129445 Numbers k > 0 such that k^2 is a centered triangular number.

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A080855 a(n) = (9n^2 - 3n + 2)/2.