A069133 -id:A069133 - cp's OEIS frontend

A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004

Row n gives the centered figurate numbers of the n-gon.

Antidiagonal sums are in A101338.

			The upper left corner of the infinite array T is
|0| 1   1   1   1   1   1   1   1   1   1 ... A000012
|1| 1   2   4   7  11  16  22  29  37  46 ... A000124
|2| 1   3   7  13  21  31  43  57  73  91 ... A002061
|3| 1   4  10  19  31  46  64  85 109 136 ... A005448
|4| 1   5  13  25  41  61  85 113 145 181 ... A001844
|5| 1   6  16  31  51  76 106 141 181 226 ... A005891
|6| 1   7  19  37  61  91 127 169 217 271 ... A003215
|7| 1   8  22  43  71 106 148 197 253 316 ... A069099
|8| 1   9  25  49  81 121 169 225 289 361 ... A016754
|9| 1  10  28  55  91 136 190 253 325 406 ... A060544

Cf. A000217, A001263, A101338.

A101321 := proc(n,k)
        n*k*(k+1)/2+1 ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_] := 1 + n m (m + 1)/2;
Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 23 2020 *)

T(n,m) = 1 + n*m*(m+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).
T(10,m) = A062786(m+1).
T(11,m) = A069125(m+1).
T(12,m) = A003154(m+1).
T(13,m) = A069126(m+1).
T(14,m) = A069127(m+1).
T(15,m) = A069128(m+1).
T(16,m) = A069129(m+1).
T(17,m) = A069130(m+1).
T(18,m) = A069131(m+1).
T(19,m) = A069132(m+1).
T(20,m) = A069133(m+1).
T(n+1,m) = T(n,m) + m*(m+1)/2. - Gary W. Adamson and Michel Marcus, Oct 13 2015

Edited by R. J. Mathar, Oct 21 2009

A069190 Centered 24-gonal numbers.

Original entry on oeis.org

1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673

Offset: 1

Terrel Trotter, Jr., Apr 10 2002

Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011

			a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Odd Ordered Star Perimeters.
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 k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

FoldList[#1 + #2 &, 1, 24 Range@ 45] (* Robert G. Wilson v *)
Table[12n^2-12n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,25,73},50] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=12*n^2-12*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 12*n^2 - 12*n + 1.
a(n) = 24*n + a(n-1) - 24 with a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011
G.f.: x*(1+22*x+x^2)/(1-x)^3. - Harvey P. Dale, Jul 17 2011
Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).
Sum_{n>=1} a(n)/n! = 13*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)
E.g.f.: exp(x)*(1 + 12*x^2) - 1. - Stefano Spezia, May 31 2022

More terms from Harvey P. Dale, Jul 17 2011

A195142 Concentric 10-gonal numbers.

Original entry on oeis.org

0, 1, 10, 21, 40, 61, 90, 121, 160, 201, 250, 301, 360, 421, 490, 561, 640, 721, 810, 901, 1000, 1101, 1210, 1321, 1440, 1561, 1690, 1821, 1960, 2101, 2250, 2401, 2560, 2721, 2890, 3061, 3240, 3421, 3610, 3801, 4000, 4201, 4410, 4621, 4840, 5061, 5290

Offset: 0

Omar E. Pol, Sep 17 2011

Also concentric decagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 10, ..., and the same line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Main axis, perpendicular to A028895 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

A033583 and A069133 interleaved.
Cf. A028895, A032527, A032528, A077221, A085787, A195042, A195143, A195145, A195146, A195147, A195148, A195149.
Cf. A090771 (first differences).
Column 10 of A195040. - Omar E. Pol, Sep 28 2011

a195142 n = a195142_list !! n
a195142_list = scanl (+) 0 a090771_list
-- Reinhard Zumkeller, Jan 07 2012

[(10*n^2+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+10(n-1)},a[n],{n,50}] (* or *) LinearRecurrence[{2,0,-2,1},{0,1,10,21},50] (* Harvey P. Dale, Sep 29 2011 *)

G.f.: -x*(1+8*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = -a(n-1) + 5*n^2 - 5*n + 1, a(0)=0. - Vincenzo Librandi, Sep 27 2011
From Bruno Berselli, Sep 27 2011: (Start)
a(n) = a(-n) = (10*n^2 + 3*(-1)^n - 3)/4.
a(n) = a(n-2) + 10*(n-1). (End)
a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=10, a(3)=21. - Harvey P. Dale, Sep 29 2011
Sum_{n>=1} 1/a(n) = Pi^2/60 + tan(sqrt(3/5)*Pi/2)*Pi/(2*sqrt(15)). - Amiram Eldar, Jan 16 2023

A195317 Centered 40-gonal numbers.

Original entry on oeis.org

1, 41, 121, 241, 401, 601, 841, 1121, 1441, 1801, 2201, 2641, 3121, 3641, 4201, 4801, 5441, 6121, 6841, 7601, 8401, 9241, 10121, 11041, 12001, 13001, 14041, 15121, 16241, 17401, 18601, 19841, 21121, 22441, 23801, 25201, 26641, 28121, 29641, 31201, 32801, 34441, 36121

Offset: 1

Omar E. Pol, Sep 16 2011

Also centered tetracontagonal numbers or centered tetrakaicontagonal numbers. Also sequence found by reading the line from 1, in the direction 1, 41, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semi-axis opposite to A195322 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195148.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195316, A195318.
Cf. A195162, A195322.

[20*n^2 - 20*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 21 2011

Table[20*n^2 - 20*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

a(n)=20*n^2-20*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 20*n^2 - 20*n + 1.
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(5))/(8*sqrt(5)). - Amiram Eldar, Feb 11 2022
G.f.: -x*(1+38*x+x^2)/(x-1)^3. - R. J. Mathar, May 07 2024
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(20*x^2 + 1) - 1.
a(n) = 2*A069133(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.  (End)

A195314 Centered 28-gonal numbers.

Original entry on oeis.org

1, 29, 85, 169, 281, 421, 589, 785, 1009, 1261, 1541, 1849, 2185, 2549, 2941, 3361, 3809, 4285, 4789, 5321, 5881, 6469, 7085, 7729, 8401, 9101, 9829, 10585, 11369, 12181, 13021, 13889, 14785, 15709, 16661, 17641, 18649, 19685, 20749, 21841, 22961, 24109, 25285, 26489

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 29, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Semi-axis opposite to A144555 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195145.
Cf. A003154, A069129, A069133, A069190, A195315, A195316, A195317, A195318.
Cf. A069127, A118277, A144555.

[(14*n^2-14*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[14n^2-14n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,29,85},50]

a(n)=14*n^2-14*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - 14*n + 1.
G.f.: -x*(1 + 26*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Oct 01 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5/7)*Pi/2)/(2*sqrt(35)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(14*x^2 + 1) - 1.
a(n) = 2*A069127(n) - 1. (End)

A195315 Centered 32-gonal numbers.

Original entry on oeis.org

1, 33, 97, 193, 321, 481, 673, 897, 1153, 1441, 1761, 2113, 2497, 2913, 3361, 3841, 4353, 4897, 5473, 6081, 6721, 7393, 8097, 8833, 9601, 10401, 11233, 12097, 12993, 13921, 14881, 15873, 16897, 17953, 19041, 20161, 21313, 22497, 23713, 24961, 26241, 27553, 28897, 30273

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Semi-axis opposite to A016802 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195146.
Cf. A003154, A069129, A069133, A069190, A195314, A195316, A195317, A195318.
Cf. A016802, A074377.

[(16*n^2-16*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[16*n^2 - 16*n + 1, {n, 1, 41}] (* Amiram Eldar, Feb 11 2022 *)
LinearRecurrence[{3,-3,1},{1,33,97},50] (* Harvey P. Dale, Feb 11 2024 *)

a(n)=16*n^2-16*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 16*n^2 - 16*n + 1.
G.f.: -x*(1 + 30*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3)*Pi/4)/(8*sqrt(3)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(16*x^2 + 1) - 1.
a(n) = 2*A069129(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A195316 Centered 36-gonal numbers.

Original entry on oeis.org

1, 37, 109, 217, 361, 541, 757, 1009, 1297, 1621, 1981, 2377, 2809, 3277, 3781, 4321, 4897, 5509, 6157, 6841, 7561, 8317, 9109, 9937, 10801, 11701, 12637, 13609, 14617, 15661, 16741, 17857, 19009, 20197, 21421, 22681, 23977, 25309, 26677, 28081, 29521, 30997, 32509

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 37, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195321 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
John Elias, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195147.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195317, A195318.
Cf. A069131, A195160, A195321.

[(18*n^2-18*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[18*n^2 - 18*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

a(n)=18*n^2-18*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 18*n^2 - 18*n + 1.
G.f.: -x*(1 + 34*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7)*Pi/6)/(6*sqrt(7)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(18*x^2 + 1) - 1.
a(n) = 2*A069131(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A195318 Centered 44-gonal numbers.

Original entry on oeis.org

1, 45, 133, 265, 441, 661, 925, 1233, 1585, 1981, 2421, 2905, 3433, 4005, 4621, 5281, 5985, 6733, 7525, 8361, 9241, 10165, 11133, 12145, 13201, 14301, 15445, 16633, 17865, 19141, 20461, 21825, 23233, 24685, 26181, 27721, 29305, 30933, 32605, 34321, 36081, 37885, 39733

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 45, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195323 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195149.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195316, A195317.
Cf. A069173, A195313, A195323.

[22*n^2 - 22*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 21 2011

Table[22n^2-22n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,45,133},50] (* Harvey P. Dale, Mar 16 2019 *)

a(n)=22*n^2-22*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 22*n^2 - 22*n + 1.
Sum_{n>=1} 1/a(n) = Pi*tan(3*Pi/(2*sqrt(11)))/(6*sqrt(11)). - Amiram Eldar, Feb 11 2022
G.f.: -x*(1+42*x+x^2)/(x-1)^3. - R. J. Mathar, May 07 2024
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(22*x^2 + 1) - 1.
a(n) = 2*A069173(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A010010 a(0) = 1, a(n) = 20*n^2 + 2 for n>0.

Original entry on oeis.org

1, 22, 82, 182, 322, 502, 722, 982, 1282, 1622, 2002, 2422, 2882, 3382, 3922, 4502, 5122, 5782, 6482, 7222, 8002, 8822, 9682, 10582, 11522, 12502, 13522, 14582, 15682, 16822, 18002, 19222, 20482, 21782, 23122, 24502, 25922, 27382, 28882, 30422, 32002, 33622

Offset: 0

N. J. A. Sloane

Bruno Berselli, Table of n, a(n) for n = 0..1000
Volodymyr Mazorchuk and Xiaoyu Zhu, Combinatorics of infinite rank module categories over finite dimensional sl_3-modules in Lie-algebraic context, arXiv:2501.00291 [math.RT], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

[1] cat [20*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 20 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {22, 82, 182}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

a(n) = A033571(n)+A158186(n) = A158187(n)*2 for n>0. - Reinhard Zumkeller, Mar 13 2009
G.f.: (1+x)*(1+18*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*20+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(10)/40*Pi*coth(Pi/sqrt(10)) = 1.0772981051444036327... - R. J. Mathar, May 07 2024
a(n) = A069133(n)+A069133(n+1). - R. J. Mathar, May 07 2024

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 13, 7, 1, 1, 15, 25, 22, 9, 1, 1, 21, 41, 46, 33, 11, 1, 1, 28, 61, 79, 73, 46, 13, 1, 1, 36, 85, 121, 129, 106, 61, 15, 1, 1, 45, 113, 172, 201, 191, 145, 78, 17, 1, 1, 55, 145, 232, 289, 301, 265, 190, 97, 19, 1

Offset: 0

Peter Luschny, Mar 21 2023

A combination of a multiset M is an unordered selection of k objects of M, where every object can appear at most as many times as it appears in M.
A(n, k) = Cardinality(Union_{j=0..k} Combination(MultiSet(1^[j*n], 0^[(k-j)*n]))), where MultiSet(r^[s], u^[v]) denotes a set that contains the element r with multiplicity s and the element u with multiplicity v; thus the multisets under consideration have n*k elements. Since the base set is {1, 0} the elements can be represented as binary strings. Applying the combination operator to the multisets results in a set of binary strings where '0' resp. '1' can appear at most j*n resp. (k-j)*n times. 'At most' means that they do not have to appear; in other words, the resulting set always includes the empty string ''.
In contrast to the procedure in A361045 we consider here the cardinality of the set union and not the sum of the individual cardinalities. If you want to exclude the empty string, you will find the sequences listed in A361521. The same construction with multiset permutations instead of multiset combinations results in A361043.
A different view can be taken if one considers the hypergeometric representation, hypergeom([-k, -m], [1], n). This is a family of arrays that includes the 'rascal' triangle: the all 1's array A000012 (m = 0), the rascal array A077028 (m = 1), this array (m = 2), and A361731 (m = 3).

			Array A(n, k) starts:
   [0] 1,  1,   1,    1,   1,   1,   1,    1, ...  A000012
   [1] 1,  3,   6,   10,  15,  21,  28,   36, ...  A000217
   [2] 1,  5,  13,   25,  41,  61,  85,  113, ...  A001844
   [3] 1,  7,  22,   46,  79, 121, 172,  232, ...  A038764
   [4] 1,  9,  33,   73, 129, 201, 289,  393, ...  A081585
   [5] 1, 11,  46,  106, 191, 301, 436,  596, ...  A081587
   [6] 1, 13,  61,  145, 265, 421, 613,  841, ...  A081589
   [7] 1, 15,  78,  190, 351, 561, 820, 1128, ...  A081591
   000012  | A028872 | A239325 |
       A005408    A100536   A069133
.
Triangle T(n, k) starts:
   [0] 1;
   [1] 1,  1;
   [2] 1,  3,   1;
   [3] 1,  6,   5,   1;
   [4] 1, 10,  13,   7,   1;
   [5] 1, 15,  25,  22,   9,   1;
   [6] 1, 21,  41,  46,  33,  11,   1;
   [7] 1, 28,  61,  79,  73,  46,  13,  1;
   [8] 1, 36,  85, 121, 129, 106,  61, 15,  1;
   [9] 1, 45, 113, 172, 201, 191, 145, 78, 17, 1.
.
Row 4 of the triangle:
A(0, 4) =  1 = card('').
A(1, 3) = 10 = card('', 0, 00, 000, 1, 10, 100, 11, 110, 111).
A(2, 2) = 13 = card('', 0, 00, 000, 0000, 1, 10, 100, 11, 110, 1100, 111, 1111).
A(3, 1) =  7 = card('', 0, 00, 000, 1, 11, 111).
A(4, 0) =  1 = card('').

Rows: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591.
Columns: A000012, A005408, A028872, A100536, A239325, A069133.
Cf. A239592 (main diagonal), A239331 (transposed array).
Cf. A361045, A361043, A361521, A361731, A077028.

A := (n, k) -> 1 + n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;
# Alternative:
ogf := n -> (1 + (n - 1)*x)^2 / (1 - x)^3:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..7);

def A(m: int, steps: int) -> int:
    if m == 0: return 1
    size = m * steps
    cset = set()
    for a in range(0, size + 1, m):
        S = [str(int(i < a)) for i in range(size)]
        C = Combinations(S)
        cset.update("".join(i for i in c) for c in C)
    return len(cset)
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size + 1)]
for n in range(8): print(ARow(n, 7))

A(n, k) = 1 + n*k*(4 + n*(k - 1))/2.
T(n, k) = 1 + k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = [x^k] (1 + (n - 1)*x)^2 / (1 - x)^3.
A(n, k) = hypergeom([-k, -2], [1], n).
A(n, k) = A361521(n, k) + 1.

cp's OEIS Frontend

A101321 Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.

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Maple

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A069190 Centered 24-gonal numbers.

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Mathematica

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A195142 Concentric 10-gonal numbers.

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Haskell

Magma

Mathematica

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A195317 Centered 40-gonal numbers.

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Magma

Mathematica

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A195314 Centered 28-gonal numbers.

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Magma

Mathematica

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A195315 Centered 32-gonal numbers.

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Magma

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A195316 Centered 36-gonal numbers.

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A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.