A008815 -id:A008815 - cp's OEIS frontend

A008810 a(n) = ceiling(n^2/3).

0, 1, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046

Offset: 0

N. J. A. Sloane

a(n+1) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 3*w = 2*x + y. - Clark Kimberling, Jun 04 2012

a(n) is also the number of L-shapes (3-cell polyominoes) packing into an n X n square. See illustration in links. - Kival Ngaokrajang, Nov 10 2013

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, number of red blocks in Fig 2.5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000290, A056105, A056109.

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), this sequence (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a008810 = ceiling . (/ 3) . fromInteger . a000290
a008810_list = [0,1,2,3,6] ++ zipWith5
               (\u v w x y -> 2 * u - v + w - 2 * x + y)
   (drop 4 a008810_list) (drop 3 a008810_list) (drop 2 a008810_list)
   (tail a008810_list) a008810_list
-- Reinhard Zumkeller, Dec 20 2012

[Ceiling(n^2/3): n in [0..60]]; // G. C. Greubel, Sep 12 2019

seq(ceil(n^2/3), n=0..60); # G. C. Greubel, Sep 12 2019

Ceiling[Range[0,60]^2/3] (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,6},60] (* Harvey P. Dale, Jun 20 2011 *)

a(n)=ceil(n^2/3) /* Michael Somos, Aug 03 2006 */

[ceil(n^2/3) for n in (0..60)] # G. C. Greubel, Sep 12 2019

a(-n) = a(n) = ceiling(n^2/3).
G.f.: x*(1 + x^3)/((1 - x)^2*(1 - x^3)) = x*(1 - x^6)/((1 - x)*(1 - x^3))^2.
From Michael Somos, Aug 03 2006: (Start)
Euler transform of length 6 sequence [ 2, 0, 2, 0, 0, -1].
a(3n-1) = A056105(n).
a(3n+1) = A056109(n). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4. - Harvey P. Dale, Jun 20 2011
a(A008585(n)) = A033428(n). - Reinhard Zumkeller, Dec 20 2012
9*a(n) = 4 + 3*n^2 - 2*A099837(n+3). - R. J. Mathar, May 02 2013
a(n) = n^2 - 2*A000212(n). - Wesley Ivan Hurt, Jul 07 2013
Sum_{n>=1} 1/a(n) = Pi^2/18 + sqrt(2)*Pi*sinh(2*sqrt(2)*Pi/3)/(1+2*cosh(2*sqrt(2)*Pi/3)). - Amiram Eldar, Aug 13 2022
E.g.f.: (exp(x)*(4 + 3*x*(1 + x)) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 28 2022

A008812 Expansion of (1+x^5)/((1-x)^2*(1-x^5)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 14, 17, 20, 25, 30, 35, 40, 45, 52, 59, 66, 73, 80, 89, 98, 107, 116, 125, 136, 147, 158, 169, 180, 193, 206, 219, 232, 245, 260, 275, 290, 305, 320, 337, 354, 371, 388, 405, 424, 443, 462, 481, 500, 521, 542, 563, 584, 605, 628, 651, 674

Offset: 0

N. J. A. Sloane

Number of 0..n arrays of six elements with zero second differences. - R. H. Hardin, Nov 16 2011
Also number of ordered triples (w,x,y) with all terms in {1,...,n+1} and w + 4*x = 5*y. Also the number of 3-tuples (w,x,y) with all terms in {1,...,n+1} and 5*w = 2*x +3*y. - Clark Kimberling, Apr 15 2012 [Corrected by Pontus von Brömssen, Jan 26 2020]
a(n) is also the number of 5 boxes polyomino (zig-zag patterns) packing into (n+3) X (n+3) square. See illustration in links. - Kival Ngaokrajang, Nov 10 2013
Also, number of ordered pairs (x,y) with both terms in {1,...,n+1} and x+4*y divisible by 5; or number of ordered pairs (x,y) with both terms in {1,...,n+1} and 2*x+3*y divisible by 5. - Pontus von Brömssen, Jan 26 2020

			For n = 5 there are 8 0..5 arrays of six elements with zero second differences: [0,0,0,0,0,0], [0,1,2,3,4,5], [1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3,3,3], [4,4,4,4,4,4], [5,4,3,2,1,0], [5,5,5,5,5,5].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A130497 (first differences).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), this sequence (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,8,11];; for n in [8..65] do a[n]:=2*a[n-1]-a[n-2] +a[n-5]-2*a[n-6]+a[n-7]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1+x^5)/((1-x)^2*(1-x^5)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^5)/((1-x)^2*(1-x^5)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^5)/(1-x)^2/(1-x^5),{x,0,65}],x] (* or *) LinearRecurrence[{2,-1,0,0,1,-2,1}, {1,2,3,4,5,8,11}, 65] (* Harvey P. Dale, Apr 17 2015 *)

Vec((1+x^5)/(1-x)^2/(1-x^5)+O(x^65)) \\ Charles R Greathouse IV, Sep 25 2012

def A008812_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^5)/((1-x)^2*(1-x^5))).list()
A008812_list(65) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^5)/((1-x)^2*(1-x^5)).

a(n) = 2*a(n-1) -a(n-2) +a(n-5) -2*a(n-6) +a(n-7). - R. H. Hardin, Nov 16 2011

More terms added by G. C. Greubel, Sep 12 2019

A008811 Expansion of x(1+x^4)/((1-x)^2(1-x^4)).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 10, 13, 16, 21, 26, 31, 36, 43, 50, 57, 64, 73, 82, 91, 100, 111, 122, 133, 144, 157, 170, 183, 196, 211, 226, 241, 256, 273, 290, 307, 324, 343, 362, 381, 400, 421, 442, 463, 484, 507, 530, 553, 576, 601, 626, 651, 676, 703, 730, 757, 784, 813

Offset: 0

N. J. A. Sloane

Number of 0..n-1 arrays of 5 elements with zero 2nd differences. - R. H. Hardin, Nov 15 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Daniel Gabric and Joe Sawada, Investigating the discrepancy property of de Bruijn sequences, University of Guelph (Canada, 2020).
János Pach and Pankaj K. Agarwal, Combinatorial Geometry, p. 220, 1995, Problem 13.10.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A129756 (first differences).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), this sequence (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[0,1,2,3,4,7];; for n in [7..60] do a[n]:=2*a[n-1]-a[n-2] +a[n-4]-2*a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1+x^4)/((1-x)^2*(1-x^4)) )); // G. C. Greubel, Sep 12 2019

f := n->n^2/4+3*n/2+g(n);
g := n->if n mod 2 = 0 then 3 elif n mod 4 = 1 then 9/4 else 13/4; fi;
seq(f(n), n=-3..50);

CoefficientList[Series[x*(1+x^4)/((1-x)^2*(1-x^4)), {x,0,60}], x] (* G. C. Greubel, Sep 12 2019 *)

concat([0], Vec(x*(1+x^4)/((1-x)^2*(1-x^4))+O(x^60))) \\ Charles R Greathouse IV, Sep 26 2012, modified by G. C. Greubel, Sep 12 2019

def A008811_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+x^4)/((1-x)^2*(1-x^4))).list()
A008811_list(60) # G. C. Greubel, Sep 12 2019

G.f.: x*(1+x^4)/((1-x)^2*(1-x^4)).
a(n) = 2*a(n-1) -a(n-2) +a(n-4) -2*a(n-5) +a(n-6). - R. H. Hardin, Nov 15 2011
a(n) = (-2*(1+(-1)^n)*(-1)^floor(n/2) + 2*n^2 + 5 - (-1)^n)/8. - Tani Akinari, Jul 24 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x) - 2*cos(x))/4. - Stefano Spezia, May 26 2021
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/2)*Pi/4 + tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - Amiram Eldar, Aug 25 2022
a(n) = 2*floor((n^2 + 4)/8) + (n mod 2). - Ridouane Oudra, Sep 08 2023

A008813 Expansion of (1+x^6)/((1-x)^2*(1-x^6)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 12, 15, 18, 21, 24, 29, 34, 39, 44, 49, 54, 61, 68, 75, 82, 89, 96, 105, 114, 123, 132, 141, 150, 161, 172, 183, 194, 205, 216, 229, 242, 255, 268, 281, 294, 309, 324, 339, 354, 369, 384, 401, 418, 435, 452, 469, 486, 505, 524, 543, 562

Offset: 0

N. J. A. Sloane

Number of 0..n arrays of 7 elements with zero second differences. - R. H. Hardin, Nov 16 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), this sequence (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,9,12];; for n in [9..70] do a[n]:=2*a[n-1]-a[n-2] +a[n-6]-2*a[n-7]+a[n-8]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^6)/((1-x)^2*(1-x^6)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^6)/((1-x)^2*(1-x^6)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^6)/(1-x)^2/(1-x^6), {x,0,70}], x] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1}, {1,2,3,4,5,6,9,12}, 70] (* Harvey P. Dale, Oct 13 2012 *)

Vec((1+x^6)/((1-x)^2*(1-x^6)) +O(x^70)) \\ Charles R Greathouse IV, Sep 26 2012

def A008813_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^6)/((1-x)^2*(1-x^6))).list()
A008813_list(70) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^6)/((1-x)^2*(1-x^6)).

a(n) = 2*a(n-1) -a(n-2) +a(n-6) -2*a(n-7) +a(n-8). - R. H. Hardin, Nov 16 2011

More terms added by G. C. Greubel, Sep 12 2019

A008814 Expansion of (1+x^7)/((1-x)^2*(1-x^7)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 13, 16, 19, 22, 25, 28, 33, 38, 43, 48, 53, 58, 63, 70, 77, 84, 91, 98, 105, 112, 121, 130, 139, 148, 157, 166, 175, 186, 197, 208, 219, 230, 241, 252, 265, 278, 291, 304, 317, 330, 343, 358, 373, 388, 403, 418, 433, 448, 465, 482, 499

Offset: 0

N. J. A. Sloane

Number of 0..n arrays of 8 elements with zero second differences. - R. H. Hardin, Nov 16 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), this sequence (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,7,10,13];; for n in [10..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-7]-2*a[n-8]+a[n-9]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^7)/((1-x)^2*(1-x^7)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^7)/((1-x)^2*(1-x^7)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^7)/(1-x)^2/(1-x^7), {x,0,70}], x] (* or *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,10,13}, 70] (* Harvey P. Dale, Dec 18 2012 *)

a(n)=(n*(n+2)+[7,11,13,13,11,7,1][n%7+1])/7 \\ Charles R Greathouse IV, Nov 16 2011

a(n)=(n*(n+2)+13-6*(n%7==6))\7  \\ Tani Akinari, Jul 25 2013

def A008814_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^7)/((1-x)^2*(1-x^7))).list()
A008814_list(70) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^7)/((1-x)^2*(1-x^7)).

a(n) = 2*a(n-1) -a(n-2) +a(n-7) -2*a(n-8) +a(n-9). - R. H. Hardin, Nov 16 2011

More terms added by G. C. Greubel, Sep 12 2019

A008816 Expansion of (1+x^9)/((1-x)^2*(1-x^9)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 88, 95, 102, 109, 116, 123, 130, 137, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 236, 247, 258, 269, 280, 291, 302, 313, 324, 337, 350, 363, 376, 389, 402

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,-2,1).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), this sequence (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,7,8,9,12,15];; for n in [12..70] do a[n]:=2*a[n-1] -a[n-2]+a[n-9]-2*a[n-10]+a[n-11]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^9)/((1-x)^2*(1-x^9)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^9)/((1-x)^2*(1-x^9)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Sep 12 2019

LinearRecurrence[{2,-1,0,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,9,12,15}, 70] (* or *) CoefficientList[Series[(1+x^9)/((1-x)^2*(1-x^9)), {x,0, 70}], x] (* G. C. Greubel, Sep 12 2019 *)

my(x='x+O('x^70)); Vec((1+x^9)/((1-x)^2*(1-x^9))) \\ G. C. Greubel, Sep 12 2019

def A008815_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^8)/((1-x)^2*(1-x^8))).list()
A008815_list(70) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^9)/((1-x)^2*(1-x^9)). - G. C. Greubel, Sep 12 2019

More terms added by G. C. Greubel, Sep 12 2019

A008817 Expansion of (1+x^10)/((1-x)^2*(1-x^10)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 97, 104, 111, 118, 125, 132, 139, 146, 153, 160, 169, 178, 187, 196, 205, 214, 223, 232, 241, 250, 261, 272, 283, 294, 305, 316, 327, 338, 349, 360

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), this sequence (m=10).

a:=[1,2,3,4,5,6,7,8,9,10, 13,16];; for n in [13..80] do a[n]:=2*a[n-1]-a[n-2]+a[n-10]-2*a[n-11]+a[n-12]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x^10)/((1-x)^2*(1-x^10)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^10)/((1-x)^2*(1-x^10)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^10)/(1-x)^2/(1-x^10), {x,0,80}], x] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,9,10, 13,16}, 80] (* Harvey P. Dale, Jul 31 2014 *)

my(x='x+O('x^80)); Vec((1+x^10)/((1-x)^2*(1-x^10))) \\ G. C. Greubel, Sep 12 2019

def A008817_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^10)/((1-x)^2*(1-x^10))).list()
A008817_list(80) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^10)/((1-x)^2*(1-x^10)).

a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=5, a(5)=6, a(6)=7, a(7)=8, a(8)=9, a(9)=10, a(10)=13, a(11)=16, a(n) = 2*a(n-1) - a(n-2) + a(n-10) - 2*a(n-11) + a(n-12). - Harvey P. Dale, Jul 31 2014

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 14, 14, 11, 5, 6, 16, 22, 24, 22, 16, 6, 7, 22, 32, 37, 37, 32, 22, 7, 8, 29, 44, 53, 56, 53, 44, 29, 8, 9, 37, 58, 72, 79, 79, 72, 58, 37, 9, 10, 46, 74, 94, 106, 110, 106, 94, 74, 46, 10, 11, 56, 92, 119

Offset: 0

Franck Maminirina Ramaharo, Mar 05 2018

Antidiagonal sums are given by 2*A055795.
Rows/columns n are binomial transform of {n, A152947(n+1), n, 0, 0, 0, ...}.
Some primes in the array are
n = 1: {2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, ...} = A055469, primes of the form k*(k + 1)/2 + 1;
n = 3: {3, 7, 37, 53, 479, 653, 1249, 1619, 2503, 3727, 4349, 5737, 7109, 8179, 9803, 11839, 12107, ...};
n = 4: {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, ...} = A188382, primes of the form 8*(2*k - 1)^2 + 2*(2*k - 1) + 1.

			The array T(n,k) begins
0     1    2    3    4     5     6     7     8     9    10    11  ...
1     2    4    7   11    16    22    29    37    46    56    67  ...
2     4    8   14   22    32    44    58    74    92   112   134  ...
3     7   14   24   37    53    72    94   119   147   178   212  ...
4    11   22   37   56    79   106   137   172   211   254   301  ...
5    16   32   53   79   110   146   187   233   284   340   401  ...
6    22   44   72  106   146   192   244   302   366   436   512  ...
7    29   58   94  137   187   244   308   379   457   542   634  ...
8    37   74  119  172   233   302   379   464   557   658   767  ...
9    46   92  147  211   284   366   457   557   666   784   911  ...
10   56  112  178  254   340   436   542   658   784   920  1066  ...
11   67  134  212  301   401   512   634   767   911  1066  1232  ...
12   79  158  249  352   467   594   733   884  1047  1222  1409  ...
13   92  184  289  407   538   682   839  1009  1192  1388  1597  ...
14  106  212  332  466   614   776   952  1142  1346  1564  1796  ...
15  121  242  378  529   695   876  1072  1283  1509  1750  2006  ...
16  137  274  427  596   781   982  1199  1432  1681  1946  2227  ...
17  154  308  479  667   872  1094  1333  1589  1862  2152  2459  ...
18  172  344  534  742   968  1212  1474  1754  2052  2368  2702  ...
19  191  382  592  821  1069  1336  1622  1927  2251  2594  2956  ...
20  211  422  653  904  1175  1466  1777  2108  2459  2830  3221  ...
...
The inverse binomial transforms of the columns are
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
1     1    2    4    7    11    22    29    37    45    56    67  ...  A152947
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
...

Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

Cf. A000124, A001477, A006000, A008815, A014206, A051601, A055469, A077028, A081436, A084849, A131074, A134394, A139600, A141387, A179000, A188377, A188382, A273465.

T := (n, k) -> n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1);
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1);
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2018 *)

T(n, k) := n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1)$
for n:0 thru 20 do
  print(makelist(T(n, k), k, 0, 20));

T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1);
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 12 2018

T(n,k) = T(k,n) = n*A152947(k+1) + k*A152947(n+1).
T(n,0) = A001477(n).
T(n,1) = A000124(n).
T(n,2) = A014206(n).
T(n,3) = A273465(3*n+2).
T(n,4) = A084849(n+1).
T(n,n) = A179000(n-1,n), n >= 1.
T(2*n,2*n) = 8*A081436(n-1), n >= 1.
T(2*n+1,2*n+1) = 2*A006000(2*n+1).
T(n,n+1) = A188377(n+3).
T(n,n+2) = A188377(n+2), n >= 1.
Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).

A365539 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 + x^k)/((1 - x)^2*(1 - x^k)), with k > 0.

Original entry on oeis.org

1, 4, 1, 9, 2, 1, 16, 5, 2, 1, 25, 8, 3, 2, 1, 36, 13, 6, 3, 2, 1, 49, 18, 9, 4, 3, 2, 1, 64, 25, 12, 7, 4, 3, 2, 1, 81, 32, 17, 10, 5, 4, 3, 2, 1, 100, 41, 22, 13, 8, 5, 4, 3, 2, 1, 121, 50, 27, 16, 11, 6, 5, 4, 3, 2, 1, 144, 61, 34, 21, 14, 9, 6, 5, 4, 3, 2, 1

Offset: 0

Stefano Spezia, Sep 08 2023

			Array begins:
   1,  1,  1,  1,  1,  1,  1, ...
   4,  2,  2,  2,  2,  2,  2, ...
   9,  5,  3,  3,  3,  3,  3, ...
  16,  8,  6,  4,  4,  4,  4, ...
  25, 13,  9,  7,  5,  5,  5, ...
  36, 18, 12, 10,  8,  6,  6, ...
  49, 25, 17, 13, 11,  9,  7, ...
  64, 32, 22, 16, 14, 12, 10, ...
  ...

Cf. A000027 (main diagonal and superdiagonals), A000290 (k=1), A000982 (k=2), A008810 (k=3), A008811 (k=4), A008812 (k=5), A008813 (k=6), A008814 (k=7), A008815 (k=8), A008816 (k=9), A008817 (k=10).

Cf. A365540 (antidiagonal sums).

A[n_,k_]:=SeriesCoefficient[(1+x^k)/((1-x)^2*(1-x^k)),{x,0,n}]; Table[A[n-k,k],{n,0,12},{k,n}]//Flatten

A343953 Square array T(n,k), n>=1, k>=0, read by antidiagonals, where row n is the expansion of x(1+x^n)/((1-x)^2(1-x^n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 0, 1, 2, 9, 0, 1, 2, 5, 16, 0, 1, 2, 3, 8, 25, 0, 1, 2, 3, 6, 13, 36, 0, 1, 2, 3, 4, 9, 18, 49, 0, 1, 2, 3, 4, 7, 12, 25, 64, 0, 1, 2, 3, 4, 5, 10, 17, 32, 81, 0, 1, 2, 3, 4, 5, 8, 13, 22, 41, 100, 0, 1, 2, 3, 4, 5, 6, 11, 16, 27, 50, 121, 0, 1, 2, 3, 4, 5, 6, 9, 14, 21, 34, 61, 144

Offset: 0

Jean-François Alcover, May 05 2021

			Square array begins:
0, 1, 4, 9,16,25,36,49,64,81,100,121,   ... (A000290)
0, 1, 2, 5, 8,13,18,25,32,41, 50, 61,   ... (A000982)
0, 1, 2, 3, 6, 9,12,17,22,27, 34, 41,   ... (A008810)
0, 1, 2, 3, 4, 7,10,13,16,21, 26, 31,   ... (A008811)
0, 1, 2, 3, 4, 5, 8,11,14,17, 20, 25,   ... (A008812)
0, 1, 2, 3, 4, 5, 6, 9,12,15, 18, 21,   ... (A008813)
0, 1, 2, 3, 4, 5, 6, 7,10,13, 16, 19,   ... (A008814)
0, 1, 2, 3, 4, 5, 6, 7, 8,11, 14, 17,   ... (A008815)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15,   ... (A008816)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13,   ... (A008817)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14,... not in the OEIS
...

Cf. A000290, A000982, A008810, A008811, A008812, A008813, A008814, A008815, A008816, A008817.

nmax = 15;
ro[n_] := ro[n] = CoefficientList[x(1+x^n)/((1-x)^2 (1-x^n))+O[x]^nmax, x];
T[n_, k_] := ro[n][[k+1]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}]  // Flatten

G.f. of row n: x*(1+x^n)/((1-x)^2*(1-x^n)), some cross-referenced sequences omitting the factor x and the initial term 0.

cp's OEIS Frontend

A008810 a(n) = ceiling(n^2/3).

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A008812 Expansion of (1+x^5)/((1-x)^2*(1-x^5)).

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A008811 Expansion of x*(1+x^4)/((1-x)^2*(1-x^4)).

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A008813 Expansion of (1+x^6)/((1-x)^2*(1-x^6)).

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A008814 Expansion of (1+x^7)/((1-x)^2*(1-x^7)).

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A008811 Expansion of x(1+x^4)/((1-x)^2(1-x^4)).

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

A343953 Square array T(n,k), n>=1, k>=0, read by antidiagonals, where row n is the expansion of x(1+x^n)/((1-x)^2(1-x^n)).