A008816 -id:A008816 - 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

A008815 Expansion of (1+x^8)/((1-x)^2*(1-x^8)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 32, 37, 42, 47, 52, 57, 62, 67, 72, 79, 86, 93, 100, 107, 114, 121, 128, 137, 146, 155, 164, 173, 182, 191, 200, 211, 222, 233, 244, 255, 266, 277, 288, 301, 314, 327, 340, 353, 366, 379, 392, 407, 422

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-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), this sequence (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,7,8,11,14];; for n in [11..50] do a[n]:=2*a[n-1] -a[n-2]+a[n-8]-2*a[n-9]+a[n-10]; od; a; # G. C. Greubel, Sep 12 2019

I:=[1,2,3,4,5,6,7,8,11,14]; [n le 10 select I[n] else 2*Self(n-1) -Self(n-2)+Self(n-8)-2*Self(n-9)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, May 14 2019

seq(coeff(series((1+x^8)/((1-x)^2*(1-x^8)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^8)/(1-x)^2/(1-x^8), {x, 0, 50}], x] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,11,14}, 50] (* Harvey P. Dale, Dec 17 2016 *)

a(n)=(n*(n+2)+14+4*(n%4-1)*(-1)^(n\4))\8  \\ Tani Akinari, Jul 25 2013

def A008815_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^8)/((1-x)^2*(1-x^8))).list()
A008815_list(50) # G. C. Greubel, Sep 12 2019

G.f.: (1 + x^8)/((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = floor( (n*(n+2) + 14 + 4*((n mod 4) - 1)*(-1)^floor(n/4))/8 ). - Tani Akinari, Jul 25 2013
a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10). - Vincenzo Librandi, May 14 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

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)).

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)).