A008811 -id:A008811 - 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

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

A085046 a(n) = n^2 - (1 + (-1)^n)/2.

Original entry on oeis.org

1, 3, 9, 15, 25, 35, 49, 63, 81, 99, 121, 143, 169, 195, 225, 255, 289, 323, 361, 399, 441, 483, 529, 575, 625, 675, 729, 783, 841, 899, 961, 1023, 1089, 1155, 1225, 1295, 1369, 1443, 1521, 1599, 1681, 1763, 1849, 1935, 2025, 2115, 2209, 2303, 2401, 2499, 2601

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 20 2003

Sequence pattern looks like this 1*1, 1*3, 3*3, 3*5, 5*5, 5*7, 7*7, 7*9, 9*9, 9*11, 11*11, ... = A109613(n-1)*A109613(n).
a(n+1) is the determinant of the n X n matrix M_(i, i)=3, M_(i, j)=2 if (i+j) is even, M_(i, j)=0 if (i+j) is odd. - Benoit Cloitre, Aug 06 2003
a(n) is also the longest path, in number of cells, between diagonally opposite corners of an n X n matrix if diagonal movement between adjacent cells is not allowed and no cell is used more than once. - Ray G. Opao, Jul 02 2007
(-1)^n*a(n) appears to be the Hankel transform of A141222. - Paul Barry, Jun 14 2008
Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. 4*a(n) is the total number of unit edges in each figure (see example and cf. A255840, A255876). - Wesley Ivan Hurt, Mar 09 2015

			4*a(n) is the number of unit edges in the pattern below (see comments).
                                                                 _
                                                               _|_|_
                            _              _ _               _|_|_|_|_
                          _|_|_          _|_|_|_           _|_|_|_|_|_|_
              _ _       _|_|_|_|_      _|_|_|_|_|_       _|_|_|_|_|_|_|_|_
    _        |_|_|     |_|_|_|_|_|    |_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
   |_|       |_|_|       |_|_|_|      |_|_|_|_|_|_|       |_|_|_|_|_|_|_|
                           |_|          |_|_|_|_|           |_|_|_|_|_|
                                          |_|_|               |_|_|_|
                                                                |_|
   n=1        n=2          n=3             n=4                  n=5
- _Wesley Ivan Hurt_, Mar 09 2015

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

Cf. A109613. [Bruno Berselli, Sep 17 2013]

Cf. A008811, A141222, A255840, A255876.

[n^2-(1+(-1)^n)/2 : n in [1..100]]; // Wesley Ivan Hurt, Mar 09 2015

A085046:=n->n^2-(1+(-1)^n)/2: seq(A085046(n), n=1..100); # Wesley Ivan Hurt, Mar 09 2015

Table[n^2-1/2 (1+(-1)^n), {n, 60}] (* Bruno Berselli, Sep 17 2013 *)
LinearRecurrence[{2,0,-2,1},{1,3,9,15},70] (* Harvey P. Dale, Oct 25 2015 *)

a(1) = 1, a(2) = 3, then a(2n) = (a(2n-1)*a(2n+1))^1/2 and a(2n+1) = {a(2n) + a(2n+2)}/2. Even-indexed terms are the geometric mean, and odd-indexed terms are the arithmetic mean, of their neighbors.
a(2n+1) = (2n+1)^2 and a(2n) = 4n^2 - 1.
a(n) = A008811(2n) - 1. - N. J. A. Sloane, Jun 12 2004
From Bruno Berselli, Sep 17 2013: (Start)
G.f.: x*(1 + x + 3*x^2 - x^3)/((1+x)*(1-x)^3).
a(n) = n^2 - (1 + (-1)^n)/2. (End)
a(1)=1, a(2)=3, a(3)=9, a(4)=15, a(n) = 2*a(n-1) + 0*a(n-2) -  2*a(n-3) + a(n-4). - Harvey P. Dale, Oct 25 2015
E.g.f.: 1 - cosh(x) + x*(1 + x)*(cosh(x) + sinh(x)). - Stefano Spezia, May 26 2021
Sum_{n>=1} 1/a(n) = Pi^2/8 + 1/2. - Amiram Eldar, Aug 25 2022

More terms from Benoit Cloitre, Aug 06 2003
Formula added in the first comment by Bruno Berselli, Sep 17 2013
Replaced name with Sep 17 2013 formula from Bruno Berselli [Wesley Ivan Hurt, May 17 2020]

A200272 T(n,k) is the number of 0..k arrays x(0..n+1) of n+2 elements with zero n-1st differences.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 7, 6, 5, 2, 1, 7, 10, 9, 8, 5, 2, 1, 8, 13, 14, 13, 8, 5, 2, 1, 9, 16, 25, 18, 13, 8, 7, 2, 1, 10, 21, 36, 37, 26, 21, 16, 3, 2, 1, 11, 26, 47, 64, 53, 34, 31, 4, 3, 2, 1, 12, 31, 64, 97, 112, 53, 52, 19, 10, 7, 2, 1, 13, 36, 87, 140, 197, 118, 83

Offset: 1

R. H. Hardin Nov 15 2011

Each row is an Ehrhart quasi-polynomial. - Robert Israel, May 04 2021

			Some solutions for n=7 k=6
..3....2....3....3....5....2....3....4....0....6....3....5....1....6....0....1
..3....2....3....3....1....2....3....0....4....2....3....1....5....2....4....5
..6....1....3....4....0....0....2....1....3....4....5....3....5....5....2....1
..6....1....3....4....1....0....2....2....2....5....5....4....4....6....1....0
..3....2....3....3....3....2....3....2....2....4....3....3....3....4....2....3
..0....3....3....2....5....4....4....2....2....3....1....2....2....2....3....6
..0....3....3....2....6....4....4....3....1....4....1....3....1....3....2....5
..3....2....3....3....5....2....3....4....0....6....3....5....1....6....0....1
..3....2....3....3....1....2....3....0....4....2....3....1....5....2....4....5
Table starts
.1.1..1..1...1...1...1....1....1....1.....1.....1.....1.....1......1.....1
.2.3..4..5...6...7...8....9...10...11....12....13....14....15.....16....17
.2.3..4..7..10..13..16...21...26...31....36....43....50....57.....64....73
.2.3..6..9..14..25..36...47...64...87...110...143...176...209....258...311
.2.5..8.13..18..37..64...97..140..207...286...399...528...687....878..1127
.2.5..8.13..26..53.112..197..302..465...688..1013..1406..1995...2790..3759
.2.5..8.21..34..53.118..267..516..901..1454..2249..3264..5135...7616.11061
.2.7.16.31..52..83.228..515.1014.1903..3236..5351..8174.13719..21660.33377
.2.3..4.19..34.119.236..589.1236.2707..5062.10081.17712.30497..51072.83605
.2.3.10.37.104.261.574.1181.2560.5137.10922.23531.44970.81485.148250

R. H. Hardin, Table of n, a(n) for n = 1..291

Row 3 is A008811(n+1).

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

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

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

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