A008810 -id:A008810 - cp's OEIS frontend

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

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

A200082 T(n,k)=Number of 0..k arrays x(0..n) of n+1 elements with zero n-1st differences.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 6, 3, 2, 1, 6, 9, 8, 7, 2, 1, 7, 12, 17, 14, 9, 2, 1, 8, 17, 26, 27, 18, 9, 2, 1, 9, 22, 43, 58, 37, 24, 15, 2, 1, 10, 27, 64, 111, 108, 85, 56, 7, 2, 1, 11, 34, 89, 182, 245, 202, 169, 26, 3, 2, 1, 12, 41, 122, 279, 454, 429, 394, 151, 26, 11, 2, 1, 13, 48

Offset: 1

R. H. Hardin Nov 13 2011

Table starts
.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
.2..3..6...9..12...17....22....27....34.....41.....48.....57......66......75
.2..3..8..17..26...43....64....89...122....163....208....269.....334.....407
.2..7.14..27..58..111...182...279...404....617....872...1199....1580....2045
.2..9.18..37.108..245...454...759..1172...2001...3144...4663....6568....8945
.2..9.24..85.202..429..1046..2145..4022...6955..11438..17927...26868...41817
.2.15.56.169.394..855..2546..6179.12710..23899..41522..68427..106948..183797
.2..7.26.151.468.1863..5056.12965.29904..64603.124728.243309..432190..748301
.2..3.26.219.848.3573.11638.31507.84560.198435.418330.878657.1704398.3107463
T(n,k) is the number of integer lattice points in k*C(n) where C(n) is the polytope in R^(n+1) defined by two linear equations and the bounds 0 <= x_i <= 1.  Since the vertices of this polytope have rational coordinates, T(n,k) for each fixed n is an Ehrhart quasi-polynomial of degree n-1. - Robert Israel, Nov 11 2019

			Some solutions for n=7 k=6
..1....4....5....4....5....0....2....6....0....4....6....1....2....3....1....4
..5....2....3....5....0....1....1....1....4....5....4....3....0....6....0....0
..2....6....4....4....5....3....2....3....6....3....0....5....2....3....4....2
..1....6....3....3....5....3....3....5....6....2....0....4....4....0....6....3
..3....3....1....3....1....2....3....5....5....3....2....2....4....0....5....2
..5....2....1....4....0....2....2....4....4....5....2....2....2....3....3....1
..4....5....4....5....5....3....1....4....3....6....0....4....0....6....2....2
..1....4....5....4....5....0....2....6....0....4....6....1....2....3....1....4

R. H. Hardin, Table of n, a(n) for n = 1..268

Row 3 is A008810(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

A143977 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m] X [0,n].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 3, 4, 5, 6, 5, 4, 3, 3, 5, 6, 7, 7, 6, 5, 3, 3, 6, 7, 8, 9, 8, 7, 6, 3, 4, 6, 8, 10, 10, 10, 10, 8, 6, 4, 4, 7, 9, 11, 12, 12, 12, 11, 9, 7, 4, 4, 8, 10, 12, 14, 14, 14, 14, 12, 10, 8, 4, 5, 8, 11, 14, 15, 16, 17, 16, 15, 14, 11, 8, 5

Offset: 1

Clark Kimberling, Sep 06 2008

Rows numbered 3,6,9,12,15,... are, except for initial terms, multiples of (1,2,3,4,5,6,7,...) = A000027.

			Northwest corner:
  1  1  1  2  2  2  3
  1  2  2  3  4  4  5
  1  2  3  4  5  6  7
  2  3  4  6  7  8 10
  2  4  5  7  9 10 12

Stefano Spezia, First 140 antidiagonals of the array, flattened

Diagonals: A008810, A007984, A000212, A128422.
Rows and columns: A002264, A004523, A000027, A004772, A047212, et al.
Cf. A143974, A143976, A143979.

T[m_,n_]:=Ceiling[m n/3];Flatten[Table[T[m-n+1,n],{m,13},{n,m}]] (* Stefano Spezia, Oct 27 2022 *)

R(m,n) = ceiling(m*n/3). [Corrected by Stefano Spezia, Oct 27 2022]

A023133 Signature sequence of Pi (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 2, 6, 3, 7, 4, 1, 8, 5, 2, 9, 6, 3, 10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3, 13, 10, 7, 4, 1, 14, 11, 8, 5, 2, 15, 12, 9, 6, 3, 16, 13, 10, 7, 4, 1, 17, 14, 11, 8, 5, 2, 18, 15, 12, 9, 6, 3, 19, 16, 13, 10, 7, 4, 1, 20, 17, 14, 11, 8, 5, 2, 21, 18, 15, 12, 9, 6, 3

Offset: 1

Clark Kimberling

nonn

C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
C. Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A001840, A008810.

lista(nn) = {v = vector(nn^2, k, kij = k+nn-1; i = 1+(kij % nn); j = kij\nn; i+j*Pi); vs = vecsort(v, , 1); for (k=1, #vs, print1(curi = 1+((vs[k]+nn-1) % nn), ", "); if (curi == nn, break));} \\ Michel Marcus, Apr 10 2015

A280984 Minimum number of dominoes on an n X n chessboard needed to prevent placement of another domino.

Original entry on oeis.org

0, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 122, 134, 147, 163, 178, 192, 210, 227, 243, 263, 282, 300, 322, 343, 363

Offset: 1

Rick L. Shepherd, Jan 11 2017, Aug 06 2017

Each domino must cover exactly two adjacent squares of a row or column. Sequence inspired by question for 8 X 8 case in "Minimum Guard Problem" link.
A.k.a. lower matching number of the n X n grid graph. - Eric W. Weisstein, Dec 16 2024
a(n) = 0 for n = 1, a(n) = ceiling(n^2/3) + 1 for n = 19, 22, 23, 25, 26, 28, 29, 31, 32, and a(n) = ceiling(n^2/3) for other n <= 32. - Eric W. Weisstein, Dec 16 2024

Andrejs Cibulis and Walter Trump, Domino Exclusion Problem, Baltic J. Modern Computing, Vol. 8 (2020), No. 4, 496-519.
A. Gyárfás, J. Lehel, and Zs. Tuza, Clumsy packing of dominoes, Discrete Mathematics, Volume 71, Issue 1 (1988), 33-46.
Peter Kagey, Minimum number of dominoes on an n X n chessboard to prevent placement of another domino.
Mathematics Stack Exchange user "Manin", Minimum Guard Problem.
Walter Trump, Minimum Domino Packing
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Lower Matching Number.

Cf. A008810 (maximum number of L-shaped trominoes with the same orientation in an n X n square, a.k.a. ceil(n^2/3)).
Cf. A378763 (lower matching number of the n X n torus grid graph).
Cf. A379177 (lower matching number of the n X n X n grid graph).

Proved: a(n) >= A008810(n) for n>1; when n = 0 (mod 3), a(n) = A008810(n). - Andrey Zabolotskiy, Oct 22 2017

a(n) > n^2/3 + n/111 for large n not congruent to 0 (mod 3) [from Gyárfás, Lehel, Tuza]. - Peter Kagey, May 22 2019

a(10)-a(14) from Lars Blomberg, Aug 08 2017
a(15) from Andrey Zabolotskiy, Oct 20 2017
a(16)-a(17) from Rob Pratt (see the link to Peter Kagey's question) and a(18) added by Andrey Zabolotskiy, Feb 13 2020
a(19)-a(33) from Walter Trump, Jun 14 2020

A225215 Floor of the Euclidean distance of a point on the (1, 1, 1; 1, 1, 1) 3D walk.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47, 47, 48, 49, 49, 50, 50

Offset: 1

Jon Perry, May 02 2013

Consider a standard 3-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 1 along the positive y-axis, 1 along the positive z-axis, 1 along the positive x-axis, and so on. After 3, 6, 9, 12, 15 etc steps we have returned to the space diagonal (with equal x, y and z coordinates).

This sequence gives the floor of the Euclidean distance to the origin after n steps.

Cf. A213172, A224985.

p = new Array(0, 0, 0);
for (a = 1; a < 100; a++) {
p[a%3] += 1;
document.write(Math.floor(Math.sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2])) + ", ");
}

a(n) ~ n/sqrt(3). - Charles R Greathouse IV, May 02 2013

a(n) = floor(sqrt(A008810(n))), where A008810(n) is the squared Euclidean distance after n steps. - R. J. Mathar, May 02 2013

cp's OEIS Frontend

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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A200082 T(n,k)=Number of 0..k arrays x(0..n) of n+1 elements with zero n-1st differences.

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

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

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Magma