A247918 -id:A247918 - cp's OEIS frontend

A247907 Expansion of (1 + x) / ((1 - x^4) * (1 - x - x^5)) in powers of x.

1, 2, 2, 2, 3, 5, 7, 9, 12, 16, 21, 28, 38, 51, 67, 88, 117, 156, 207, 274, 363, 481, 637, 844, 1119, 1483, 1964, 2601, 3446, 4566, 6049, 8013, 10615, 14062, 18628, 24677, 32691, 43307, 57369, 75997, 100675, 133367, 176674, 234043, 310041, 410717, 544084

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 12*x^8 + ...

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

Cf. A003520, A133872, A247918.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 +x)/((1-x^4)*(1-x-x^5))));  // G. C. Greubel, Aug 04 2018

CoefficientList[Series[(1 + x)/((1 - x^4) (1 - x - x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)

{a(n) = if( n<0, n=-8-n; polcoeff( -1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 + x - x^3)) + x * O(x^n), n), polcoeff( 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 - x^2 - x^3)) + x * O(x^n), n))};

G.f.: 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 - x^2 - x^3)).
a(n) = -A247918(-8-n) for all n in Z.
Convolution of A003520 and A133872.
0 = a(n) + a(n+4) - a(n+5) + mod(floor((n-1) / 2), 2) for all n in Z.
0 = a(n) - a(n+1) + a(n+2) - a(n+3) + a(n+4) - 2*a(n+5) + 2*a(n+6) - 2*a(n+7) + a(n+8) for all n in Z.

A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1, 0, 0, 2, -2, 1, 1, -2, 4, -3, 1, 3, -6, 7, -3, -2, 9, -13, 11, -1, -11, 22, -23, 12, 10, -33, 46, -35, 2, 43, -78, 81, -37, -41, 122, -159, 118, 4, -162, 281, -277, 114, 167, -443, 558, -391, -52, 610, -1001, 949, -338, -662

Offset: 0

Michael Somos, Apr 28 2015

			G.f. = 1 + x^5 + x^8 - x^9 + x^10 + 2*x^13 - 2*x^14 + x^15 + x^16 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,0,1,-1).

Cf. A104769, A233522, A247917, A247918.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 / ((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 02 2018

a[ n_] := SeriesCoefficient[ If[ n >= 0, 1 / (1 - x^5 - x^8 + x^9), -x^9 /(1 - x - x^4 + x^9)], {x, 0, Abs@n}];

{a(n) = if( n>=0, polcoeff( 1 / (1 - x^5 - x^8 + x^9) + x * O(x^n), n), polcoeff( -x^9 / (1 - x - x^4 + x^9) + x * O(x^-n), -n))};

G.f.: 1 / ((1 - x^4) * (1 + x^4 - x^5)) = (1 + x) / ((1 + x^3) * (1 - x^4) * (1 + x - x^3)).
a(n) = a(n-5) + a(n-8) - a(n-9) for all n in Z.
a(n) - a(n+2) - a(n+3) has period 12.
a(n) - a(n+12) = A104769(n+5) = -A247917(n+4) for all n in Z.
a(n) + a(n+1) = A247918(n) for all n in Z.
a(n) = -A233522(-9 - n) for all n in Z.

cp's OEIS Frontend

A247907 Expansion of (1 + x) / ((1 - x^4) * (1 - x - x^5)) in powers of x.

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