A029000 -id:A029000 - cp's OEIS frontend

A069183 Expansion of 1/((1-x)(1-x^2)^2(1-x^3)(1-x^6)).

1, 1, 3, 4, 7, 9, 15, 18, 27, 33, 45, 54, 72, 84, 108, 126, 156, 180, 220, 250, 300, 340, 400, 450, 525, 585, 675, 750, 855, 945, 1071, 1176, 1323, 1449, 1617, 1764, 1960, 2128, 2352, 2548, 2800, 3024, 3312, 3564, 3888, 4176, 4536, 4860, 5265, 5625, 6075

Offset: 0

Rick L. Shepherd, Apr 10 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,3,0,-3,1,2,1,-2,-1,1).

Cf. A029000.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^6)) )); // G. C. Greubel, May 26 2024

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^6)(1-x^2)^2), {x, 0, 100}], x] (* Jinyuan Wang, Mar 15 2020 *)

a(n) = polcoeff(1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^6)+x*O(x^n)), n);

def A069183_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^6)) ).list()
A069183_list(60) # G. C. Greubel, May 26 2024

G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^6)).

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 3*a(n-6) - 3*a(n-8) + a(n-9) + 2*a(n-10) + a(n-11) - 2*a(n-12) - a(n-13) + a(n-14). - Wesley Ivan Hurt, May 24 2024

A258133 Expansion of tri-digit zeros interlaced with an arithmetic progression of positive and negative numbers.

Original entry on oeis.org

1, 0, 0, 0, 2, -2, 2, 0, 0, 0, 3, -3, 3, 0, 0, 0, 4, -4, 4, 0, 0, 0, 5, -5, 5, 0, 0, 0, 6, -6, 6, 0, 0, 0, 7, -7, 7, 0, 0, 0, 8, -8, 8, 0, 0, 0, 9, -9, 9, 0, 0, 0, 10, -10, 10, 0, 0, 0, 11, -11, 11, 0, 0, 0, 12, -12, 12, 0, 0, 0, 13, -13, 13, 0, 0, 0, 14

Offset: 0

Avi Friedlich, May 21 2015

This sequence is observed as the second difference in the expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^6)) (A029000), a sequence noted for its interlaced and structural coordination numbers.

			G.f. = 1 + 2*x^4 - 2*x^5 + 2*x^6 + 3*x^10 - 3*x^11 + 3*x^12 + 4*x^16 + ...

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

a[ n_] := With[ {m=n-1}, If[ OddQ[ Quotient[ m, 3]], Quotient[ m+9, 6] (-1)^Mod[m, 3], 0]]; (* Michael Somos, Jun 07 2015 *)

Vec((x^9-x^7-x^6+x^4-x^3+x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, May 24 2015

{a(n) = n--; if(n\3%2, (n+9)\6 * (-1)^(n%3), 0)}; /* Michael Somos, Jun 07 2015 */

a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-6) + a(n-7) - a(n-9) - a(n-10). a(6*k-2) = -a(6*k-1) = a(6*k) = k+1 for k >= 1.
G.f.: (x^9-x^7-x^6+x^4-x^3+x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2). - Colin Barker, May 24 2015
a(n) = -a(-14-n) for all n in Z. - Michael Somos, Jun 07 2015

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A069183 Expansion of 1/((1-x)(1-x^2)^2(1-x^3)(1-x^6)).

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A258133 Expansion of tri-digit zeros interlaced with an arithmetic progression of positive and negative numbers.

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