A100212 Expansion of 4*x^4*(2 + x)/(1 - 2*x + 2*x^2 - 4*x^4 + 8*x^5 - 8*x^6).

0, 0, 0, 0, 8, 20, 24, 8, 0, 0, 0, 0, 128, 320, 384, 128, 0, 0, 0, 0, 2048, 5120, 6144, 2048, 0, 0, 0, 0, 32768, 81920, 98304, 32768, 0, 0, 0, 0, 524288, 1310720, 1572864, 524288, 0, 0, 0, 0, 8388608, 20971520, 25165824, 8388608, 0, 0, 0, 0, 134217728, 335544320

Offset: 0

Author

Creighton Dement, Nov 08 2004

Keywords

• nonn
• easy

Comments

a(n) = 0 iff n == {0, 1, 2 or 3} (mod 8). - Robert G. Wilson v, Nov 12 2004

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,0,4,-8,8).

Crossrefs

Cf. A038503, A009116, A100213.

Programs

Magma

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( 4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6) )); // G. C. Greubel, Apr 01 2024

Mathematica

CoefficientList[ Series[4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6), {x, 0, 55}], x] (* Robert G. Wilson v, Nov 12 2004 *)
LinearRecurrence[{2,-2,0,4,-8,8},{0,0,0,0,8,20},60] (* Harvey P. Dale, Oct 10 2012 *)

PARI

Vec(4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

SageMath

def A100212_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6) ).list()
A100212_list(60) # G. C. Greubel, Apr 01 2024

Formula

a(8n+4) = a(8n+7) = 2^(4n+3), a(8n+5) = (5/2)*2^(4n+3), a(8n+6) = 3*2^(4n+3), a(8n+8) = 0, a(8n+9) = 0, a(8n+10) = 0, a(8n+11) = 0.

(a(n)) = negseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)

a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=8, a(5)=20, a(n) = 2*a(n-1) - 2*a(n-2) + 4*a(n-4) - 8*a(n-5) + 8*a(n-6). - Harvey P. Dale, Oct 10 2012

Extensions

More terms from Robert G. Wilson v, Nov 12 2004