A008936 Expansion of (1 - 2*x -x^4)/(1 - 2*x)^2 in powers of x.

1, 2, 4, 8, 15, 28, 52, 96, 176, 320, 576, 1024, 1792, 3072, 5120, 8192, 12288, 16384, 16384, 0, -65536, -262144, -786432, -2097152, -5242880, -12582912, -29360128, -67108864, -150994944, -335544320, -738197504, -1610612736, -3489660928, -7516192768, -16106127360, -34359738368

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Formula

From Michael Somos, Aug 19 2014: (Start)

a(n) = 2^n for all n<4.

a(n) = 2^n - (n-3) * 2^(n-4) for all n>=4.

a(n) = 4*(a(n-1) - a(n-2)) for all n in Z except n=4.

a(n) = 2*a(n-1) - 2^(n-4).

0 = a(n)*(-8*a(n+1) + 8*a(n+2) - 2*a(n+3)) + a(n+1)*(+4*a(n+1) - 4*a(n+2) + a(n+3)) for all n in Z. (End)

E.g.f.: ( -3 -4*x -2*x^2 + (19 - 2*x)*exp(2*x) )/16. - G. C. Greubel, Sep 13 2019

Example

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 28*x^5 + 52*x^6 + 96*x^7 + 176*x^8 + ...

Maple

A008936 := proc(n) option remember; if n <= 3 then 2^n else 2*A008936(n-1)-2^(n-4); fi; end;

Mathematica

a[ n_]:= 2^n - 2^(n-4) Max[0, n-3]; (* Michael Somos, Aug 19 2014 *)
Table[If[n < 4, 2^n, 2^(n-4)*(19 - n)], {n, 0, 40}] (* G. C. Greubel, Sep 13 2019 *)

Prog

(PARI) {a(n) = 2^n - 2^(n-4) * max(n-3, 0)}; /* Michael Somos, Jan 12 2000 */
(PARI) Vec((1-2*x-x^4)/(1-2*x)^2 +O(x^40)) \\ Charles R Greathouse IV, Sep 26 2012
(Magma) [n lt 4 select 2^n else 2^(n-4)*(19-n): n in [0..40]]; // G. C. Greubel, Sep 13 2019
(Sage) [1, 2, 4, 8]+[2^(n-4)*(19 - n) for n in (4..40)] # G. C. Greubel, Sep 13 2019
(GAP) a:=[1, 2];; for n in [3..40] do a[n]:=4*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Sep 13 2019

Crossrefs

Sequence in context: A114833 A065617 A062065 * A320452 A073769 A008937

Adjacent sequences: A008933 A008934 A008935 * A008937 A008938 A008939

Keyword

sign,easy

Author

N. J. A. Sloane, Alejandro Teruel (teruel(AT)usb.ve)

Extensions

Better description from Michael Somos, Jan 12 2000

More terms added by G. C. Greubel, Sep 13 2019

Status

approved