A008936 - OEIS

%I #27 Mar 03 2024 04:01:23

%S 1,2,4,8,15,28,52,96,176,320,576,1024,1792,3072,5120,8192,12288,16384,

%T 16384,0,-65536,-262144,-786432,-2097152,-5242880,-12582912,-29360128,

%U -67108864,-150994944,-335544320,-738197504,-1610612736,-3489660928,-7516192768,-16106127360,-34359738368

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

%H G. C. Greubel, <a href="/A008936/b008936.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).

%F From _Michael Somos_, Aug 19 2014: (Start)

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

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

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

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

%F 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)

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

%e 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 + ...

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

%t a[ n_]:= 2^n - 2^(n-4) Max[0, n-3]; (* _Michael Somos_, Aug 19 2014 *)

%t Table[If[n < 4, 2^n, 2^(n-4)*(19 - n)], {n,0,40}] (* _G. C. Greubel_, Sep 13 2019 *)

%o (PARI) {a(n) = 2^n - 2^(n-4) * max(n-3, 0)}; /* _Michael Somos_, Jan 12 2000 */

%o (PARI) Vec((1-2*x-x^4)/(1-2*x)^2 +O(x^40)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) [n lt 4 select 2^n else 2^(n-4)*(19-n): n in [0..40]]; // _G. C. Greubel_, Sep 13 2019

%o (Sage) [1,2,4,8]+[2^(n-4)*(19 - n) for n in (4..40)] # _G. C. Greubel_, Sep 13 2019

%o (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

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_, Alejandro Teruel (teruel(AT)usb.ve)

%E Better description from _Michael Somos_, Jan 12 2000

%E More terms added by _G. C. Greubel_, Sep 13 2019