A100131 a(n) = Sum_{k=0..floor(n/4)} binomial(n-2k, 2k)*2^(n-4k).
1, 2, 4, 8, 17, 38, 88, 208, 497, 1194, 2876, 6936, 16737, 40398, 97520, 235424, 568353, 1372114, 3312564, 7997224, 19306993, 46611190, 112529352, 271669872, 655869073, 1583407994, 3822685036, 9228778040, 22280241089, 53789260190, 129858761440, 313506783040
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).
Programs
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Magma
I:=[1, 2, 4, 8]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012
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Maple
with(combinat): seq((n+1+fibonacci(n+1, 2))/2, n=0..30); # Zerinvary Lajos, Jun 02 2008
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Mathematica
CoefficientList[Series[(1-2x)/((1-2x)^2-x^4),{x,0,40}],x] (* Harvey P. Dale, Mar 22 2011 *) LinearRecurrence[{4,-4,0,1},{1,2,4,8},40] (* Vincenzo Librandi, Jun 25 2012 *)
Formula
G.f.: (1-2x)/((1-2x)^2-x^4) = (1-2x)/((1-x)^2(1-2x-x^2));
a(n) = 4a(n-1) - 4a(n-2) + a(n-4);
a(n) = ((sqrt(2)+1)^(n+1) + (sqrt(2)-1)^(n+1)(-1)^n)/(4*sqrt(2)) + (n+1)/2;
a(n) = Sum_{k=0..n} (1-k)*A000129(n-k+1).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} binomial(k, j)*binomial(n-j, k)*((j+1) mod 2). - Paul Barry, May 31 2005
a(n) = (1/2)*(Pell(n+1) + n + 1), where Pell(n) = A000129(n). - Ralf Stephan, May 15 2007 [corrected by Jon E. Schoenfield, Feb 19 2019]
Comments