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A100131
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a(n) = Sum_{k=0..floor(n/4)} binomial(n-2k, 2k)*2^(n-4k).
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6
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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
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OFFSET
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0,2
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COMMENTS
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Binomial transform of 1,1,1,1,2,2,4,4,8,8,... (g.f.: (1-x)(1+x)^2/(1-2x^2)).
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LINKS
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FORMULA
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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
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MAPLE
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with(combinat):seq((n+fibonacci(n, 2))/2, n=1..30); # Zerinvary Lajos, Jun 02 2008
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MATHEMATICA
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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 *)
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PROG
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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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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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