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A100334 An inverse Catalan transform of F(2n). 6
0, 1, 2, 2, 0, -5, -13, -21, -21, 0, 55, 144, 233, 233, 0, -610, -1597, -2584, -2584, 0, 6765, 17711, 28657, 28657, 0, -75025, -196418, -317811, -317811, 0, 832040, 2178309, 3524578, 3524578, 0, -9227465, -24157817, -39088169, -39088169, 0, 102334155, 267914296, 433494437, 433494437, 0, -1134903170 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..45.

Index entries for linear recurrences with constant coefficients, signature (3,-4,2,-1).

FORMULA

G.f.: x(1-x)/(1-3x+4x^2-2x^3+x^4); a(n)=(phi)^n*sqrt(2/5+2sqrt(5)/25)sin(pi*(n+1)/5) -(1/phi)^n*sqrt(2/5-2sqrt(5)/25)sin(2pi*(n+1)/5), where phi=(1+sqrt(5))/2; a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*sum{j=0..n-k, C(n-k, j)F(j)}}; a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*F(2n-2k)}.

a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4). - Paul Curtz, May 13 2008

a(n)=Sum_{k, 0<=k<=n} A109466(n,k)*A001906(k). [From Philippe Deléham, Oct 30 2008]

MATHEMATICA

Table[FullSimplify[GoldenRatio^n*Sqrt[2/5 + 2*Sqrt[5]/25]*Sin[Pi*n/5 + Pi/5] - (1/GoldenRatio)^n*Sqrt[2/5 - 2*Sqrt[5]/25]*Sin[2*Pi*n/5 + 2*Pi/5]], {n, 0, 41}] (* Arkadiusz Wesolowski, Oct 26 2012 *)

CROSSREFS

Sequence in context: A222128 A088972 A168505 * A277295 A254749 A129936

Adjacent sequences:  A100331 A100332 A100333 * A100335 A100336 A100337

KEYWORD

easy,sign

AUTHOR

Paul Barry, Nov 17 2004

STATUS

approved

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Last modified February 23 23:56 EST 2018. Contains 299595 sequences. (Running on oeis4.)