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 A103311 A transform of the Fibonacci numbers. 6
 0, 1, 1, 0, -2, -5, -8, -8, 0, 21, 55, 89, 89, 0, -233, -610, -987, -987, 0, 2584, 6765, 10946, 10946, 0, -28657, -75025, -121393, -121393, 0, 317811, 832040, 1346269, 1346269, 0, -3524578, -9227465, -14930352, -14930352, 0, 39088169, 102334155, 165580141, 165580141, 0, -433494437, -1134903170 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Apply the Chebyshev transform (1/(1+x^2),x/(1+x^2)) followed by the binomial involution (1/(1-x),-x/(1-x)) (expressed as Riordan arrays) to -Fib(n). Conjecture : all elements in absolute value are Fibonacci numbers. LINKS FORMULA G.f.: x(1-x)^2/(1-3x+4x^2-2x^3+x^4); a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4); a(n)=(sqrt(5)/2-1/2)^n(sqrt(2sqrt(5)/25+1/5)sin(2*pi*n/5)-sqrt(5)cos(2*pi*n/5)/5)+ (sqrt(5)/2+1/2)^n(sqrt(5)cos(pi*n/5)/5+sqrt(1/5-2sqrt(5)/25)sin(pi*n/5)); a(n)=-sum{j=0..n, (-1)^j*C(n, j)*sum{k=0..floor(j/2), (-1)^k*C(n-k, k)Fib(j-2k)}}. CROSSREFS Cf. A000045. Sequence in context: A154127 A138371 A140053 * A019824 A019772 A046825 Adjacent sequences:  A103308 A103309 A103310 * A103312 A103313 A103314 KEYWORD easy,sign AUTHOR Paul Barry, Jan 30 2005 STATUS approved

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