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Search: id:A103311
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| A103311 |
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A transform of the Fibonacci numbers. |
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+0
7
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| 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; listen)
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OFFSET
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0,5
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COMMENT
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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.
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FORMULA
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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)}}.
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CROSSREFS
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Cf. A000045.
Sequence in context: A154127 A138371 A140053 this_sequence A019824 A019772 A046825
Adjacent sequences: A103308 A103309 A103310 this_sequence A103312 A103313 A103314
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KEYWORD
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easy,sign
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
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