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A054204
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Integers expressible as sums of distinct even-subscripted Fibonacci numbers.
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5
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1, 3, 4, 8, 9, 11, 12, 21, 22, 24, 25, 29, 30, 32, 33, 55, 56, 58, 59, 63, 64, 66, 67, 76, 77, 79, 80, 84, 85, 87, 88, 144, 145, 147, 148, 152, 153, 155, 156, 165, 166, 168, 169, 173, 174, 176, 177, 199, 200, 202, 203, 207, 208, 210, 211, 220, 221, 223, 224, 228, 229
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OFFSET
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1,2
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COMMENTS
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Number of partitions of a(n) into sums of distinct Fibonacci numbers is (n+1)st term of Stern's Diatomic series A002487. This sequence has A046815 as a subsequence.
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LINKS
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FORMULA
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Subscripts in Zeckendorf representation of a(n) are 2(e+1) where e is exponent used to write n as sum of powers of 2.
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EXAMPLE
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a(9)=22 since 9=2^3+2^0 and 22=F(2(3+1)) + F(2(0+1)) = F(8) + F(2).
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MATHEMATICA
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fibEvenCount[n_] := Plus @@ (Reverse @ IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]; evenIndexed = fibEvenCount /@ Select[Range[10000], BitAnd[#, 2 #] == 0 &]; Position[evenIndexed, _?(# == 0 &)] // Flatten (* Amiram Eldar, Jan 20 2020*)
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PROG
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(PARI) my(m=Mod('x, 'x^2-3*'x+1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x, m)), 'x, 3); \\ Kevin Ryde, Nov 25 2020
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CROSSREFS
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Distinct additive closure of A001906.
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KEYWORD
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nonn
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AUTHOR
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Marjorie Bicknell-Johnson (marjohnson(AT)earthlink.net), Apr 30 2000
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STATUS
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approved
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