A051656 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2*i).

0, 1, 3, 12, 21, 77, 168, 609, 987, 3572, 7755, 28059, 47376, 171409, 372099, 1346268, 2178309, 7881197, 17108664, 61899729, 104512485, 378129724, 820851717, 2969869413, 4809706272, 17401680769, 37775923491, 136674575148

Offset: 0

Author

Antti Karttunen, Nov 30 1999

Keywords

• nonn
• nice

Comments

Positions in the first column (A003622) of Wythoff array of the terms which have their Zeckendorf Expansion patterned as row[2n+1] in Pascal's Triangle computed modulo 2 (A047999)

References

• Proof in preparation, to be published (see A048757).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A048757, A047999, A035513, A038183, A051256. First row of A050609, First column of A050610.

a(n) = A019586[A048757[n]]. A048757[n] = SS(Athis_sequence[n])+1, where SSx means the second Fibonacci Successor of x (= x's Z.E. shifted left twice).

Cf. A001906.

Programs

Haskell

a051656 = sum . zipWith (*) a001906_list . a047999_row
-- Reinhard Zumkeller, Feb 27 2011

Mathematica

Table[Sum[Mod[Binomial[n,i],2]Fibonacci[2i],{i,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 30 2011 *)

PARI

a(n)=sum(i=0,n,if(!bitand(i,n-i),fibonacci(2*i))) \\ Charles R Greathouse IV, Jan 04 2013

Formula

a(n) = sum_{i=0..n} (C(2n, 2i) mod 2)*F(2*i) = FL(n)product_{i=0..inf} L(2^i)^bit(n, i) where L is n-th Lucas number (A000032) and FL is defined as in A048757: FL(n) = n-th fibonacci number if n even, n-th Lucas number if n odd.