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%I #18 May 01 2014 02:49:12
%S 0,1,3,12,21,77,168,609,987,3572,7755,28059,47376,171409,372099,
%T 1346268,2178309,7881197,17108664,61899729,104512485,378129724,
%U 820851717,2969869413,4809706272,17401680769,37775923491,136674575148
%N Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2*i).
%C 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)
%D Proof in preparation, to be published (see A048757).
%H Reinhard Zumkeller, <a href="/A051656/b051656.txt">Table of n, a(n) for n = 0..1000</a>
%F 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.
%t Table[Sum[Mod[Binomial[n,i],2]Fibonacci[2i],{i,0,n}],{n,0,30}] (* _Harvey P. Dale_, Oct 30 2011 *)
%o (Haskell)
%o a051656 = sum . zipWith (*) a001906_list . a047999_row
%o -- _Reinhard Zumkeller_, Feb 27 2011
%o (PARI) a(n)=sum(i=0,n,if(!bitand(i,n-i),fibonacci(2*i))) \\ _Charles R Greathouse IV_, Jan 04 2013
%Y Cf. A048757, A047999, A035513, A038183, A051256. First row of A050609, First column of A050610.
%Y 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).
%Y Cf. A001906.
%K nonn,nice
%O 0,3
%A _Antti Karttunen_, Nov 30 1999
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