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A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2). 3
1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 6, 5, 8, 2, 4, 4, 6, 4, 8, 6, 10, 3, 6, 6, 9, 5, 10, 8, 13, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 3, 6, 6, 9, 6, 12, 9, 15, 5, 10, 10, 15, 8, 16, 13, 21, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 4, 8, 8, 12, 8, 16, 12, 20, 6, 12, 12, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the Run Length Transform of S(n) = Fibonacci(n+2).

The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

a(n) = Sum_{k=0..n} ({binomial(3k,k)*binomial(n,k)} mod 2). - Chai Wah Wu, Oct 19 2016

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..8191

Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

MAPLE

with(combinat); ans:=[];

for n from 0 to 100 do lis:=[]; t1:=convert(n, base, 2); L1:=nops(t1); out1:=1; c:=0;

for i from 1 to L1 do

   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;

   elif out1 = 0 and t1[i] = 1 then c:=c+1;

   elif out1 = 1 and t1[i] = 0 then c:=c;

   elif out1 = 0 and t1[i] = 0 then lis:=[c, op(lis)]; out1:=1; c:=0;

   fi;

   if i = L1 and c>0 then lis:=[c, op(lis)]; fi;

                   od:

a:=mul(fibonacci(i+2), i in lis);

ans:=[op(ans), a];

od:

ans;

PROG

(PARI) a(n)=my(s=1, k); while(n, n>>=valuation(n, 2); k=valuation(n+1, 2); s*=fibonacci(k+2); n>>=k); s \\ Charles R Greathouse IV, Oct 21 2016

CROSSREFS

Cf. A245562, A000045, A001045, A071053, A245565, A246028.

Sequence in context: A286622 A286589 A246029 * A214126 A205378 A323889

Adjacent sequences:  A245561 A245562 A245563 * A245565 A245566 A245567

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Aug 10 2014; revised Sep 05 2014

STATUS

approved

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Last modified November 13 23:48 EST 2019. Contains 329106 sequences. (Running on oeis4.)