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A246028 a(n) = Product_{i in row n of A245562} Fibonacci(i+1). 3
1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 4, 3, 3, 5, 8, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 5, 2, 2, 2, 4, 2, 2, 4, 6, 3, 3, 3, 6, 5, 5, 8, 13, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 4, 3, 3, 5, 8, 2, 2, 2, 4, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This is the Run Length Transform of S(n) = Fibonacci(n+1).

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(n-k,2k)*binomial(n,k)} mod 2). - Chai Wah Wu, Oct 19 2016

LINKS

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

N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb 05 2015: Part 1, Part 2

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

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

Index entries for sequences related to cellular automata

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+1), 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); if(k>1, s*=fibonacci(k+1)); n>>=k); s \\ Charles R Greathouse IV, Oct 21 2016

(PARI) a(n)=sum(k=0, n, !bitand(n-3*k, 2*k) && !bitand(n-k, k)) \\ Charles R Greathouse IV, Oct 21 2016

CROSSREFS

Cf. A245562, A000045, A001045, A071053, A245565, A245564.

Sequence in context: A272604 A284580 A227349 * A232186 A325757 A161161

Adjacent sequences:  A246025 A246026 A246027 * A246029 A246030 A246031

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Aug 15 2014; revised Sep 05 2014

STATUS

approved

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Last modified November 14 01:24 EST 2019. Contains 329108 sequences. (Running on oeis4.)