A246028 a(n) = Product_{i in row n of A245562} Fibonacci(i+1).

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

Offset: 0

Author

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

Keywords

• nonn
• easy

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).

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

Crossrefs

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

Programs

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;

Mathematica

a[n_] := Sum[Mod[Binomial[n-k, 2k] Binomial[n, k], 2], {k, 0, n}];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 28 2020, after Chai Wah Wu *)

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

Python

def A246028(n): return sum(int(not (~(n-k) & 2*k) | (~n & k)) for k in range(n+1)) # Chai Wah Wu, Sep 27 2021

Formula

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