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

%I A246028 #32 Sep 28 2021 01:21:44
%S A246028 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,
%T A246028 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,
%U A246028 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
%N A246028 a(n) = Product_{i in row n of A245562} Fibonacci(i+1).
%C A246028 This is the Run Length Transform of S(n) = Fibonacci(n+1).
%C A246028 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).
%H A246028 Alois P. Heinz, <a href="/A246028/b246028.txt">Table of n, a(n) for n = 0..8191</a>
%H A246028 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: <a href="https://vimeo.com/119073818">Part 1</a>, <a href="https://vimeo.com/119073819">Part 2</a>
%H A246028 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168, 2015
%H A246028 Chai Wah Wu, <a href="https://arxiv.org/abs/1610.06166">Sums of products of binomial coefficients mod 2 and run length transforms of sequences</a>, arXiv:1610.06166 [math.CO], 2016.
%H A246028 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A246028 a(n) = Sum_{k=0..n} ((binomial(n-k,2k)*binomial(n,k)) mod 2). - _Chai Wah Wu_, Oct 19 2016
%p A246028 with(combinat); ans:=[];
%p A246028 for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
%p A246028 for i from 1 to L1 do
%p A246028 if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
%p A246028 elif out1 = 0 and t1[i] = 1 then c:=c+1;
%p A246028 elif out1 = 1 and t1[i] = 0 then c:=c;
%p A246028 elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
%p A246028 fi;
%p A246028 if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
%p A246028 od:
%p A246028 a:=mul(fibonacci(i+1), i in lis);
%p A246028 ans:=[op(ans),a];
%p A246028 od:
%p A246028 ans;
%t A246028 a[n_] := Sum[Mod[Binomial[n-k, 2k] Binomial[n, k], 2], {k, 0, n}];
%t A246028 a /@ Range[0, 100] (* _Jean-François Alcover_, Feb 28 2020, after _Chai Wah Wu_ *)
%o A246028 (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
%o A246028 (PARI) a(n)=sum(k=0,n, !bitand(n-3*k,2*k) && !bitand(n-k,k)) \\ _Charles R Greathouse IV_, Oct 21 2016
%o A246028 (Python)
%o A246028 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
%Y A246028 Cf. A245562, A000045, A001045, A071053, A245565, A245564.
%K A246028 nonn,easy
%O A246028 0,4
%A A246028 _N. J. A. Sloane_, Aug 15 2014; revised Sep 05 2014