This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A245564 #54 Jul 09 2025 15:39:20
%S A245564 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,
%T A245564 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,
%U A245564 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
%N A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).
%C A245564 This is the Run Length Transform of S(n) = Fibonacci(n+2).
%C A245564 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).
%C A245564 Also the number of sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference. The maximal case is A384883. For prime instead of binary indices we have A166469. - _Gus Wiseman_, Jul 05 2025
%H A245564 Alois P. Heinz, <a href="/A245564/b245564.txt">Table of n, a(n) for n = 0..8191</a>
%H A245564 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.
%F A245564 a(n) = Sum_{k=0..n} ({binomial(3k,k)*binomial(n,k)} mod 2). - _Chai Wah Wu_, Oct 19 2016
%e A245564 From _Gus Wiseman_, Jul 05 2025: (Start)
%e A245564 The binary indices of 11 are {1,2,4}, with sparse subsets {{},{1},{2},{4},{1,4},{2,4}}, so a(11) = 6.
%e A245564 The maximal runs of binary indices of 11 are ((1,2),(4)), with lengths (2,1), so a(11) = F(2+2)*F(1+2) = 6.
%e A245564 The a(0) = 1 through a(12) = 3 sparse subsets are:
%e A245564 0 1 2 3 4 5 6 7 8 9 10 11 12
%e A245564 ------------------------------------------------------------------
%e A245564 {} {} {} {} {} {} {} {} {} {} {} {} {}
%e A245564 {1} {2} {1} {3} {1} {2} {1} {4} {1} {2} {1} {3}
%e A245564 {2} {3} {3} {2} {4} {4} {2} {4}
%e A245564 {1,3} {3} {1,4} {2,4} {4}
%e A245564 {1,3} {1,4}
%e A245564 {2,4}
%e A245564 The greatest number whose set of binary indices is a member of column n above is A374356(n).
%e A245564 (End)
%p A245564 with(combinat); ans:=[];
%p A245564 for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
%p A245564 for i from 1 to L1 do
%p A245564 if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
%p A245564 elif out1 = 0 and t1[i] = 1 then c:=c+1;
%p A245564 elif out1 = 1 and t1[i] = 0 then c:=c;
%p A245564 elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
%p A245564 fi;
%p A245564 if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
%p A245564 od:
%p A245564 a:=mul(fibonacci(i+2), i in lis);
%p A245564 ans:=[op(ans),a];
%p A245564 od:
%p A245564 ans;
%t A245564 a[n_] := Sum[Mod[Binomial[3k, k] Binomial[n, k], 2], {k, 0, n}];
%t A245564 a /@ Range[0, 100] (* _Jean-François Alcover_, Feb 29 2020, after _Chai Wah Wu_ *)
%t A245564 spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
%t A245564 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A245564 Table[Length[spars[bpe[n]]],{n,0,30}] (* _Gus Wiseman_, Jul 05 2025 *)
%o A245564 (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
%o A245564 (Python)
%o A245564 # use RLT function from A278159
%o A245564 from sympy import fibonacci
%o A245564 def A245564(n): return RLT(n,lambda m: fibonacci(m+2)) # _Chai Wah Wu_, Feb 04 2022
%Y A245564 Cf. A245562, A000045, A001045, A071053, A245565, A246028.
%Y A245564 A034839 counts subsets by number of maximal runs, strict partitions A116674.
%Y A245564 A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
%Y A245564 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A245564 Cf. A000071, A001629, A010049, A053538, A166469, A202023, A202064, A208342, A374356, A384883.
%K A245564 nonn,easy
%O A245564 0,2
%A A245564 _N. J. A. Sloane_, Aug 10 2014; revised Sep 05 2014