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 A115364 #32 Jun 25 2024 11:42:25
%S A115364 1,3,1,6,1,3,1,10,1,3,1,6,1,3,1,15,1,3,1,6,1,3,1,10,1,3,1,6,1,3,1,21,
%T A115364 1,3,1,6,1,3,1,10,1,3,1,6,1,3,1,15,1,3,1,6,1,3,1,10,1,3,1,6,1,3,1,28,
%U A115364 1,3,1,6,1,3,1,10,1,3,1,6,1,3,1,15,1,3,1,6,1,3,1,10,1,3,1,6,1
%N A115364 a(n) = A000217(A001511(n)), where A001511 is one more than the 2-adic valuation of n, and A000217(n) is the n-th triangular number, binomial(n+1, 2).
%C A115364 Row sums of A115363. In general, the row sums of ((1,x) - m(x,x^2))^(-2) are obtained by following the ruler function A001511(n) by the solution of the recurrence a(1)=1, a(n) = n*m^(n-1) + a(n-1), n > 1.
%C A115364 The Stephan formula says this is the Dirichlet convolution of A000012 with A104117. - _R. J. Mathar_, Feb 07 2011
%H A115364 Antti Karttunen, <a href="/A115364/b115364.txt">Table of n, a(n) for n = 1..16383</a>
%F A115364 a(n) = binomial(A007814(n)+2, 2) = binomial(A001511(n)+1, 2).
%F A115364 Dirichlet g.f.: zeta(s)*(2^s/(2^s-1))^2. - _Ralf Stephan_, Jun 17 2007
%F A115364 Multiplicative with a(2^k) = A000217(k+1), a(p^k) = 1 for odd primes p. - _Antti Karttunen_, Nov 02 2018
%F A115364 O.g.f.: Sum_{k >= 1} k*x^(2^(k-1))/(1 - x^(2^(k-1))). More generally, if f(n) is an arithmetic function and g(n) := Sum_{k = 1..n} f(k), then Sum_{k >= 1} f(k)*x^(2^(k-1))/(1 - x^(2^(k-1))) = Sum_{n >= 1} g(A001511(n))*x^n. This is the case f(n) = n. - _Peter Bala_, Mar 26 2019
%F A115364 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - _Amiram Eldar_, Oct 22 2022
%F A115364 More precise asymptotics: Sum_{k=1..n} a(k) ~ 4*n - log(n)*(log(n) + 2*log(4*Pi))/(4*log(2)^2). - _Vaclav Kotesovec_, Jun 25 2024
%t A115364 Array[PolygonalNumber[IntegerExponent[#, 2] + 1] &, 93] (* _Michael De Vlieger_, Nov 02 2018 *)
%o A115364 (PARI) A115364(n) = binomial(valuation(n,2)+2,2); \\ _Antti Karttunen_, Nov 02 2018
%o A115364 (Python)
%o A115364 def A115364(n): return (m:=((~n & n-1).bit_length()+1))*(m+1)>>1 # _Chai Wah Wu_, Jul 02 2022
%Y A115364 Cf. A000217, A001511, A007814, A104117, A115363.
%Y A115364 Cf. also A094290.
%K A115364 easy,mult,nonn
%O A115364 1,2
%A A115364 _Paul Barry_, Jan 21 2006
%E A115364 Formula corrected and the name changed by _Antti Karttunen_, Nov 02 2018