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URL: https://oeis.org/A115364

⇱ A115364 - OEIS

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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).
7
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, 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, 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
OFFSET
1,2
COMMENTS
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.
The Stephan formula says this is the Dirichlet convolution of A000012 with A104117. - R. J. Mathar, Feb 07 2011
LINKS
FORMULA
a(n) = binomial(A007814(n)+2, 2) = binomial(A001511(n)+1, 2).
Dirichlet g.f.: zeta(s)*(2^s/(2^s-1))^2. - Ralf Stephan, Jun 17 2007
Multiplicative with a(2^k) = A000217(k+1), a(p^k) = 1 for odd primes p. - Antti Karttunen, Nov 02 2018
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
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Oct 22 2022
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
MATHEMATICA
Array[PolygonalNumber[IntegerExponent[#, 2] + 1] &, 93] (* Michael De Vlieger, Nov 02 2018 *)
PROG
(PARI) A115364(n) = binomial(valuation(n, 2)+2, 2); \\ Antti Karttunen, Nov 02 2018
(Python)
def A115364(n): return (m:=((~n & n-1).bit_length()+1))*(m+1)>>1 # Chai Wah Wu, Jul 02 2022
CROSSREFS
KEYWORD
easy,mult,nonn
AUTHOR
Paul Barry, Jan 21 2006
EXTENSIONS
Formula corrected and the name changed by Antti Karttunen, Nov 02 2018
STATUS
approved