A115362 Row sums of ((1,x) + (x,x^2))^(-1)*((1,x)-(x,x^2))^(-1) (using Riordan array notation).

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3

Offset: 0

Author

Paul Barry, Jan 21 2006

Keywords

• nonn
• easy

Comments

Row sums of the matrix product A115358*A115361.

Generalized Ruler Function for k=4. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

a(n) is 1 + the 4-adic valuation of n+1. - Joerg Arndt, Oct 07 2015

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16384
• Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=4.)

Crossrefs

Cf. A053737, A115358, A115361, quadrisection of A235127.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, 1 + IntegerExponent[n + 1, 4]]; (* Michael Somos, Jul 19 2017 *)

PARI

a(n) = 1 + valuation(n+1,4); \\ Joerg Arndt, Oct 07 2015

PARI

{a(n) = if( n<0, 0, n%4==3, 1 + a((n - 3) / 4), 1)}; /* Michael Somos, Jul 13 2017 */

Sage

[(1/3)*(4-sum(n.digits(4))+sum((n-1).digits(4))) for n in [1..96]] # Tom Edgar, Oct 06 2015

Formula

G.f.: Sum_{k>=0} x^(4^k)/(1-x^(4^k)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

Dirichlet g.f. (conjectured): zeta(s)/(1-2^(-2s)). - Ralf Stephan, Mar 27 2015

a(n) = (1/3)*(4 + A053737(n) - A053737(n+1)). - Tom Edgar, Oct 06 2015

a(4*n) = a(4*n+1) = a(4*n+2) = 1, a(4*n+3) = 1+a(n), if n >= 0. - Michael Somos, Jul 13 2017

a(n) = 1 + A235127(1+n). - Antti Karttunen, Nov 18 2017, after Joerg Arndt's Oct 07 2015 comment.