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

0,4

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.)

Formula

G.f.: A(x) = 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.

Mathematica

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

Prog

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

Crossrefs

Cf. A053737, A115358, A115361, A235127.

Sequence in context: A161102 A276329 A161101 * A365632 A340853 A277872

Adjacent sequences: A115359 A115360 A115361 * A115363 A115364 A115365

Keyword

nonn,easy

Author

Paul Barry, Jan 21 2006

Status

approved