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URL: https://oeis.org/A115362

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A115362
Row sums of ((1,x) + (x,x^2))^(-1)*((1,x)-(x,x^2))^(-1) (using Riordan array notation).
12
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
Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, Vol. 45, No. 10 (December 1938), pages 694-696. (The A indices in P at equations 1' and 2' for p=4.)
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.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 4/3. - Amiram Eldar, Feb 19 2026
MATHEMATICA
a[ n_] := If[ n < 0, 0, 1 + IntegerExponent[n + 1, 4]]; (* Michael Somos, Jul 19 2017 *)
PROG
(SageMath) [(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, quadrisection of A235127.
Sequence in context: A161102 A276329 A161101 * A365632 A340853 A277872
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Jan 21 2006
STATUS
approved