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%I #49 Jun 06 2020 21:40:47
%S 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,
%T 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,
%U 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
%N Row sums of ((1,x) + (x,x^2))^(-1)*((1,x)-(x,x^2))^(-1) (using Riordan array notation).
%C Row sums of the matrix product A115358*A115361.
%C Generalized Ruler Function for k=4. - _Frank Ruskey_ and Chris Deugau (deugaucj(AT)uvic.ca)
%C a(n) is 1 + the 4-adic valuation of n+1. - _Joerg Arndt_, Oct 07 2015
%H Antti Karttunen, <a href="/A115362/b115362.txt">Table of n, a(n) for n = 0..16384</a>
%H Joseph Rosenbaum, <a href="https://doi.org/10.2307/2302451">Elementary Problem E319</a>, 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.)
%F G.f.: A(x) = Sum_k>=0} x^(4^k)/(1-x^(4^k)). - _Frank Ruskey_ and Chris Deugau (deugaucj(AT)uvic.ca)
%F Dirichlet g.f. (conjectured): zeta(s)/(1-2^(-2s)). - _Ralf Stephan_, Mar 27 2015
%F a(n) = (1/3)*(4 + A053737(n) - A053737(n+1)). - _Tom Edgar_, Oct 06 2015
%F 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
%F a(n) = 1 + A235127(1+n). - _Antti Karttunen_, Nov 18 2017, after _Joerg Arndt_'s Oct 07 2015 comment.
%t a[ n_] := If[ n < 0, 0, 1 + IntegerExponent[n + 1, 4]]; (* _Michael Somos_, Jul 19 2017 *)
%o (Sage) [(1/3)*(4-sum(n.digits(4))+sum((n-1).digits(4))) for n in [1..96]] # _Tom Edgar_, Oct 06 2015
%o (PARI) a(n) = 1 + valuation(n+1,4); \\ _Joerg Arndt_, Oct 07 2015
%o (PARI) {a(n) = if( n<0, 0, n%4==3, 1 + a((n - 3) / 4), 1)}; /* _Michael Somos_, Jul 13 2017 */
%Y Cf. A053737, A115358, A115361, A235127.
%K nonn,easy
%O 0,4
%A _Paul Barry_, Jan 21 2006
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