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A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

%I A317931 #14 Sep 23 2018 21:28:17
%S A317931 1,1,1,3,3,1,3,5,3,3,5,3,5,3,1,35,5,3,7,9,5,5,7,5,19,5,5,9,7,1,5,63,1,
%T A317931 5,9,9,11,7,5,15,11,5,13,15,13,7,9,35,27,19,7,15,13,5,7,15,3,7,11,3,9,
%U A317931 5,-7,231,-1,1,11,15,7,9,13,15,15,11,47,21,19,5,13,105,27,11,19,15,27,13,11,25,17,13,23,21,11,9,1,63
%N A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.
%H A317931 Antti Karttunen, <a href="/A317931/b317931.txt">Table of n, a(n) for n = 1..16384</a>
%H A317931 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F A317931 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A002487(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317931 (PARI)
%o A317931 A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
%o A317931 A317931perA317932(n) = if(1==n,n,(A002487(n)-sumdiv(n,d,if((d>1)&&(d<n),A317931perA317932(d)*A317931perA317932(n/d),0)))/2);
%o A317931 A317931(n) = numerator(A317931perA317932(n));
%o A317931 (PARI)
%o A317931 \\ Memoized implementation:
%o A317931 memo = Map();
%o A317931 A317931perA317932(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A002487(n)-sumdiv(n,d,if((d>1)&&(d<n),A317931perA317932(d)*A317931perA317932(n/d),0)))/2); mapput(memo,n,v); (v)));
%Y A317931 Cf. A002487, A317932 (denominators, conjectured).
%Y A317931 Cf. also A299151, A317927, A317839, A317843.
%K A317931 sign,frac,look
%O A317931 1,4
%A A317931 _Antti Karttunen_, Aug 11 2018