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%I #26 Jul 06 2022 19:37:35
%S 1,0,2,0,0,0,4,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N a(n) = A036987(n)*(n+1)/2.
%C Guy Steele defines a family of 36 integer sequences, denoted here by GS(i,j) for 1 <= i, j <= 6, as follows. a[1]=1; a[2n] = i-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}; a[2n+1] = j-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}. The present sequence is GS(1,5).
%C The full list of 36 sequences:
%C GS(1,1) = A000007
%C GS(1,2) = A000035
%C GS(1,3) = A036987
%C GS(1,4) = A007814
%C GS(1,5) = A135416 (the present sequence)
%C GS(1,6) = A135481
%C GS(2,1) = A135528
%C GS(2,2) = A000012
%C GS(2,3) = A000012
%C GS(2,4) = A091090
%C GS(2,5) = A135517
%C GS(2,6) = A135521
%C GS(3,1) = A036987
%C GS(3,2) = A000012
%C GS(3,3) = A000012
%C GS(3,4) = A000120
%C GS(3,5) = A048896
%C GS(3,6) = A038573
%C GS(4,1) = A135523
%C GS(4,2) = A001511
%C GS(4,3) = A008687
%C GS(4,4) = A070939
%C GS(4,5) = A135529
%C GS(4,6) = A135533
%C GS(5,1) = A048298
%C GS(5,2) = A006519
%C GS(5,3) = A080100
%C GS(5,4) = A087808
%C GS(5,5) = A053644
%C GS(5,6) = A000027
%C GS(6,1) = A135534
%C GS(6,2) = A038712
%C GS(6,3) = A135540
%C GS(6,4) = A135542
%C GS(6,5) = A054429
%C GS(6,6) = A003817
%C (with a(0)=1): Moebius transform of A038712.
%H Antti Karttunen, <a href="/A135416/b135416.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F G.f.: sum{k>=1, 2^(k-1)*x^(2^k-1) }.
%F Recurrence: a(2n+1) = 2a(n), a(2n) = 0, starting a(1) = 1.
%p GS:=proc(i,j,M) local a,n; a:=array(1..2*M+1); a[1]:=1;
%p for n from 1 to M do
%p a[2*n] :=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][i];
%p a[2*n+1]:=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][j];
%p od: a:=convert(a,list); RETURN(a); end;
%p GS(1,5,200):
%t i = 1; j = 5; Clear[a]; a[1] = 1; a[n_?EvenQ] := a[n] = {0, 1, a[n/2], a[n/2]+1, 2*a[n/2], 2*a[n/2]+1}[[i]]; a[n_?OddQ] := a[n] = {0, 1, a[(n-1)/2], a[(n-1)/2]+1, 2*a[(n-1)/2], 2*a[(n-1)/2]+1}[[j]]; Array[a, 105] (* _Jean-François Alcover_, Sep 12 2013 *)
%o (PARI)
%o A048298(n) = if(!n,0,if(!bitand(n,n-1),n,0));
%o A135416(n) = (A048298(n+1)/2); \\ _Antti Karttunen_, Jul 22 2018
%o (Python)
%o def A135416(n): return int(not(n&(n+1)))*(n+1)>>1 # _Chai Wah Wu_, Jul 06 2022
%Y Equals A048298(n+1)/2. Cf. A036987, A182660.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, based on a message from Guy Steele and _Don Knuth_, Mar 01 2008
%E Formulae and comments by _Ralf Stephan_, Jun 20 2014
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