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A151552
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G.f.: Product_{k>=1} (1 + x^(2^k-1) + x^(2^k)).
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18
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1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 4, 3, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 2, 3, 4, 4, 5, 7, 7, 5, 5
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OFFSET
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0,5
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LINKS
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FORMULA
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a(n) = 1 for 0 <= n <= 3; thereafter write n = 2^i + j, with 0 <= j < 2^i, then a(n) = a(j) + a(j+1), except that a(2^(i+1)-2) = a(2^(i+1)-1) = 1.
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EXAMPLE
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Written as a triangle:
1;
1;
1,1;
2,2,1,1;
2,2,2,3,4,3,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,3,4,4,5,7,7,5,5,...
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MAPLE
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G := mul( 1 + x^(2^n-1) + x^(2^n), n=1..20);
wt := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
f:=proc(n) local t1, k; global wt; t1:=0; for k from 0 to 20 do if n+k mod 2 = 0 then t1:=t1+binomial(wt(n+k), k); fi; od; t1; end;
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MATHEMATICA
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a[n_] := Sum[If[EvenQ[n + k], Binomial[DigitCount[n + k, 2, 1], k], 0], {k, 0, Floor[Log2[n + 1]]}]; Array[a, 100, 0] (* Amiram Eldar, Jul 29 2023 *)
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CROSSREFS
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For generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k)) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
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KEYWORD
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AUTHOR
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STATUS
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approved
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