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A173920
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Triangle read by rows: T(n,k) = convolution of n with k in binary representation, 0<=k<=n.
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7
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0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2
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OFFSET
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0,10
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COMMENTS
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T(n,k) = SUM(bn(i)*bk(L-i-1): 0<=i<L), where L=A070939(n), n=SUM(bn(i)*2^i:0<=i<L), and k=SUM(bk(i)*2^i:0<=i<L);
T(n,2*k+1) = T(n,2*k) + 1;
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LINKS
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FORMULA
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T(n,k) = c(A030101(n),k,0) with c(x,y,z) = if y=0 then z else c([x/2],[y/2],z+(x mod 2)*(y mod 2)).
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EXAMPLE
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T(13,10) = T('1101','1010') = 1*0 + 1*1 + 0*0 + 1*1 = 2;
T(13,11) = T('1101','1011') = 1*1 + 1*1 + 0*0 + 1*1 = 3;
T(13,12) = T('1101','1100') = 1*0 + 1*0 + 0*1 + 1*1 = 1;
T(13,13) = T('1101','1101') = 1*1 + 1*0 + 0*1 + 1*1 = 2.
Triangle begins:
0;
0, 1;
0, 1, 0;
0, 1, 1, 2;
0, 1, 0, 1, 0;
0, 1, 0, 1, 1, 2;
...
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MATHEMATICA
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T[n_, k_] := Module[{bn, bk, lg},
bn = IntegerDigits[n, 2];
bk = IntegerDigits[k, 2];
lg = Max[Length[bn], Length[bk]];
ListConvolve[PadLeft[bn, lg], PadLeft[bk, lg]]][[1]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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