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A089401
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a(n) = m - A089398(2^m + n) for m>=n.
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2
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1, 1, 3, 2, 4, 5, 6, 5, 7, 8, 11, 9, 11, 12, 13, 12, 14, 15, 18, 18, 19, 20, 21, 20, 22, 23, 26, 24, 26, 27, 28, 27, 29, 30, 33, 33, 36, 36, 37, 36, 38, 39, 42, 40, 42, 43, 44, 43, 45, 46, 49, 49, 50, 51, 52, 51, 53, 54, 57, 55, 57, 58, 59, 58, 60, 61, 64, 64, 67, 69, 69, 68, 70
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OFFSET
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1,3
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COMMENTS
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A089398(n) = n-th column sum of binary digits of k*2^(k-1), where summation is over all k>=1, without carrying from columns sums that may exceed 2.
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LINKS
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FORMULA
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a(n) = n/2+1/2*sum(k=1, n, (-1)^floor((n-k)/2^(k-1))). - Benoit Cloitre, Mar 28 2005
Let a(0)=0; when n - 2^[log_2(n)] <= [log_2(n)] then a(n) = a(n - 2^[log_2(n)]) + n - [log_2(n)], else a(n) = a(n - 2^[log_2(n)]) + 2^[log_2(n)] - 1. Thus a(2^m) = 2^m - m for all m>=0; for 0<=k<=m: a(2^m + k) = a(k) + 2^m + k - m; for m<k<=2^m: a(2^m + k) = a(k) + 2^m - 1. - Paul D. Hanna, Mar 28 2005
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EXAMPLE
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a(6)=5 since 7 - A089398(2^7 + 6) = 7 - 2 = 5.
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MATHEMATICA
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f[n_] := Block[{lg = Floor[Log[2, n]] + 1}, Sum[ Join[ Reverse[ IntegerDigits[n - i + 1, 2]], {0}][[i]], {i, lg}]]; Table[n - f[2^n + n] + 2, {n, 0, 72}] (* Robert G. Wilson v, Mar 29 2005 *)
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PROG
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(PARI) a(n)=n/2+1/2*sum(k=1, n, (-1)^floor((n-k)/2^(k-1))) \\ Benoit Cloitre
(PARI) {a(n)=if(n<=0, 0, m=floor(log(n)/log(2)); if(n-2^m<=m, n-m+a(n-2^m), 2^m-1+a(n-2^m)))} \\ Paul D. Hanna, Mar 28 2005
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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