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%I #15 Apr 19 2015 22:23:43
%S 0,1,1,2,2,1,2,3,3,2,2,3,1,2,3,4,4,3,3,3,2,2,4,2,3,3,4,1,2,3,4,5,5,4,
%T 4,4,3,3,4,3,3,3,5,2,2,4,3,4,2,4,5,3,3,2,3,4,4,5,1,2,3,4,5,6,6,5,5,5,
%U 4,4,5,4,4,4,5,3,3,4,4,4,3,4,6,3,3,3,3,5,5,4,2,2,4,3,5,3,4,5,6,2,4,4,4,5,3,4,3,3,2,5,5,3,6,2,4,4,3,4,5,5,6,1,2,3,4,5,6,7,7
%N a(0) = 0; and for n >= 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).
%C a(n) tells how many nonzero terms of A005187 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n. This count includes both n (in case it is a term of A005187) and 1 (but not 0). See also comments in A256479 and A256991.
%C The 1's (seem to) occur at positions given by A000325.
%H Antti Karttunen, <a href="/A256478/b256478.txt">Table of n, a(n) for n = 0..16384</a>
%F a(0) = 0; and for n >= 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).
%F a(n) = A000120(A233277(n)). [Binary weight of A233277(n).]
%F Other identities and observations. For all n >= 1:
%F a(n) = 1 + A257248(n) = 1 + A080791(A233275(n)).
%F a(n) = A070939(n) - A256479(n).
%F a(n) >= A255559(n).
%o (Scheme, with memoization-macro definec)
%o (definec (A256478 n) (if (< n 1) n (+ (A079559 n) (A256478 (if (zero? (A079559 n)) (A234017 n) (+ -1 (A213714 n)))))))
%o ;; Alternative definitions:
%o (define (A256478 n) (A000120 (A233277 n)))
%o (define (A256478 n) (if (zero? n) n (+ 1 (A080791 (A233275 n)))))
%Y One more than A257248.
%Y Cf. A000120, A000325, A005187, A070939, A079559, A080791, A213714, A234017, A233275, A233276, A233277, A255559, A256479, A256991.
%K nonn
%O 0,4
%A _Antti Karttunen_, Apr 15 2015
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