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%I #19 Nov 07 2019 21:17:18
%S 1,3,4,11,20,50,93,185,368,920,1748,3453,6876,13743,27479,54957,
%T 109912,274780,522082,1030428,2053989,4104555,8207405,16413982,
%U 32827412,65654641,131309190,262618337,525236644,1050473279,2100946551,4201893101,8403786200,21009465500,39917984450
%N a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n -1 <= 2^(p+1), with a(1) = 1, a(2) = 3, and a(3) = 4.
%F From _Petros Hadjicostas_, Nov 07 2019: (Start)
%F a(n) = a(2^ceiling(log_2(n-1)) + 2 - n) + Sum_{i = 1..n-1} a(i) for n >= 4.
%F a(n) = a(n - 1 - A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4. (End)
%e From _Petros Hadjicostas_, Nov 07 2019: (Start)
%e a(4) = a(2^ceiling(log_2(4-1)) + 2 - 4) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 11.
%e a(5) = a(2^ceiling(log_2(5-1)) + 2 - 5) + a(1) + a(2) + a(3) + a(4) = a(1) + a(1) + a(2) + a(3) + a(4) = 20.
%e a(6) = a(2^ceiling(log_2(6-1)) + 2 - 6) + a(1) + a(2) + a(3) + a(4) + a(5) = a(4) + a(1) + a(2) + a(3) + a(4) + a(5) = 50.
%e a(7) = a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = a(3) + Sum_{i = 1..6} a(i) = 93.
%e a(8) = a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = a(2) + Sum_{i = 1..7} a(i) = 185. (End)
%p s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
%p a := proc(n) option remember; `if`(n < 2, 1, `if`(n < 3, 3,
%p `if`(n < 4, 4, s(n - 1) + a(Bits:-Iff(n - 2, n - 2) + 3 - n))))
%p end proc:
%p seq(a(n), n = 1 .. 40); # _Petros Hadjicostas_, Nov 07 2019
%Y Cf. A006257, A049933, A049937, A049945.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Name edited by and more terms from _Petros Hadjicostas_, Nov 07 2019
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