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%I #29 May 06 2022 13:12:16
%S 1,1,2,5,10,20,40,81,165,326,652,1305,2613,5231,10472,20964,41969,
%T 83858,167716,335433,670869,1341743,2683496,5367012,10734065,21468214,
%U 42936589,85873504,171747661,343496630,686995878,1373996997,2748004486,5495988009,10991976018,21983952037,43967904077
%N a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), with a(1) = a(2) = 1 and a(3) = 2.
%F a(n) = a(n - 1 - 2^ceiling(-1 + log_2(n-1))) + Sum_{i = 1..n-1} a(i) = a((1 + A006257(n-2))/2) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = a(2) = 1 and a(3) = 2. - _Petros Hadjicostas_, Oct 01 2019
%e From _Petros Hadjicostas_, Oct 01 2019: (Start)
%e a(4) = a(4 - 1 - 2^ceiling(-1 + log_2(3))) + a(1) + a(2) + a(3) = a(1) + a(1) + a(2) + a(3) = 5.
%e a(5) = a(5 - 1 - 2^ceiling(-1 + log_2(4))) + a(1) + a(2) + a(3) + a(4) = a(2) + a(1) + a(2) + a(3) + a(4) = 10.
%e a(6) = a(6 - 1 - 2^ceiling(-1 + log_2(5))) + a(1) + a(2) + a(3) + a(4) + a(5) = a(1) + a(1) + a(2) + a(3) + a(4) + a(5) = 20.
%e (End)
%p a := proc(n) local i; option remember; if n < 4 then return [1, 1, 2][n]; end if; add(a(i), i = 1 .. n - 1) + a(n - 3/2 - 1/2*Bits:-Iff(n - 2, n - 2)); end proc; # _Petros Hadjicostas_, Oct 01 2019
%o (PARI) lista(nn) = { nn = max(nn, 3); my(va = vector(nn)); va[1] = 1; va[2] = 1; va[3] = 2; my(sa = vecsum(va)); for (n=4, nn, va[n] = sa + va[n - 1 - 2^ceil(-1 + log(n-1)/log(2))]; sa += va[n]; ); va; } \\ _Petros Hadjicostas_, Apr 27 2020
%Y Cf. A006257, A049890 (similar, but with minus a(m)), A049891 (similar, but with minus a(2*m)), A049939 (similar, but with plus a(2*m)), A049940, A049960, A049964, A049978.
%K nonn
%O 1,3
%A _Clark Kimberling_
%E More terms from _Petros Hadjicostas_, Oct 01 2019
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