A049978 - OEIS

%I #20 Sep 28 2019 02:07:32

%S 1,3,4,9,20,38,78,157,319,630,1262,2525,5055,10121,20260,40560,81199,

%T 162242,324486,648973,1297951,2595913,5191844,10383728,20767535,

%U 41535232,83070775,166142182,332285627,664573784,1329152634,2658315407,5316651114,10633261669,21266523340,42533046681,85066093367,170132186745

%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) = 1, a(2) = 3, and a(3) = 4.

%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) = 1, a(2) = 3, and a(3) = 4. - _Petros Hadjicostas_, Sep 27 2019

%e From _Petros Hadjicostas_, Sep 27 2019: (Start)

%e a(4) = a(4-1-2^ceiling(-1 + log_2(4-1))) + a(1) + a(2) + a(3) = a(1) + a(1) + a(2) + a(3) = 9.

%e a(5) = a(5-1-2^ceiling(-1 + log_2(5-1))) + a(1) + a(2) + a(3) + a(4) = a(2) + a(1) + a(2) + a(3) + a(4) = 20.

%e a(6) = a(6-1-2^ceiling(-1 + log_2(6-1))) + a(1) + a(2) + a(3) + a(4) + a(5) = a(1) + a(1) + a(2) + a(3) + a(4) + a(5) = 38.

%e (End)

%p a := proc(n) local i; option remember; if n < 4 then return [1, 3, 4][n]; end if; add(a(i), i = 1 .. n - 1) + a(n - 3/2 - 1/2*Bits:-Iff(n - 2, n - 2)); end proc;

%p seq(a(n), n = 1 .. 37); # _Petros Hadjicostas_, Sep 27 2019 using a modification of a program by _Peter Luschny_

%Y Cf. A006257, A049939, A049940, A049960, A049964.

%K nonn

%O 1,2

%A _Clark Kimberling_

%E Name edited by and more terms from _Petros Hadjicostas_, Sep 27 2019