A049933 - OEIS

%I #24 May 06 2022 13:11:36

%S 1,1,1,4,8,19,35,70,140,349,663,1310,2609,5214,10425,20850,41700,

%T 104249,198073,390935,779265,1557231,3113815,6227316,12454423,

%U 24908776,49817517,99635018,199270025,398540046,797080089,1594160178,3188320356,7970800889,15144521689

%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), starting with a(1) = a(2) = a(3) = 1.

%F From _Petros Hadjicostas_, Nov 06 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 06 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) = 4.

%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) = 8.

%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) = 19.

%e a(7) =  a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = a(3) +  Sum_{i = 1..6} a(i) = 35.

%e a(8) =  a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = a(2) +  Sum_{i = 1..7} a(i) = 70. (End)

%p s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)); end proc;

%p a := proc(n) option remember;

%p   `if`(n < 4, 1, s(n - 1) + a(Bits:-Iff(n - 2, n - 2) + 3 - n));

%p end proc;

%p seq(a(n), n = 1 .. 30);

%t b[n_] := Module[{p}, For[p = 0, True, p++, If[2^p < n - 1 <= 2^(p + 1), Return[p]]]];

%t a[n_] := a[n] = If[n < 4, 1, With[{m = 2^(b[n] + 1) + 2 - n}, Total[ Array[a, n - 1]] + a[m]]];

%t Array[a, 35] (* _Jean-François Alcover_, Apr 24 2020 *)

%Y Cf. A006257, A049885 (similar, but with minus a(m)), A049937, A049945.

%K nonn

%O 1,4

%A _Clark Kimberling_

%E Name edited by and more terms from _Petros Hadjicostas_, Nov 06 2019