A049963 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n-1 <= 2^(p+1), with a(1) = 1, a(2) = 2 and a(3) = 4.

1, 2, 4, 9, 25, 43, 93, 220, 617, 1016, 2039, 4112, 8401, 17598, 38292, 90070, 252612, 415156, 830319, 1660672, 3321521, 6643838, 13290772, 26595030, 53262532, 106850150, 214945816, 434874798, 889700788, 1859656696

Offset

1,2

Comments

The number m in the definition of the sequence equals 2*n - 2 - x, where x is the smallest power of 2 >= n-1. It turns out that m = 1 + A006257(n-2), where the sequence b(n) = A006257(n) satisfies b(2*n) = 2*b(n) - 1 and b(2*n + 1) = 2*b(n) + 1, and it is related to the so-called Josephus problem. - Petros Hadjicostas, Sep 25 2019

Links

Table of n, a(n) for n=1..30.

Index entries for sequences related to the Josephus Problem

Formula

a(n) = a(1 + A006257(n-2)) + Sum_{i = 1..n-1} a(i) for n >= 4 with a(1) = 1, a(2) = 2 and a(3) = 4. - Petros Hadjicostas, Sep 25 2019

Example

From Petros Hadjicostas, Sep 25 2019: (Start)
a(4) = a(1 + A006257(4-2)) + a(1) + a(2) + a(3) = a(2) + a(1) + a(2) + a(3) = 9.
a(7) = a(1 + A006257(7-2)) + a(1) + a(2) + a(3) + a(4) + a(5) + a(6) = a(4) + a(1) + a(2) + a(3) + a(4) + a(5) + a(6) = 93.
(End)

Maple

a := proc(n) local i; option remember; if n < 4 then return [1, 2, 4][n]; end if; add(a(i), i = 1 .. n - 1) + a(2*n - 3 - Bits:-Iff(n - 2, n - 2)); end proc;
seq(a(n), n = 1..40); # Petros Hadjicostas, Sep 25 2019, courtesy of Peter Luschny

Crossrefs

Cf. A006257, A049920, A049939, A049960, A049964, A049979.

Cf. A049914 (similar, but with minus a(m/2)), A049915 (similar, but with minus a(m)), A049962 (similar, but with plus a(m/2)).

Sequence in context: A013091 A013168 A013183 * A368458 A114110 A337693

Adjacent sequences: A049960 A049961 A049962 * A049964 A049965 A049966

Keyword

nonn

Author

Clark Kimberling

Extensions

Name edited by Petros Hadjicostas, Sep 25 2019

Status

approved