A049971 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), starting with a(1) = 1, a(2) = 3, and a(3) = 2.

1, 3, 2, 9, 24, 42, 90, 213, 597, 984, 1974, 3981, 8133, 17037, 37071, 87198, 244557, 401919, 803844, 1607721, 3215613, 6431997, 12866991, 25747038, 51564237, 103443195, 208092192, 421008660, 861332361, 1800360879, 3918287223, 9215926665, 25847419116, 42478911570, 84957823146

Offset

1,2

Links

David A. Corneth, Table of n, a(n) for n = 1..1000

Example

a(5) = a(1) + a(2) + a(3) + a(4) + a(m) = 1 + 3 + 2 + 9 + a(m) = 15 + a(m). where m = 2*n - 2 - 2^(p+1) and 2^p < n - 1 = 4 <= 2^(p+1). We have p = 1 giving m = 2*5 - 2 - 4 = 4. As a(m) = a(4) = 9, we have a(5) = 15 + 9 = 24. - David A. Corneth, Apr 26 2020

Maple

s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
a := proc(n) option remember;
`if`(n < 4, [1, 3, 2][n], s(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 2)):
end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Apr 25 2020
# Alternative, uses A062050:
a := proc(n) option remember; if n < 4 then [1, 3, 2][n] else
add(a(i), i = 1..n-1 ) + a(2*A062050(n-2)) fi end:
seq(a(n), n = 1..35); # Peter Luschny, Aug 06 2021

Mathematica

a[1] = 1; a[2] = 3; a[3] = 2; a[n_] := a[n] = Sum[a[k], {k, 1, n - 1}] + a[2*n - 2 - 2^Floor[1 + Log[2, n - 2]]]; Table[a[n], {n, 1, 30}] (* Vaclav Kotesovec, Apr 26 2020 *)

Prog

(PARI) lista(nn) = { my(va = vector(nn)); va[1] = 1; va[2] = 3; va[3] = 2; my(sa = vecsum(va)); for (n=4, nn, va[n] = sa + va[2*n - 2 - 2*2^logint(n-2, 2)]; sa += va[n]; ); va; } \\ Michel Marcus, Apr 26 2020 (with nn > 2)
(PARI) first(n) = {n = max(n, 3); my(res = vector(n), s = 6, p = 1); res[1]  = 1; res[2] = 3; res[3] = 2; for(i = 4, n, if(i - 1 > 1 << (p + 1), p++); res[i] = s + res[2*i-2-2^(p+1)]; s += res[i]) ; res} \\ David A. Corneth, Apr 26 2020

Crossrefs

Cf. A049922 (similar, but with minus a(m/2)), A049923 (similar, but with minus a(m)), A049970 (similar, but with plus a(m/2)).

Cf A062050.

Sequence in context: A234840 A234743 A284989 * A234748 A156528 A234744

Adjacent sequences: A049968 A049969 A049970 * A049972 A049973 A049974

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Name edited by Petros Hadjicostas, Apr 25 2020

More terms from David A. Corneth, Apr 26 2020

Status

approved