A049971 - OEIS

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.

%I #55 May 06 2022 13:12:16

%S 1,3,2,9,24,42,90,213,597,984,1974,3981,8133,17037,37071,87198,244557,

%T 401919,803844,1607721,3215613,6431997,12866991,25747038,51564237,

%U 103443195,208092192,421008660,861332361,1800360879,3918287223,9215926665,25847419116,42478911570,84957823146

%N 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.

%H David A. Corneth, <a href="/A049971/b049971.txt">Table of n, a(n) for n = 1..1000</a>

%e 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

%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, 3, 2][n], s(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 2)):

%p end proc:

%p seq(a(n), n = 1..40); # _Petros Hadjicostas_, Apr 25 2020

%p # Alternative, uses A062050:

%p a := proc(n) option remember; if n < 4 then [1, 3, 2][n] else

%p add(a(i), i = 1..n-1 ) + a(2*A062050(n-2)) fi end:

%p seq(a(n), n = 1..35); # _Peter Luschny_, Aug 06 2021

%t 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 *)

%o (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)

%o (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

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

%Y Cf A062050.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_

%E Name edited by _Petros Hadjicostas_, Apr 25 2020

%E More terms from _David A. Corneth_, Apr 26 2020