A049933
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.
Original entry on oeis.org
1, 1, 1, 4, 8, 19, 35, 70, 140, 349, 663, 1310, 2609, 5214, 10425, 20850, 41700, 104249, 198073, 390935, 779265, 1557231, 3113815, 6227316, 12454423, 24908776, 49817517, 99635018, 199270025, 398540046, 797080089, 1594160178, 3188320356, 7970800889, 15144521689
Offset: 1
From _Petros Hadjicostas_, Nov 06 2019: (Start)
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.
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.
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.
a(7) = a(7 - 1 - A006257(7-2)) + Sum_{i = 1..6} a(i) = a(3) + Sum_{i = 1..6} a(i) = 35.
a(8) = a(8 - 1 - A006257(8-2)) + Sum_{i = 1..7} a(i) = a(2) + Sum_{i = 1..7} a(i) = 70. (End)
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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, s(n - 1) + a(Bits:-Iff(n - 2, n - 2) + 3 - n));
end proc;
seq(a(n), n = 1 .. 30);
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b[n_] := Module[{p}, For[p = 0, True, p++, If[2^p < n - 1 <= 2^(p + 1), Return[p]]]];
a[n_] := a[n] = If[n < 4, 1, With[{m = 2^(b[n] + 1) + 2 - n}, Total[ Array[a, n - 1]] + a[m]]];
Array[a, 35] (* Jean-François Alcover, Apr 24 2020 *)
A049893
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) = 1 and a(3) = 3.
Original entry on oeis.org
1, 1, 3, 4, 8, 13, 27, 56, 112, 169, 367, 748, 1501, 3006, 6013, 12028, 24056, 36085, 78185, 159377, 320259, 641271, 1282923, 2566044, 5132145, 10264346, 20528721, 41057456, 82114917, 164229838, 328459677, 656919356, 1313838712, 1970758069, 4269975817, 8704181473
Offset: 1
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s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
a := proc(n) option remember;
`if`(n < 4, [1,1,3][n], s(n - 1) - a(2^ceil(log[2](n - 1)) + 2 - n)):
end proc:
seq(a(n), n = 1..34); # Petros Hadjicostas, Nov 11 2019
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nn = 36; a[1] = a[2] = 1; a[3] = 3; s = 5; Do[Set[k, s - a[2^(Ceiling@ Log2[n - 1]) + 2 - n]]; a[n] = k; s += k, {n, 4, nn}]; Array[a, nn] (* Michael De Vlieger, Oct 21 2024 *)
A049897
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) = 1 and a(3) = 4.
Original entry on oeis.org
1, 1, 4, 5, 10, 16, 33, 69, 138, 208, 452, 921, 1848, 3701, 7403, 14809, 29618, 44428, 96262, 196226, 394305, 789537, 1579543, 3159330, 6318730, 12637529, 25275094, 50550205, 101100416, 202200837, 404401675, 808803353, 1617606706
Offset: 1
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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, 1, 4][n], s(n - 1) - a(2^ceil(log[2](n - 1)) + 2 - n)):
end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Nov 14 2019
A049901
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) = 1, a(2) = 2, and a(3) = 1.
Original entry on oeis.org
1, 2, 1, 2, 5, 9, 19, 37, 75, 114, 246, 502, 1008, 2019, 4039, 8077, 16155, 24234, 52506, 107032, 215075, 430656, 861568, 1723268, 3446575, 6893188, 13786394, 27572798, 55145600, 110291203, 220582407, 441164813, 882329627, 1323494442
Offset: 1
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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, 2, 1][n], s(n - 1) - a(2^ceil(log[2](n - 1)) + 2 - n)):
end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Nov 14 2019
Showing 1-4 of 4 results.