A050029 -id:A050029 - cp's OEIS frontend

A096118 Duplicate of A050029.

1, 1, 2, 3, 4, 7, 9, 10, 11, 21, 30, 37, 41, 44, 46, 47, 48, 95, 141, 185, 226, 263, 293, 314

Offset: 1

dead

A096119 A050029(2^n + 1).

1, 2, 4, 11, 48, 362, 5030, 133924, 6977521, 719087781, 147394353130, 60255915944715, 49197429536084417, 80280819225274033666, 261914438169525117048056

Offset: 0

Amarnath Murthy, Jul 01 2004

nonn

Extended and description corrected by Antti Karttunen, Aug 25 2006

A050049 a(n) = a(n-1) + a(m) for n >= 3, 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 and a(2) = 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 14, 16, 17, 33, 47, 58, 64, 69, 72, 74, 75, 149, 221, 290, 354, 412, 459, 492, 509, 525, 539, 550, 556, 561, 564, 566, 567, 1133, 1697, 2258, 2814, 3364, 3903, 4428, 4937, 5429, 5888, 6300, 6654, 6944, 7165, 7314

Offset: 1

Clark Kimberling

nonn

a(1) = 1 and a(2) = 2; subsequent terms are generated like this: if a(s) is the last term available, say a(2), then a(s+1) = a(s) + a(s-1), a(s+2) = a(s) + a(s-1) + a(s-2), ..., a(2*s-1) = a(s) + a(s-1) + a(s-2) + ... + a(2) + a(1), a(2*s) = a(2*s-1) + a(2*s-2), and so on. - Amarnath Murthy, Aug 01 2005
From Petros Hadjicostas, Nov 13 2019: (Start)
We explain further the process introduced by Amarnath Murthy above. The terms a(s) that are the "last term[s] available" are those that correspond to s = A000051(k) = 2^k + 1 for k >= 0. Thus, they are the terms a(2), a(3), a(5), a(9), a(17), a(33), and so on. See the example below.
In the Mathematica program below, the author of the program starts with a(1) = 1, a(2) = 2, and a(3) = 3, but that is not necessary. We may start with a(1) = 1 and a(2) = 2 and still get the same sequence. (End)

			From _Petros Hadjicostas_, Nov 13 2019: (Start)
We explain _Amarnath Murthy_'s process (see the Comments above).
a(3) = a(2) + a(1) = 3. [Now a(3) is the last term available.]
a(4) = a(3) + a(2) = 5.
a(5) = a(3) + a(2) + a(1) = 6. [Now a(5) is the last term available.]
a(6) = a(5) + a(4) = 11.
a(7) = a(5) + a(4) + a(3) = 14.
a(8) = a(5) + a(4) + a(3) + a(2) = 16.
a(9) = a(5) + ... + a(1) = 17. [Now a(9) is the last term available.]
a(10) = a(9) + a(8) = 33.
a(11) = a(9) + a(8) + a(7) = 47.
...
a(17) = a(9) + a(8) + ... + a(1) = 75. [Now a(17) is the last term available.]
a(18) = a(17) + a(16) = 149. (End)

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. A000051 (index of "available" terms as described above), A110428 (a multiplicative version of this sequence).

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050033 (1,1,3), A050037 (1,1,4), A050041 (1,2,1), A050045 (1,2,2), A050053 (1,2,4), A050057 (1,3,1), A050061 (1,3,2), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 3, [1, 2][n], a(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)); end proc;
seq(a(n), n = 1..50); # Petros Hadjicostas, Nov 13 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 2, 3}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 13 2019

A050069 a(n) = a(n-1) + a(m) for n >= 3, 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 and a(2) = 3.

Original entry on oeis.org

1, 3, 4, 7, 8, 15, 19, 22, 23, 45, 64, 79, 87, 94, 98, 101, 102, 203, 301, 395, 482, 561, 625, 670, 693, 715, 734, 749, 757, 764, 768, 771, 772, 1543, 2311, 3075, 3832, 4581, 5315, 6030, 6723, 7393, 8018, 8579, 9061, 9456, 9757

Offset: 1

Clark Kimberling

nonn

In the Mathematica program below, the author of the program uses a(1) = 1, a(2) = 3, and a(3) = 4 as initial conditions. This is not necessary. We get the same sequence using only a(1) = 1 and a(2) = 3 as initial conditions. - Petros Hadjicostas, Nov 13 2019

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050033 (1,1,3), A050037 (1,1,4), A050041 (1,2,1), A050045 (1,2,2), A050049 (1,2,3), A050053 (1,2,4), A050057 (1,3,1), A050061 (1,3,2), A050065 (1,3,3).

a := proc(n) option remember;
`if`(n < 3, [1, 3][n], a(n - 1) + a(Bits:-Iff(n - 2, n - 2) + 3 - n)); end proc;
seq(a(n), n = 1 .. 48); # Petros Hadjicostas, Nov 08 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 3, 4}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 08 2015 *)

Name edited by Petros Hadjicostas, Nov 08 2019

A050033 a(n) = 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, 5, 9, 12, 13, 14, 27, 39, 48, 53, 57, 60, 61, 62, 123, 183, 240, 293, 341, 380, 407, 421, 434, 446, 455, 460, 464, 467, 468, 469, 937, 1404, 1868, 2328, 2783, 3229, 3663, 4084, 4491, 4871, 5212, 5505, 5745, 5928, 6051

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050037 (1,1,4), A050041 (1,2,1), A050045 (1,2,2), A050049 (1,2,3), A050053 (1,2,4), A050057 (1,3,1), A050061 (1,3,2), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 1, 3][n], a(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)); end proc;
seq(a(n), n = 1 .. 48); # Petros Hadjicostas, Nov 08 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 1, 3}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 08 2019

A050037 a(n) = 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, 6, 11, 15, 16, 17, 33, 48, 59, 65, 70, 74, 75, 76, 151, 225, 295, 360, 419, 467, 500, 517, 533, 548, 559, 565, 570, 574, 575, 576, 1151, 1725, 2295, 2860, 3419, 3967, 4500, 5017, 5517, 5984, 6403, 6763, 7058, 7283, 7434

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050033 (1,1,3), A050041 (1,2,1), A050045 (1,2,2), A050049 (1,2,3), A050053 (1,2,4), A050057 (1,3,1), A050061 (1,3,2), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 1, 4][n], a(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)); end proc;
seq(a(n), n = 1 .. 48); # Petros Hadjicostas, Nov 08 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 1, 4}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 08 2019

A050041 a(n) = 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, 3, 4, 7, 8, 10, 11, 21, 29, 36, 40, 43, 44, 46, 47, 93, 137, 180, 220, 256, 285, 306, 317, 327, 335, 342, 346, 349, 350, 352, 353, 705, 1055, 1404, 1750, 2092, 2427, 2754, 3071, 3377, 3662, 3918, 4138, 4318, 4455, 4548, 4595

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050033 (1,1,3), A050037 (1,1,4), A050045 (1,2,2), A050049 (1,2,3), A050053 (1,2,4), A050057 (1,3,1), A050061 (1,3,2), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 2, 1][n], a(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)); end proc;
seq(a(n), n = 1..50); # Petros Hadjicostas, Nov 11 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 2, 1}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 11 2019

A050045 a(n) = 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 and a(2) = a(3) = 2.

Original entry on oeis.org

1, 2, 2, 4, 5, 9, 11, 13, 14, 27, 38, 47, 52, 56, 58, 60, 61, 121, 179, 235, 287, 334, 372, 399, 413, 426, 437, 446, 451, 455, 457, 459, 460, 919, 1376, 1831, 2282, 2728, 3165, 3591, 4004, 4403, 4775, 5109, 5396, 5631, 5810, 5931

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050033 (1,1,3), A050037 (1,1,4), A050041 (1,2,1), A050049 (1,2,3), A050053 (1,2,4), A050057 (1,3,1), A050061 (1,3,2), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 2, 2][n], a(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)):
end proc:
seq(a(n), n = 1..60); # Petros Hadjicostas, Nov 14 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 2, 2}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 14 2019

A050053 a(n) = 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) = 4.

Original entry on oeis.org

1, 2, 4, 6, 7, 13, 17, 19, 20, 39, 56, 69, 76, 82, 86, 88, 89, 177, 263, 345, 421, 490, 546, 585, 605, 624, 641, 654, 661, 667, 671, 673, 674, 1347, 2018, 2685, 3346, 4000, 4641, 5265, 5870, 6455, 7001, 7491, 7912, 8257, 8520, 8697

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050033 (1,1,3), A050037 (1,1,4), A050041 (1,2,1), A050045 (1,2,2), A050049 (1,2,3), A050057 (1,3,1), A050061 (1,3,2), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 2, 4][n], a(n - 1) + a(2^ceil(log[2](n - 1)) + 2 - n)); end proc;
seq(a(n), n = 1 .. 48); # Petros Hadjicostas, Nov 09 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 2, 4}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 08 2015 *)

Name edited by Petros Hadjicostas, Nov 09 2019

A050057 a(n) = 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) = 3, and a(3) = 1.

Original entry on oeis.org

1, 3, 1, 4, 5, 9, 10, 13, 14, 27, 37, 46, 51, 55, 56, 59, 60, 119, 175, 230, 281, 327, 364, 391, 405, 418, 428, 437, 442, 446, 447, 450, 451, 901, 1348, 1794, 2236, 2673, 3101, 3519, 3924, 4315, 4679, 5006, 5287, 5517, 5692, 5811

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050025 (1,1,1), A050029 (1,1,2), A050033 (1,1,3), A050037 (1,1,4), A050041 (1,2,1), A050045 (1,2,2), A050049 (1,2,3), A050053 (1,2,4), A050061 (1,3,2), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 3, 1][n], a(n - 1) + a(Bits:-Iff(n - 2, n - 2) + 3 - n)); end proc;
seq(a(n), n = 1..48); # Petros Hadjicostas, Nov 08 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 3, 1}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 08 2015 *)

Name edited by Petros Hadjicostas, Nov 08 2019

cp's OEIS Frontend

A096118 Duplicate of A050029.

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A096119 A050029(2^n + 1).

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A050049 a(n) = a(n-1) + a(m) for n >= 3, 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 and a(2) = 2.

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A050069 a(n) = a(n-1) + a(m) for n >= 3, 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 and a(2) = 3.

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Maple

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A050033 a(n) = 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.

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Maple

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A050037 a(n) = 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.

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Maple

Mathematica

Extensions

A050041 a(n) = 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.

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Maple

Mathematica

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A050045 a(n) = 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 and a(2) = a(3) = 2.

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Programs

Maple

Mathematica

Extensions

A050053 a(n) = 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) = 4.

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