A050049 -id:A050049 - cp's OEIS frontend

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.

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

A050061 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) = 2.

Original entry on oeis.org

1, 3, 2, 5, 6, 11, 13, 16, 17, 33, 46, 57, 63, 68, 70, 73, 74, 147, 217, 285, 348, 405, 451, 484, 501, 517, 530, 541, 547, 552, 554, 557, 558, 1115, 1669, 2221, 2768, 3309, 3839, 4356, 4857, 5341, 5792, 6197, 6545, 6830, 7047, 7194

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), A050057 (1,3,1), A050065 (1,3,3), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 3, 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 11 2019

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

Name edited by Petros Hadjicostas, Nov 11 2019

A050065 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) = 3.

Original entry on oeis.org

1, 3, 3, 6, 7, 13, 16, 19, 20, 39, 55, 68, 75, 81, 84, 87, 88, 175, 259, 340, 415, 483, 538, 577, 597, 616, 632, 645, 652, 658, 661, 664, 665, 1329, 1990, 2648, 3300, 3945, 4577, 5193, 5790, 6367, 6905, 7388, 7803, 8143, 8402, 8577

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), A050057 (1,3,1), A050061 (1,3,2), A050069 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 3, 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, 3}, Flatten@Table[k, {n, 5}, {k, 2^n, 1, -1}]] (* Ivan Neretin, Sep 08 2015 *)

Name edited by Petros Hadjicostas, Nov 08 2019

A110428 a(1) = 1 and a(2) = 2. Subsequent terms are generated like this: if a(m) is the last term available -- say a(2) -- then a(m+1) = a(m) * a(m-1), a(m+2) = a(m) * a(m-1) * a(m-2), ..., a(2m-1) = a(m) a(m-1) * a(m-2) * ... * a(2) * a(1), a(2m) = a(2m-1) * a(2*m-2), and so on.

Original entry on oeis.org

1, 2, 2, 4, 4, 16, 32, 64, 64, 4096, 131072, 2097152, 8388608, 33554432, 67108864, 134217728, 134217728, 18014398509481984, 1208925819614629174706176, 40564819207303340847894502572032, 340282366920938463463374607431768211456

Offset: 1

Amarnath Murthy, Aug 01 2005

By choosing appropriate values for a(1) and a(2), many such sequences can be generated.

			a(3) = a(2)*a(1) = 2. [Now a(3) is the last term available.]
a(4) = a(3)*a(2) = 4.
a(5) = a(3)*a(2)*a(1) = 4. [Now a(5) is the last term available.]
a(6) = a(5)*a(4) = 16.
a(7) = a(5)*a(4)*a(3) = 32.
a(8) = a(5)*a(4)*a(3)*a(2) = 64.
a(9) = a(5)*a(4)*a(3)*a(2)*a(1) = 64. [Now a(9) is the last term available.]
a(10) = a(9)*a(8) = 4096.
a(11) = a(9)*a(8)*a(7) = 131072.
...
a(17) = a(9)*a(8)*...*a(1) = 134217728. [Now a(17) is the last term available.]
a(18) = a(17)*a(16) = 18014398509481984.
[Example extended by _Petros Hadjicostas_, Nov 13 2019]

Cf. A000051 (index of "available" terms as described above), A050049 (an additive version of this sequence), A329474 (log[2] of this sequence).

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..25); # Petros Hadjicostas, Nov 13 2019

From Petros Hadjicostas, Nov 13 2019: (Start)
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.
a(A000051(n)) = a(2^n + 1) = a(2^n) for n >= 1.
a(A000051(n) + 1) = a(2^n + 2) = a(2^n + 1) * a(2^n) = a(2^n)^2 for n >= 1.
log[2](a(n)) = A329474(n) for n >= 1. (End)

More terms from Joshua Zucker, May 10 2006

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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.

Views

Author

Keywords

Links

Crossrefs

Programs

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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.

Views

Author

Keywords

Links

Crossrefs

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A050061 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) = 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A110428 a(1) = 1 and a(2) = 2. Subsequent terms are generated like this: if a(m) is the last term available -- say a(2) -- then a(m+1) = a(m) * a(m-1), a(m+2) = a(m) * a(m-1) * a(m-2), ..., a(2m-1) = a(m) a(m-1) * a(m-2) * ... * a(2) * a(1), a(2m) = a(2m-1) * a(2*m-2), and so on.