A295862 -id:A295862 - cp's OEIS frontend

A295961 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 10, 19, 35, 61, 104, 175, 290, 477, 780, 1271, 2066, 3353, 5436, 8808, 14264, 23093, 37379, 60495, 97898, 158418, 256342, 414787, 671157, 1085973, 1757160, 2843164, 4600356, 7443553, 12043944, 19487533, 31531514, 51019085, 82550638, 133569763

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

See A295862 for a guide to related sequences.

			a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 10
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A295961 *)

A295962 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

3, 4, 11, 20, 37, 64, 109, 182, 302, 496, 811, 1321, 2147, 3484, 5648, 9150, 14818, 23989, 38829, 62841, 101694, 164560, 266280, 430867, 697175, 1128071, 1825276, 2953378, 4778686, 7732097, 12510817, 20242949, 32753803, 52996790, 85750632, 138747462

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

See A295862 for a guide to related sequences.

			a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 11
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A295962 *)

A295963 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 7, 14, 28, 50, 87, 147, 245, 404, 663, 1082, 1761, 2860, 4639, 7518, 12177, 19716, 31915, 51654, 83593, 135272, 218891, 354191, 573111, 927332, 1500474, 2427838, 3928345, 6356217, 10284597, 16640850, 26925484, 43566372, 70491895, 114058307, 184550243

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

See A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A295963 *)

A295964 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 4, 9, 18, 33, 58, 100, 168, 279, 459, 751, 1224, 1990, 3230, 5238, 8487, 13745, 22253, 36020, 58296, 94340, 152661, 247027, 399715, 646770, 1046514, 1693314, 2739859, 4433206, 7173099, 11606340, 18779475, 30385852, 49165365, 79551256, 128716661, 208267958

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

See A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 9
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A295964 *)

A295965 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

2, 3, 9, 17, 32, 56, 97, 163, 271, 446, 730, 1190, 1935, 3142, 5095, 8256, 13371, 21648, 35041, 56712, 91777, 148514, 240317, 388858, 629203, 1018090, 1647323, 2665445, 4312801, 6978280, 11291116, 18269432, 29560585, 47830055, 77390679, 125220774, 202611494

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

See A295862 for a guide to related sequences.

			a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 9
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A295965 *)

A295966 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 5, 9, 19, 34, 60, 103, 173, 287, 472, 772, 1258, 2045, 3319, 5381, 8719, 14120, 22860, 37002, 59885, 96911, 156821, 253758, 410606, 664392, 1075027, 1739449, 2814507, 4553988, 7368529, 11922552, 19291117, 31213706, 50504861, 81718606, 132223507, 213942154

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

See A295862 for a guide to related sequences.

			a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 9
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

a[0] = 1; a[1] = 5; b[0] = 2; b[1] = 3; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A295966 *)

A296221 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 11, 40, 146, 533, 1946, 7105, 25941, 94714, 345812, 1262601, 4609907, 16831321, 61453163, 224372837, 819212023, 2991040928, 10920647625, 39872588647, 145579582824, 531528442330, 1940673819263, 7085631873740, 25870488153041, 94456241758347

Offset: 0

Clark Kimberling, Dec 09 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) + 1 = 11
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, ...)

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A296000, A296222.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}] + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}];  (* A296221 *)
Table[b[n], {n, 0, 20}]
t = N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
d = RealDigits[Last[t], 10][[1]] (* A296222 *)

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

A296223 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 9, 34, 124, 453, 1654, 6040, 22055, 80532, 294058, 1073735, 3920679, 14316124, 52274468, 190877084, 696976221, 2544966858, 9292793804, 33932079081, 123900951107, 452416889887, 1651973131976, 6032080786047, 22025781112962, 80425818360771

Offset: 0

Clark Kimberling, Dec 10 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) - 1 = 9
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A296000, A296224.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}] - 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}]  (* A296223 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296224 *)

A296225 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) + n, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 12, 44, 161, 588, 2147, 7839, 28621, 104498, 381533, 1393015, 5086038, 18569636, 67799608, 247543185, 903805055, 3299883119, 12048205018, 43989207775, 160609019998, 586399678681, 2141004179974, 7817021504815, 28540731390577, 104205079621096

Offset: 0

Clark Kimberling, Dec 10 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) + 2 = 12
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, ...)

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A296000, A296226.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = n + Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 200}]  (* A296225 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296226 *)

A296227 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 8, 34, 146, 628, 2703, 11632, 50057, 215415, 927016, 3989317, 17167612, 73879038, 317930779, 1368182139, 5887829959, 25337665679, 109038016813, 469233798454, 2019298993572, 8689843823858, 37395841786394, 160929127296116, 692541811472532

Offset: 0

Clark Kimberling, Dec 10 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) - 2 = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A296000, A296228.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = - n + Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 200}]  (* A296227 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296228 *)

cp's OEIS Frontend

A295961 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A295962 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A295963 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A295964 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A295965 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A295966 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A296221 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A296223 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A296221 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

A296223 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

A296225 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) + n, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

A296227 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2) + ... + a(n-1)*b(0) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.