A295053 -id:A295053 - cp's OEIS frontend

A295061 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 8, 17, 38, 75, 161, 310, 655, 1252, 2633, 5022, 10547, 20104, 42206, 80435, 168844, 321761, 675398, 1287067, 2701616, 5148293, 10806490, 20593199, 43225988, 82372825, 172903982

Offset: 0

Clark Kimberling, Nov 18 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.09... and 2.09... .

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = 4 a[n - 2] + b[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, 18}]  (* A295061 *)
Table[b[n], {n, 0, 10}]

A295062 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 6, 16, 29, 71, 124, 293, 506, 1183, 2036, 4745, 8158, 18995, 32649, 75998, 130615, 304012, 522481, 1216070, 2089947, 4864304, 8359813, 19457242, 33439279, 77828996, 133757146, 311316015

Offset: 0

Clark Kimberling, Nov 18 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.71... and 2.32... .

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1]=4;
a[n_] := a[n] = 4 a[n - 2] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295062 *)
Table[b[n], {n, 0, 10}]

A295063 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1) + b(n-2), 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, 10, 21, 51, 97, 219, 405, 896, 1643, 3609, 6599, 14465, 26427, 57893, 105743, 231609, 423011, 926478, 1692089, 3705959, 6768405, 14823887, 27073673, 59295603, 108294749, 237182471

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 2.19... and 1.82... .

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1]=4;
a[n_] := a[n] = 4 a[n - 2] + b[n - 1] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295063 *)
Table[b[n], {n, 0, 10}]

A295065 Solution of the complementary equation a(n) = 8*a(n-3) + b(n-2), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 5, 12, 30, 47, 104, 249, 386, 843, 2005, 3102, 6759, 16056, 24833, 54090, 128467, 198684, 432741, 1027758, 1589495, 3461952, 8222089, 12715986, 27695643, 65776740, 101727917

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

The sequence a(n+1)/a(n) appears to have three convergent subsequences, with limits 1.54..., 2.17..., 2.37...

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

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

Cf. A295053, A295064.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2;
a[n_] := a[n] = 8 a[n - 3] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295065 *)
Table[b[n], {n, 0, 10}]

A295070 Solution of the complementary equation a(n) = a(n-2) + b(n-1) + b(n-2), 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, 11, 19, 24, 35, 43, 57, 68, 84, 97, 115, 130, 150, 168, 191, 211, 236, 259, 287, 312, 342, 369, 401, 430, 464, 495, 531, 565, 604, 640, 681, 719, 762, 802, 848, 891, 939, 984, 1034, 1081, 1133, 1182, 1236, 1287, 1343, 1396, 1454, 1510, 1571, 1629

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> 1.

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + b[ n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295070 *)
Table[b[n], {n, 0, 10}]

A295133 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1), 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, 10, 35, 111, 340, 1028, 3093, 9290, 27882, 83659, 250991, 752988, 2258980, 6776957, 20330889, 60992686, 182978078, 548934255

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n - 1] + b[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, 18}]  (* A295133 *)
Table[b[n], {n, 0, 10}]

A295134 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 1, 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, 9, 31, 98, 300, 907, 2730, 8200, 24611, 73845, 221548, 664658, 1993989, 5981983, 17945966, 53837916, 161513767

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n - 1] + b[n - 1] -1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295134 *)
Table[b[n], {n, 0, 10}]

A295135 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 2, 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, 27, 85, 260, 787, 2369, 7116, 21358, 64085, 192267, 576814, 1730456, 5191383, 15574165, 46722512, 140167554

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n - 1] + b[n - 1] -2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295135 *)
Table[b[n], {n, 0, 10}]

A295136 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 3, 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, 7, 23, 72, 221, 669, 2014, 6050, 18159, 54487, 163472, 490428, 1471297, 4413905, 13241730, 39725206, 119175635, 357526923

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n - 1] + b[n - 1] - 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295136 *)
Table[b[n], {n, 0, 10}]

A295137 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 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, 26, 80, 242, 729, 2190, 6573, 19722, 59169, 177510, 532533, 1597602, 4792809, 14378430, 43135293, 129405882

Offset: 0

Clark Kimberling, Nov 19 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

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

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

Cf. A295053.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n - 1] + b[n - 1] - n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295137 *)
Table[b[n], {n, 0, 10}]

cp's OEIS Frontend

A295061 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295062 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295063 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295065 Solution of the complementary equation a(n) = 8*a(n-3) + b(n-2), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295070 Solution of the complementary equation a(n) = a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295133 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295134 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295135 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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