A295053 -id:A295053 - cp's OEIS frontend

A295141 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 8, 22, 57, 142, 348, 847, 2052, 4962, 11988, 28951, 69904, 168774, 407468, 983727, 2374940, 5733626, 13842212, 33418071, 80678377, 194774849, 470228100

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, b(1) = 4
a(2) =2*a(1) + a(0) + b(0) = 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, A295142, A295143, A295144.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = 2 a[ n - 1] + 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}]  (* A295141 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 1 + sqrt(2).

A295142 Solution of the complementary equation a(n) = 2*a(n-1) + 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, 9, 25, 64, 159, 389, 945, 2289, 5534, 13369, 32285, 77953, 188206, 454381, 1096985, 2648369, 6393742, 15435873, 37265509, 89966913, 217199358, 524365653, 1265930690

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) = 3, b(0) = 2, b(1) = 4
a(2) =2*a(1) + a(0) + b(0) = 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, A295141, A295143, A295144.

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] = 2 a[ n - 1] + 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}]  (* A295142 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 1 + sqrt(2).

A295143 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 9, 25, 65, 162, 397, 966, 2340, 5658, 13669, 33010, 79704, 192434, 464589, 1121630, 2707868, 6537386, 15782661, 38102730, 91988144, 222079042

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, b(1) = 4
a(2) =2*a(1) + a(0) + b(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, A295141, A295142, A295144.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = 2 a[ n - 1] + 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}]  (* A295143 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 1 + sqrt(2).

A295144 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-1), 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, 11, 30, 77, 191, 467, 1134, 2745, 6636, 16030, 38710, 93465, 225656, 544794, 1315262, 3175337, 7665956, 18507270, 44680518, 107868329, 260417200, 628702754

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

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

Cf. A295053, A295141, A295142, A295143.

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] = 2 a[ n - 1] + 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}]  (* A295144 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 1 + sqrt(2).

A295145 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 7, 15, 34, 70, 146, 295, 597, 1198, 2404, 4813, 9635, 19277, 38564, 77136, 154283, 308575, 617162, 1234334, 2468681, 4937373, 9874760, 19749532, 39499079, 78998171, 157996358

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, b(1) = 4
a(2) = a(1) + 2*a(0) + b(0) = 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, A295146, A295147, A295148.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[ n - 1] + 2 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}]  (* A295145 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 2.

A295146 Solution of the complementary equation a(n) = a(n-1) + 2*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, 7, 17, 36, 76, 156, 317, 639, 1284, 2574, 5155, 10317, 20642, 41292, 82594, 165197, 330405, 660820, 1321652, 2643315, 5286643, 10573298, 21146610, 42293233, 84586481, 169172976

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

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

Cf. A295053, A295145, A295147, A295148.

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] = a[ n - 1] + 2 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}]  (* A295146 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 2.

A295147 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 8, 17, 39, 80, 167, 337, 682, 1368, 2745, 5495, 11000, 22006, 44024, 88055, 176123, 352254, 704522, 1409053, 2818121, 5636252, 11272520, 22545051, 45090119, 90180250, 180360518

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, b(1) = 4
a(2) = a(1) + 2*a(0) + b(1) = 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, A295145, A295146, A295148.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[ n - 1] + 2 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}]  (* A295147 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 2.

A295148 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), 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, 9, 20, 44, 91, 187, 379, 764, 1534, 3075, 6157, 12322, 24652, 49313, 98635, 197280, 394571, 789153, 1578318, 3156648, 6313309, 12626631, 25253276, 50506566, 101013147, 202026309

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

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

Cf. A295053, A295145, A295146, A295147.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[ n - 1] + 2 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}]  (* A295148 *)
Table[b[n], {n, 0, 10}]

a(n+1)/a(n) -> 2.

A295064 Solution of the complementary equation a(n) = 8*a(n-3) + b(n-1), 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, 14, 31, 48, 121, 258, 395, 980, 2077, 3175, 7856, 16633, 25418, 62867, 133084, 203365, 502958, 1064695, 1626944, 4023689, 8517586, 13015579, 32189540, 68140717, 104124662

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 1.52..., 2.11..., 2.47...

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

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; a[2] = 5; b[0] = 2;
a[n_] := a[n] = 8 a[n - 3] + 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}]  (* A295064 *)
Table[b[n], {n, 0, 10}]

A295066 Solution of the complementary equation a(n) = 2*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.

Original entry on oeis.org

1, 3, 6, 11, 19, 30, 47, 70, 106, 153, 226, 321, 468, 659, 954, 1338, 1929, 2698, 3881, 5420, 7787, 10866, 15601, 21760, 31231, 43551, 62494, 87135, 125022, 174305, 250080, 348647, 500198, 697333, 1000436, 1394707, 2000914, 2789457, 4001872, 5578959, 8003790

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 1.43..., 1.39... .

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

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

Cf. A295053, A295067.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = 2 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}]  (* A295066 *)
Table[b[n], {n, 0, 10}]

cp's OEIS Frontend

A295141 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295142 Solution of the complementary equation a(n) = 2*a(n-1) + 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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A295143 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295144 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-1), 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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Mathematica

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A295145 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

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Mathematica

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A295146 Solution of the complementary equation a(n) = a(n-1) + 2*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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A295147 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295148 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), 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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