A296245 -id:A296245 - cp's OEIS frontend

A296264 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 9, 28, 67, 137, 260, 477, 847, 1456, 2459, 4097, 6766, 11103, 18141, 29550, 48033, 77963, 126416, 204841, 331763, 537156, 869519, 1407325, 2277546, 3685654, 5964070, 9650654, 15615716, 25267426, 40884264, 66152880, 107038404, 173192616, 280232426

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n - 2];
j = 1; While[j < 10, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296264 *)
Table[b[n], {n, 0, 20}]     (* complement *)

A296265 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

3, 4, 9, 23, 62, 127, 245, 452, 807, 1391, 2354, 3927, 6491, 10658, 17421, 28385, 46148, 74913, 121481, 196856, 318865, 516321, 835836, 1352859, 2189451, 3543122, 5733443, 9277495, 15011930, 24290481, 39303533, 63595204, 102899997, 166496533, 269397936

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..999
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n - 2];
j = 1; While[j < 10, 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}];  (* A296265 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296267 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 14, 41, 90, 179, 332, 591, 1022, 1733, 2898, 4811, 7917, 12983, 21188, 34494, 56042, 90935, 147417, 238835, 386780, 626190, 1013594, 1640459, 2654781, 4296023, 6951644, 11248566, 18201170, 29450759, 47653017, 77104931, 124759172, 201865398, 326625938

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] b[n];
j = 1; While[j < 10, 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}];  (* A296267 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296268 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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, 15, 37, 87, 172, 322, 574, 995, 1689, 2827, 4684, 7719, 12641, 20648, 33612, 54620, 88631, 143691, 232805, 377024, 610404, 988052, 1599131, 2587911, 4187825, 6776576, 10965300, 17742836, 28709159, 46453083, 75163397, 121617704, 196782431, 318401539

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

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 - 2] b[n];
j = 1; While[j < 10, 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}];  (* A296268 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296269 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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, 10, 37, 82, 167, 312, 567, 987, 1697, 2852, 4744, 7820, 12819, 20927, 34069, 55356, 89824, 145620, 235927, 382075, 618577, 1001276, 1620528, 2622532, 4243843, 6867215, 11111957, 17980132, 29093112, 47074332, 76168599, 123244155, 199414084, 322659643

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

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 - 2] b[n];
j = 1; While[j < 10, 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}];  (* A296269 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296270 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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.

Original entry on oeis.org

2, 4, 11, 33, 79, 160, 302, 542, 952, 1624, 2744, 4563, 7531, 12349, 20168, 32840, 53368, 86607, 140415, 227505, 368448, 596528, 965600, 1562803, 2529131, 4092717, 6622688, 10716304, 17339952, 28057310, 45398382, 73456916, 118856593, 192314877, 311172913

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

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 - 2] b[n];
j = 1; While[j < 10, 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}];  (* A296270 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296271 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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, 12, 28, 75, 151, 289, 520, 908, 1558, 2620, 4373, 7217, 11845, 19350, 31518, 51228, 83145, 134813, 218441, 353782, 572798, 927204, 1500677, 2428635, 3930122, 6359656, 10290738, 16651417, 26943243, 43595815, 70540282, 114137392, 184679042, 298817877

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

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 - 2] b[n];
j = 1; While[j < 10, 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}];  (* A296271 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296273 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 3, 24, 57, 123, 236, 431, 757, 1298, 2187, 3641, 6010, 9861, 16111, 26244, 42661, 69247, 112288, 181955, 294705, 477166, 772446, 1250262, 2023410, 3274428, 5298650, 8573948, 13873528, 22448468, 36323052, 58772642, 95096884, 153870786, 248969002, 402841194

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(1)*b(2) = 24;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];
j = 1; While[j < 10, 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}];  (* A296273 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296274 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), 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, 20, 54, 116, 226, 414, 730, 1254, 2116, 3526, 5824, 9560, 15624, 25456, 41386, 67184, 108969, 176615, 286090, 463257, 749947, 1213854, 1964503, 3179113, 5144428, 8324411, 13469769, 21795172, 35265997, 57062291, 92329478, 149393029, 241723839, 391118274

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

			a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5;
a(2) = a(0) + a(1) + b(1)*b(2) = 20;
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

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] b[n];
j = 1; While[j < 10, 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}];  (* A296274 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296275 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), 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, 25, 58, 125, 239, 436, 765, 1311, 2208, 3675, 6065, 9950, 16255, 26477, 43038, 69857, 113275, 183552, 297289, 481347, 779188, 1261159, 2041049, 3302964, 5344825, 8648659, 13994414, 22644065, 36639535, 59284722, 95925447, 155211429, 251138208, 406351043

Offset: 0

Clark Kimberling, Dec 12 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 A296245 for a guide to related sequences.

			a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(1)*b(2) = 25;
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245.

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] b[n];
j = 1; While[j < 10, 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}];  (* A296275 *)
Table[b[n], {n, 0, 20}]  (* complement *)

cp's OEIS Frontend

A296264 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296265 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296267 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296268 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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.

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A296269 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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.

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A296270 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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.

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A296271 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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.

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A296273 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

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