A296245 -id:A296245 - cp's OEIS frontend

A296289 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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.

1, 3, 12, 30, 66, 131, 245, 439, 764, 1302, 2196, 3652, 6028, 9888, 16154, 26312, 42770, 69422, 112570, 182410, 295440, 478354, 774344, 1253296, 2028288, 3282284, 5311326, 8594447, 13906669, 22502073, 36409762, 58912920, 95323834, 154237975, 249563101

Offset: 0

Clark Kimberling, Dec 14 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) + 2*b(1) = 12
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 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] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
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}]; (* A296289 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296290 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 4, 11, 30, 65, 130, 243, 436, 759, 1303, 2192, 3649, 6021, 9878, 16137, 26285, 42726, 69351, 112455, 182224, 295139, 477867, 773556, 1252021, 2026225, 3278946, 5305925, 8585708, 13892529, 22479194, 36372743, 58853022, 95226917, 154081160, 249309369

Offset: 0

Clark Kimberling, Dec 14 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) + 2*b(1) = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 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] = 1; a[1] = 4; b[0] = 2; b[1] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
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}]; (* A296290 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296291 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

2, 3, 13, 31, 68, 134, 250, 447, 777, 1323, 2220, 3697, 6097, 10002, 16337, 26609, 43250, 70199, 113827, 184444, 298731, 483679, 782960, 1267237, 2050845, 3318782, 5370381, 8689973, 14061250, 22752180, 36814450, 59567715, 96383317, 155952253, 252336862

Offset: 0

Clark Kimberling, Dec 14 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) + 2*b(1) = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 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] = 3; b[0] = 1; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
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}]; (* A296291 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296293 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n), 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, 13, 33, 74, 147, 275, 492, 855, 1455, 2450, 4070, 6712, 11003, 17967, 29255, 47542, 77154, 125092, 202683, 328255, 531463, 860290, 1392374, 2253336, 3646435, 5900551, 9547823, 15449270, 24998079, 40448399, 65447594, 105897177, 171346025, 277244528

Offset: 0

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

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A296245.

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] + n*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}]; (* A296293 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296294 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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, 35, 77, 152, 283, 505, 876, 1489, 2495, 4149, 6836, 11206, 18294, 29785, 48399, 78541, 127336, 206314, 334130, 540969, 875671, 1417261, 2293604, 3711590, 6005974, 9718401, 15725271, 25444629, 41170920, 66616665, 107788769, 174406688, 282196783

Offset: 0

Clark Kimberling, Dec 14 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) + 2*b(2) = 14
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, ...)

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] + n*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}]; (* A296294 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296295 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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, 80, 157, 291, 518, 897, 1523, 2550, 4227, 6969, 11417, 18638, 30340, 49298, 79995, 129689, 210121, 340290, 550936, 891798, 1443355, 2335825, 3779905, 6116510, 9897252, 16014658, 25912867, 41928545, 67842497, 109772194, 177615945, 287389465

Offset: 0

Clark Kimberling, Dec 14 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) + 2*b(2) = 15
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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] = 4; b[0] = 2; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]; (* A296295 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296296 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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, 15, 36, 79, 155, 288, 513, 889, 1510, 2529, 4193, 6914, 11328, 18494, 30107, 48921, 79385, 128702, 208524, 337706, 546755, 885033, 1432409, 2318114, 3751248, 6070142, 9822227, 15893265, 25716449, 41610734, 67328268, 108940186, 176269708, 285211220

Offset: 0

Clark Kimberling, Dec 14 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) + 2*b(2) = 15
Complement: (b(n)) = (1, 4, 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] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]; (* A296296 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296297 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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, 16, 38, 82, 160, 296, 526, 910, 1544, 2584, 4282, 7046, 11549, 18847, 30681, 49848, 80886, 131130, 212453, 344063, 557041, 901676, 1459338, 2361686, 3821749, 6184215, 10006801, 16191912, 26199670, 42392602, 68593357, 110987111, 179581689, 290570126

Offset: 0

Clark Kimberling, Dec 14 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) + 2*b(2) = 16
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, ...)

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] + n*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}]; (* A296297 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296556 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 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, 11, 23, 45, 81, 141, 239, 400, 661, 1085, 1772, 2885, 4687, 7604, 12325, 19965, 32328, 52333, 84704, 137082, 221833, 358964, 580848, 939865, 1520768, 2460690, 3981517, 6442268, 10423848, 16866181, 27290096, 44156346, 71446513, 115602932, 187049520

Offset: 0

Clark Kimberling, Dec 20 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(2) + 2 = 11
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 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, A296493, A296565, A296566.

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] + n;
j = 1; While[j < 16, 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}];  (* A296556 *)
Table[b[n], {n, 0, 20}] (* complement *)

A296557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - n, 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, 6, 11, 21, 36, 61, 102, 168, 275, 448, 728, 1181, 1914, 3100, 5019, 8125, 13150, 21281, 34437, 55724, 90167, 145897, 236070, 381973, 618049, 1000028, 1618083, 2618117, 4236206, 6854330, 11090543, 17944880, 29035430, 46980317, 76015754, 122996078

Offset: 0

Clark Kimberling, Dec 20 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) = 2, b(0) = 3, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + b(2) - 2 = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 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, A296493, A296567, A296568.

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] - n;
j = 1; While[j < 16, 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}];  (* A296557 *)
Table[b[n], {n, 0, 20}] (* complement *)

cp's OEIS Frontend

A296289 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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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A296290 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296291 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296293 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n), 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.

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A296294 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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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A296295 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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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A296296 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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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A296297 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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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