A296245 -id:A296245 - cp's OEIS frontend

A296276 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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.

2, 4, 21, 55, 118, 229, 419, 738, 1267, 2137, 3560, 5879, 9649, 15768, 25689, 41763, 67794, 109937, 178171, 288614, 467337, 756551, 1224538, 1981791, 3207085, 5189688, 8397643, 13588261, 21986896, 35576213, 57564231, 93141634, 150707125, 243850091, 394558622

Offset: 0

Clark Kimberling, Dec 13 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(1)*b(2) = 21
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)

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 - 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}]; (* A296276 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296277 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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, 17, 51, 110, 217, 399, 706, 1215, 2053, 3424, 5659, 9293, 15192, 24773, 40307, 65460, 106187, 172109, 278802, 451463, 730865, 1182978, 1914545, 3098279, 5013636, 8112785, 13127351, 21241128, 34369535, 55611785, 89982510, 145595555, 235579397, 381176358

Offset: 0

Clark Kimberling, Dec 13 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(1)*b(2) = 17
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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] = 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] 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}]; (* A296277 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296279 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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, 44, 167, 421, 924, 1849, 3493, 6332, 11145, 19193, 32522, 54445, 90327, 148852, 244075, 398741, 649656, 1056377, 1715273, 2782276, 4509693, 7305769, 11831062, 19154381, 31005099, 50181404, 81210863, 131419237, 212659860, 344111833, 556807597, 900958700

Offset: 0

Clark Kimberling, Dec 13 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(0)*b(1)*b(2) = 44
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 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] = 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 - 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}]; (* A296279 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296280 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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, 35, 129, 374, 839, 1717, 3276, 5983, 10579, 18278, 31041, 52049, 86450, 142579, 233925, 382318, 623083, 1013381, 1645704, 2669711, 4327559, 7011070, 11354229, 18382849, 29756734, 48161507, 77942601, 126131078, 204103439, 330267253, 534406596, 864714241

Offset: 0

Clark Kimberling, Dec 13 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(1)*b(2) = 35
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 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] = 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 - 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}]; (* A296280 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296281 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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, 148, 383, 867, 1754, 3341, 6085, 10746, 18547, 31477, 52754, 87591, 144425, 236912, 387151, 630903, 1026034, 1666177, 2702837, 4381158, 7098347, 11496353, 18614356, 30132633, 48771349, 78930952, 127732061, 206695749, 334463714, 541198733, 875705287

Offset: 0

Clark Kimberling, Dec 13 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(1)*b(2) = 25
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 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] = 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 - 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}]; (* A296281 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296282 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)b(n-1)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, 21, 115, 346, 797, 1647, 3164, 5801, 10285, 17802, 30271, 50803, 84434, 139317, 228647, 373778, 609265, 991403, 1610788, 2614335, 4238923, 6868858, 11125331, 18013845, 29161100, 47199305, 76387375, 123616440, 200036551, 323688895, 523764716, 847496451

Offset: 0

Clark Kimberling, Dec 13 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(1)*b(2) = 25
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)

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 - 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}]; (* A296282 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296283 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)b(n-1)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, 17, 81, 308, 725, 1537, 2982, 5509, 9811, 17036, 29031, 48797, 81188, 134305, 220965, 362110, 591055, 962405, 1564086, 2538635, 4116521, 6670756, 10804827, 17495239, 28321990, 45841589, 74190549, 120061898, 194285183, 314382985, 508707438, 823133263

Offset: 0

Clark Kimberling, Dec 13 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(1)*b(2) = 17
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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] = 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 - 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}]; (* A296283 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296285 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 11, 25, 56, 111, 209, 376, 657, 1123, 1900, 3166, 5234, 8595, 14053, 22903, 37244, 60470, 98074, 158943, 257457, 416883, 674868, 1092349, 1767865, 2860914, 4629533, 7491257, 12121658, 19613843, 31736491, 51351388, 83088999, 134441575, 217531832

Offset: 0

Clark Kimberling, Dec 13 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) = 4, b(1) = 3, b(2) = 5
a(2) = a(0) + a(1) + 2*b(0) = 11
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 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] = 1; a[1] = 2; b[0] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]; (* A296285 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296286 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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, 8, 23, 51, 104, 197, 364, 641, 1104, 1865, 3112, 5145, 8452, 13821, 22528, 36637, 59488, 96485, 156372, 253297, 410173, 664020, 1074791, 1739459, 2814950, 4555163, 7370923, 11926954, 19298805, 31226749, 50526608, 81754477, 132282273, 214038008

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(0) = 8
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 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] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]; (* A296286 *)
Table[b[n], {n, 0, 20}]    (* complement *)

A296287 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 2, a(1) = 3, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

2, 3, 7, 22, 49, 101, 198, 362, 640, 1101, 1861, 3105, 5134, 8434, 13792, 22481, 36561, 59365, 96286, 156050, 252796, 409350, 662696, 1072644, 1735988, 2809332, 4546074, 7356216, 11903158, 19260302, 31164450, 50425806, 81591376, 132018370, 213611004

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(0) = 7
Complement: (b(n)) = (1, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ...)

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;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]; (* A296287 *)
Table[b[n], {n, 0, 20}]    (* complement *)

cp's OEIS Frontend

A296276 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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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A296277 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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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A296279 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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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A296280 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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.

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A296281 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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.

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A296282 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*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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A296283 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*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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A296285 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296279 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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.

A296280 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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.

A296281 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(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.

A296282 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)b(n-1)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.

A296283 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)b(n-1)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.