A296245 -id:A296245 - cp's OEIS frontend

A296250 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, 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.

3, 4, 32, 72, 153, 289, 523, 912, 1556, 2612, 4337, 7145, 11707, 19108, 31104, 50536, 82001, 132937, 215379, 348800, 564708, 914084, 1479417, 2394177, 3874323, 6269284, 10144448, 16414632, 26560041, 42975762, 69536959, 112513946, 182052201, 294567516

Offset: 0

Clark Kimberling, Dec 10 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(2)^2 = 32;
Complement: (b(n)) = (1, 2, 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] = 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;
j = 1; While[j < 6 , 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}]   (* A296250 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2)^2 + f(n-2)*b(3)^2 + ... + f(2)*b(n-1)^2 + f(1)*b(n)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296252 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^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, 20, 48, 104, 201, 369, 651, 1120, 1892, 3156, 5217, 8569, 14011, 22836, 37136, 60296, 97793, 158530, 256807, 415866, 673249, 1089740, 1763665, 2854134, 4618583, 7473558, 12093041, 19567560, 31661625, 51230274, 82893055, 134124554, 217018905, 351144828

Offset: 0

Clark Kimberling, Dec 11 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;
a(2) = a(0) + a(1) + b(1)^2 = 20;
Complement: (b(n)) = (2, 4, 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] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
j = 1; While[j < 6 , 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}]   (* A296252 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296253 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, 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, 14, 43, 93, 185, 342, 608, 1050, 1779, 2973, 4921, 8119, 13296, 21704, 35324, 57389, 93113, 150943, 244540, 396012, 641128, 1037765, 1679569, 2718063, 4398416, 7117320, 11516636, 18634917, 30152577, 48788583, 78942316, 127732124, 206675736, 334409229

Offset: 0

Clark Kimberling, Dec 11 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;
a(2) = a(0) + a(1) + b(1)^2 = 14;
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 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] = 4; b[0] = 2; b[1] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
j = 1; While[j < 6 , 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}]   (* A296253 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296254 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, 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, 21, 49, 106, 204, 374, 659, 1133, 1913, 3190, 5272, 8658, 14155, 23069, 37513, 60906, 98780, 160086, 259350, 419965, 679891, 1100481, 1781048, 2882258, 4664090, 7547189, 12212179, 19760329, 31973532, 51734950, 83709638, 135445813, 219156747, 354603929

Offset: 0

Clark Kimberling, Dec 11 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;
a(2) = a(0) + a(1) + b(1)^2 = 21;
Complement: (b(n)) = (1, 4, 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] = 3; b[0] = 1; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
j = 1; While[j < 6 , 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}]   (* A296254 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296255 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

2, 4, 15, 44, 95, 188, 347, 616, 1063, 1800, 3007, 4976, 8179, 13411, 21879, 35614, 57854, 93868, 152163, 246515, 399207, 646298, 1046130, 1693104, 2739963, 4433851, 7174655, 11609406, 18785022, 30395452, 49181563, 79578171, 128760959, 208340426, 337102754

Offset: 0

Clark Kimberling, Dec 11 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;
a(2) = a(0) + a(1) + b(1)^2 = 15;
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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] = 4; b[0] = 1; b[1] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
j = 1; While[j < 6 , 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}]   (* A296255 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296256 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, 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, 11, 40, 87, 176, 327, 584, 1011, 1739, 2919, 4854, 7998, 13108, 21395, 34827, 56583, 91810, 148834, 241128, 390491, 632195, 1023311, 1656182, 2680222, 4337188, 7018251, 11356339, 18375551, 29732914, 48109554, 77843624, 125954403, 203799323, 329755095

Offset: 0

Clark Kimberling, Dec 11 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;
a(2) = a(0) + a(1) + b(1)^2 = 11;
Complement: (b(n)) = (1, 2, 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] = 3; a[1] = 4; b[0] = 1; b[1] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
j = 1; While[j < 6 , 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}]   (* A296256 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296258 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^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, 27, 60, 123, 232, 436, 768, 1325, 2237, 3731, 6164, 10120, 16540, 26949, 43813, 71123, 115336, 186900, 302720, 490149, 793445, 1284219, 2078340, 3363343, 5442524, 8806767, 14250252, 23058043, 37309384, 60368583, 97679192, 158049071, 255729632

Offset: 0

Clark Kimberling, Dec 11 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;
a(2) = a(0) + a(1) + b(0)^2 = 8;
Complement: (b(n)) = (2, 4, 5, 6, 7, 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;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-2]^2;
j = 1; While[j < 6 , 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}]     (* A296258 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(0)^2 + f(n-2)*b(1)^2 + ... + f(2)*b(n-3)^2 + f(1)*b(n-2)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296259 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^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, 6, 25, 56, 130, 250, 461, 811, 1393, 2348, 3910, 6454, 10589, 17299, 28177, 45800, 74338, 120538, 195317, 316339, 512185, 829100, 1341961, 2171790, 3514535, 5687166, 9202601, 14890728, 24094353, 38986170, 63081679, 102069074, 165152049, 267222492

Offset: 0

Clark Kimberling, Dec 11 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;
a(2) = a(0) + a(1) + b(0)^2 = 6;
Complement: (b(n)) = (1, 4, 5, 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;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-2]^2;
j = 1; While[j < 6 , 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}]   (* A296259 *)
Table[b[n], {n, 0, 20}]  (* complement *)

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(0)^2 + f(n-2)*b(1)^2 + ... + f(2)*b(n-3)^2 + f(1)*b(n-2)^2, where f(n) = A000045(n), the n-th Fibonacci number.

A296262 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*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, 71, 143, 270, 485, 845, 1450, 2451, 4083, 6744, 11067, 18083, 29456, 47881, 77717, 126018, 204197, 330721, 535470, 866791, 1402911, 2270404, 3674071, 5945287, 9620257, 15566536, 25187849, 40755507, 65944546, 106701313, 172647191, 279349910

Offset: 0

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

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] = 1; a[1] = 4; b[0] = 2; 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}];  (* A296262 *)
Table[b[n], {n, 0, 20}]  (* complement *)

A296263 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) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

2, 3, 9, 32, 71, 145, 272, 497, 879, 1508, 2543, 4233, 6986, 11459, 18717, 30482, 49541, 80403, 130364, 211229, 342099, 553880, 896579, 1451109, 2348390, 3800255, 6149457, 9950582, 16100969, 26052574, 42154665, 68208429, 110364354, 178574115, 288939875

Offset: 0

Clark Kimberling, Dec 11 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) = 2, b(1) = 1, b(2) = 4
a(2) = a(0) + a(1) + b(0)*b(1) = 9
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 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;
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}];  (* A296263 *)
Table[b[n], {n, 0, 20}]      (* complement *)

cp's OEIS Frontend

A296250 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, 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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A296252 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^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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A296253 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, 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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A296254 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, 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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A296255 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, 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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A296256 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, 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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A296258 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^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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A296259 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2, where a(0) = 2, a(1) = 3, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

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