A295862 -id:A295862 - cp's OEIS frontend

A295960 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, 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.

1, 3, 8, 16, 30, 54, 93, 157, 261, 430, 704, 1148, 1868, 3033, 4919, 7971, 12910, 20902, 33834, 54759, 88617, 143401, 232044, 375472, 607544, 983046, 1590621, 2573699, 4164353, 6738086, 10902474, 17640596, 28543107, 46183741, 74726887, 120910668, 195637596

Offset: 0

Clark Kimberling, Dec 08 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 A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 8
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

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;
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}];  (* A295960 *)

A296220 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 10, 13, 16, 19, 22, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 82, 86, 91, 94, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

P. Majer proved that a(n)/n -> 4, that (a(n) - 4*n) is unbounded, and that a( ) is not a linear recurrence sequence; see the Math Overflow link and A297964. - Clark Kimberling, Feb 10 2018

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
a(2) = a(0)*b(1) + a(1)*b(0) = 10.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, ...).

Peter J. C. Moses, Table of n, a(n) for n = 0..9999
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
Pietro Majer, Limit associated with complementary sequences Math Overflow.

Cf. A296000.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[0]*b[n - 1] + a[1]*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 500}];  (* A296220 *)
Table[b[n], {n, 0, 20}]

A296215 Solution of the complementary equation a(n) = a(1)b(n-2) + a(2)b(n-3) + ... + a(n-1)*b(0), 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, 24, 87, 321, 1176, 4314, 15822, 58032, 212847, 780672, 2863317, 10501959, 38518662, 141277197, 518170812, 1900526031, 6970672818, 25566752964, 93772706622, 343935755925, 1261473710904, 4626782461218, 16969926331719, 62241612204120, 228287277978756

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

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

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

Cf. A296000, A296217.

a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}];  (* A296215 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296216 *)

A296217 Solution of the complementary equation a(n) = a(1)b(n-2) + a(2)b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 6, 26, 112, 484, 2088, 9008, 38862, 167658, 723308, 3120486, 13462360, 58079138, 250564260, 1080981064, 4663554414, 20119445656, 86799050160, 374467330636, 1615522076050, 6969664279584, 30068434774274, 129720849313094, 559639996988064, 2414391579204576

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

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

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

Cf. A296000, A296218.

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}];  (* A296217 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296218 *)

A295954 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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, 12, 23, 43, 75, 128, 214, 354, 582, 951, 1549, 2517, 4084, 6620, 10724, 17365, 28111, 45499, 73635, 119160, 192822, 312010, 504861, 816901, 1321793, 2138726, 3460552, 5599312, 9059899, 14659247, 23719183, 38378468, 62097690, 100476198, 162573929

Offset: 0

Clark Kimberling, Dec 08 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 A295862 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

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;
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}];  (* A295954 *)

A295955 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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, 13, 24, 45, 78, 133, 222, 367, 602, 984, 1602, 2603, 4223, 6845, 11088, 17954, 29064, 47041, 76129, 123196, 199352, 322576, 521957, 844563, 1366551, 2211146, 3577730, 5788910, 9366675, 15155621, 24522333, 39677992, 64200364, 103878396, 168078801

Offset: 0

Clark Kimberling, Dec 08 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 A295862 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

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;
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}];  (* A295955 *)

Definition corrected by Georg Fischer, Sep 27 2020

A295956 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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, 9, 18, 35, 62, 108, 182, 303, 499, 817, 1332, 2166, 3516, 5702, 9239, 14963, 24225, 39212, 63462, 102700, 166189, 268917, 435135, 704082, 1139248, 1843362, 2982643, 4826039, 7808717, 12634793, 20443548, 33078380, 53521968, 86600389, 140122399

Offset: 0

Clark Kimberling, Dec 08 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 A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) + 1 = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

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] + 1;
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}];  (* A295956 *)

A295957 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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, 11, 22, 41, 72, 123, 206, 342, 562, 919, 1497, 2433, 3948, 6400, 10368, 16789, 27179, 43992, 71196, 115214, 186437, 301679, 488145, 789854, 1278030, 2067916, 3345979, 5413929, 8759943, 14173908, 22933888, 37107834, 60041761, 97149635, 157191437

Offset: 0

Clark Kimberling, Dec 08 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 A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) + 1 = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

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;
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}];  (* A295957 *)

A295958 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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, 11, 21, 40, 70, 120, 201, 334, 549, 898, 1463, 2378, 3859, 6256, 10135, 16412, 26570, 43006, 69601, 112633, 182261, 294922, 477212, 772164, 1249407, 2021603, 3271043, 5292680, 8563758, 13856474, 22420269, 36276781, 58697089, 94973910, 153671041

Offset: 0

Clark Kimberling, Dec 08 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 A295862 for a guide to related sequences.

			a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) + 1 = 11
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

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;
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}];  (* A295958 *)

A295959 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 5, 11, 23, 42, 74, 126, 211, 350, 575, 940, 1531, 2488, 4037, 6544, 10601, 17166, 27789, 44978, 72792, 117796, 190615, 308439, 499083, 807552, 1306666, 2114250, 3420949, 5535233, 8956217, 14491486, 23447740, 37939264, 61387043, 99326347, 160713431

Offset: 0

Clark Kimberling, Dec 08 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 A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) + 1 = 11
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A000045, A295862.

a[0] = 1; a[1] = 5; b[0] = 2; b[1] = 3; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 1;
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}];  (* A295959 *)

cp's OEIS Frontend

A295960 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, 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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A296220 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296215 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

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A296217 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A295954 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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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A295955 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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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A295956 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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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A295957 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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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A296220 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

A296215 Solution of the complementary equation a(n) = a(1)b(n-2) + a(2)b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

A296217 Solution of the complementary equation a(n) = a(1)b(n-2) + a(2)b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.