A294532 -id:A294532 - cp's OEIS frontend

A295357 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 5, 20, 42, 83, 149, 259, 438, 730, 1204, 1973, 3219, 5237, 8504, 13792, 22350, 36200, 58612, 94878, 153559, 248509, 402143, 650730, 1052954, 1703768, 2756809, 4460667, 7217569, 11678332, 18896000, 30574434, 49470539

Offset: 0

Clark Kimberling, Nov 21 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Guide to related sequences:
***** Part 1:  initial values are a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6
A295357: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3)
A295358: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3)
A295359: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2*b(n-3)
A295360: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 3*b(n-3)
A295361: a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) - 3*b(n-3)
A295362: a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) - b(n-3)
***** Part 2: initial values as shown
A295363: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
A295364: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4
A295365: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6
A295366: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6
A295367: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
For all of these sequences, a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A293076, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2; b[1] = 4; b[2] = 6;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
z = 32; u = Table[a[n], {n, 0, z}]   (* A295357 *)
v = Table[b[n], {n, 0, 10}]  (* complement *)

A294541 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, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 7, 14, 27, 49, 85, 144, 240, 396, 649, 1060, 1725, 2802, 4545, 7366, 11931, 19318, 31271, 50612, 81907, 132544, 214477, 347049, 561555, 908634, 1470220, 2378886, 3849139, 6228059, 10077233, 16305328, 26382598, 42687964, 69070601, 111758605

Offset: 0

Clark Kimberling, Nov 04 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294541 *)
Table[b[n], {n, 0, 10}]

A294546 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n, 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, 9, 19, 38, 69, 121, 207, 347, 575, 945, 1545, 2517, 4091, 6639, 10763, 17438, 28239, 45717, 73998, 119759, 193803, 313610, 507463, 821125, 1328642, 2149823, 3478523, 5628406, 9106991, 14735461, 23842518, 38578047, 62420635, 100998755, 163419465

Offset: 0

Clark Kimberling, Nov 04 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294546 *)
Table[b[n], {n, 0, 10}]

A294553 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n, 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, 8, 16, 31, 55, 96, 163, 272, 449, 736, 1201, 1954, 3174, 5149, 8345, 13517, 21886, 35428, 57340, 92795, 150163, 242987, 393180, 636198, 1029410, 1665641, 2695086, 4360764, 7055888, 11416691, 18472619, 29889351, 48362012, 78251406, 126613462

Offset: 0

Clark Kimberling, Nov 15 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294553 *)
Table[b[n], {n, 0, 10}]

A294552 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n, 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, 12, 26, 53, 97, 171, 292, 490, 812, 1336, 2186, 3563, 5793, 9403, 15246, 24702, 40004, 64765, 104831, 169661, 274560, 444292, 718927, 1163298, 1882307, 3045690, 4928085, 7973866, 12902045, 20876008, 33778153, 54654264, 88432523, 143086896, 231519531

Offset: 0

Clark Kimberling, Nov 04 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences, and a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

The sequence is also the solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2n with initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5.

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

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

Cf. A001622, A294532.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + n;
j = 1; While[j < 5, 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}]  (* A294552 *)
Table[b[n], {n, 0, 20}] (* complement *)

Edited by Clark Kimberling, Dec 21 2017

A294556 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1, 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, 13, 28, 57, 104, 183, 312, 523, 866, 1423, 2327, 3792, 6164, 10004, 16219, 26277, 42553, 68890, 111506, 180462, 292037, 472571, 764683, 1237332, 2002097, 3239515, 5241701, 8481308, 13723104, 22204510, 35927715, 58132329, 94060151, 152192590, 246252854

Offset: 0

Clark Kimberling, Nov 15 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + n + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294556 *)
Table[b[n], {n, 0, 10}]

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

Definition corrected by Georg Fischer, Sep 27 2020

A294557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1, 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, 11, 24, 49, 90, 159, 272, 457, 759, 1250, 2046, 3336, 5425, 8807, 14281, 23140, 37476, 60674, 98211, 158949, 257228, 416249, 673552, 1089879, 1763512, 2853475, 4617074, 7470639, 12087806, 19558541, 31646446, 51205089, 82851640, 134056837, 216908588

Offset: 0

Clark Kimberling, Nov 15 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + n - 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294557 *)
Table[b[n], {n, 0, 10}]

Definition corrected by Georg Fischer, Sep 27 2020

A294558 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2*n, 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, 14, 31, 64, 118, 209, 358, 602, 999, 1644, 2690, 4386, 7133, 11580, 18778, 30427, 49278, 79782, 129141, 209008, 338238, 547339, 885674, 1433114, 2318893, 3752116, 6071122, 9823356, 15894601, 25718084, 41612816, 67331035, 108943990, 176275168, 285219305

Offset: 0

Clark Kimberling, Nov 16 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + 2*n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A294558 *)
Table[b[n], {n, 0, 10}]

Definition corrected by Georg Fischer, Sep 27 2020

A294549 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1, 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, 21, 42, 76, 133, 226, 379, 627, 1030, 1683, 2741, 4454, 7227, 11715, 18978, 30731, 49750, 80524, 130319, 210890, 341258, 552199, 893510, 1445764, 2339331, 3785154, 6124546, 9909763, 16034374, 25944204, 41978647, 67922922, 109901642, 177824639

Offset: 0

Clark Kimberling, Nov 04 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + n + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294549 *)
Table[b[n], {n, 0, 10}]

A294533 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 1, where a(0) = 1, a(1) = 2, b(0) = 3.

Original entry on oeis.org

1, 2, 7, 14, 27, 48, 84, 142, 237, 391, 641, 1046, 1703, 2766, 4487, 7272, 11779, 19072, 30873, 49968, 80865, 130858, 211749, 342634, 554412, 897076, 1451519, 2348627, 3800179, 6148840, 9949054, 16097930, 26047021, 42144989, 68192049, 110337078, 178529168

Offset: 0

Clark Kimberling, Nov 03 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622)..

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

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

Cf. A001622, A294532.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294533 *)
Table[b[n], {n, 0, 10}]

cp's OEIS Frontend

A295357 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.

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A294541 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, and (a(n)) and (b(n)) are increasing complementary sequences.

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A294546 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A294553 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A294552 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A294556 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A294557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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A294558 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

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