A294532 -id:A294532 - cp's OEIS frontend

A294534 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3.

1, 2, 8, 16, 31, 55, 95, 161, 268, 442, 724, 1181, 1921, 3119, 5059, 8198, 13278, 21498, 34799, 56321, 91145, 147492, 238664, 386184, 624877, 1011091, 1635999, 2647122, 4283155, 6930312, 11213503, 18143852, 29357393, 47501284, 76858717, 124360042, 201218801

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) + 2 = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 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] + 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294534 *)
Table[b[n], {n, 0, 10}]

A294535 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3.

Original entry on oeis.org

1, 2, 9, 18, 35, 62, 107, 180, 300, 494, 809, 1319, 2145, 3482, 5646, 9148, 14816, 23987, 38827, 62839, 101692, 164558, 266278, 430865, 697173, 1128069, 1825274, 2953376, 4778684, 7732095, 12510815, 20242947, 32753801, 52996788, 85750630, 138747460

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) + 3 = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 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] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294535 *)
Table[b[n], {n, 0, 10}]

A294536 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, 5, 10, 20, 36, 63, 107, 180, 298, 490, 801, 1305, 2121, 3442, 5580, 9040, 14640, 23701, 38363, 62087, 100474, 162586, 263086, 425699, 688813, 1114541, 1803384, 2917956, 4721372, 7639361, 12360767, 20000164, 32360968, 52361170, 84722177, 137083387

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 = 5
Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 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}]  (* A294536 *)
Table[b[n], {n, 0, 10}]

A294537 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3.

Original entry on oeis.org

1, 2, 8, 17, 34, 62, 109, 187, 314, 521, 857, 1402, 2285, 3715, 6030, 9778, 15843, 25658, 41540, 67239, 108822, 176106, 284975, 461130, 746156, 1207339, 1953550, 3160946, 5114555, 8275562, 13390180, 21665808, 35056056, 56721934, 91778062, 148500070

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) + 2 = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 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] + 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}]  (* A294537 *)
Table[b[n], {n, 0, 10}]

A294538 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2n, where a(0) = 1, a(1) = 2, b(0) = 3.

Original entry on oeis.org

1, 2, 10, 22, 45, 83, 147, 252, 424, 705, 1161, 1901, 3100, 5042, 8186, 13275, 21511, 34839, 56406, 91304, 147773, 239143, 386985, 626200, 1013260, 1639538, 2652879, 4292501, 6945467, 11238058, 18183618, 29421772, 47605489, 77027363, 124632957, 201660428

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) + 4 = 10
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 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] + 2n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294538 *)
Table[b[n], {n, 0, 10}]

A294539 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1, where a(0) = 1, a(1) = 2, b(0) = 3.

Original entry on oeis.org

1, 2, 7, 15, 30, 55, 98, 168, 283, 470, 774, 1267, 2066, 3361, 5457, 8850, 14341, 23227, 37606, 60873, 98521, 159438, 258005, 417491, 675546, 1093089, 1768689, 2861835, 4630583, 7492479, 12123125, 19615669, 31738861, 51354599, 83093531, 134448203

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, 14, 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 - 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}]  (* A294539 *)
Table[b[n], {n, 0, 10}]

A294540 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2*n - 1, where a(0) = 1, a(1) = 2, b(0) = 3.

Original entry on oeis.org

1, 2, 9, 20, 41, 76, 135, 232, 392, 652, 1075, 1761, 2873, 4674, 7590, 12310, 19949, 32311, 52316, 84686, 137064, 221815, 358947, 580833, 939854, 1520764, 2460698, 3981545, 6442329, 10423963, 16866384, 27290442, 44156924, 71447467, 115604495, 187052069

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) + 3 = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 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] + 2n - 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}]  (* A294540 *)
Table[b[n], {n, 0, 10}]

A294542 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 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, 8, 16, 31, 55, 96, 162, 270, 445, 729, 1189, 1934, 3141, 5094, 8255, 13370, 21647, 35040, 56711, 91776, 148513, 240316, 388857, 629202, 1018089, 1647322, 2665444, 4312800, 6978279, 11291115, 18269431, 29560584, 47830054, 77390678, 125220773

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 = 8.
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 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] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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}]  (* A294542 *)
Table[b[n], {n, 0, 10}]

A294543 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 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, 9, 18, 35, 62, 107, 181, 301, 496, 812, 1324, 2153, 3495, 5667, 9183, 14872, 24078, 38974, 63077, 102077, 165181, 267286, 432496, 699812, 1132339, 1832183, 2964555, 4796772, 7761362, 12558170, 20319570, 32877779, 53197389, 86075209, 139272640

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, 9, 10, 11, 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] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294543 *)
Table[b[n], {n, 0, 10}]

A294544 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 3, 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, 20, 39, 69, 119, 200, 333, 548, 897, 1462, 2377, 3858, 6255, 10134, 16411, 26569, 43005, 69600, 112632, 182260, 294921, 477211, 772163, 1249406, 2021602, 3271042, 5292679, 8563757, 13856473, 22420268, 36276780, 58697088, 94973909, 153671040

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) + 3 = 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] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294544 *)
Table[b[n], {n, 0, 10}]

cp's OEIS Frontend

A294534 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3.

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A294535 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3.

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A294536 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.

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A294537 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3.

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A294538 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2n, where a(0) = 1, a(1) = 2, b(0) = 3.

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A294539 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1, where a(0) = 1, a(1) = 2, b(0) = 3.

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A294540 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2*n - 1, where a(0) = 1, a(1) = 2, b(0) = 3.

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A294542 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 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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