cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A294367 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 9, 19, 37, 67, 117, 200, 335, 555, 912, 1491, 2429, 3948, 6407, 10387, 16829, 27253, 44121, 71415, 115579, 187039, 302665, 489753, 792469, 1282275, 2074799, 3357131, 5431989, 8789181, 14221233, 23010479, 37231779, 60242328, 97474179, 157716581
Offset: 0

Views

Author

Clark Kimberling, Oct 29 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. See A293358 for a guide to related sequences.

Examples

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

Links

Crossrefs

Cf. A001622 (golden ratio), A293765.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
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] + 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}]  (* A294367 *)
Table[b[n], {n, 0, 10}]

A294368 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 11, 23, 45, 81, 141, 239, 399, 660, 1083, 1769, 2880, 4679, 7591, 12304, 19931, 32273, 52244, 84559, 136848, 221454, 358351, 579856, 938260, 1518171, 2456488, 3974718, 6431267, 10406048, 16837380, 27243495, 44080944, 71324510, 115405527, 186730112
Offset: 0

Views

Author

Clark Kimberling, Oct 29 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. See A293358 for a guide to related sequences.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + a(0) + b(1) + 3 = 11;
b(2) is the first positive integer not already seen, namely 5.
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)

Links

Crossrefs

Cf. A001622 (golden ratio), A293765.

Programs

  • Maple
    A[0]:= 1: B[0]:= 2:
A[1]:= 3: B[1]:= 4:
Av:= {$5..200}:
for n from 2 to 100 do
  A[n]:= A[n-1]+A[n-2]+B[n-1]+n+1;
  B[n]:= min(Av minus {A[n]});
  Av:= Av minus {A[n],B[n]};
od:
seq(A[i],i=0..100); # Robert Israel, Oct 29 2017
  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
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] + 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}]  (* A294368 *)
Table[b[n], {n, 0, 10}]

Extensions

Example clarified by Robert Israel, Oct 29 2017

A294382 Solution of the complementary equation a(n) = a(n-1)*b(n-2) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 5, 19, 113, 790, 6319, 56870, 568699, 6255688, 75068255, 975887314, 13662422395
Offset: 0

Views

Author

Clark Kimberling, Oct 29 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294381 for a guide to related sequences.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1)*b(0) - 1 = 5
Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)

Links

Crossrefs

Cf. A293076, A293765, A294381.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1]*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}]  (* A294382 *)
Table[b[n], {n, 0, 10}]

A294383 Solution of the complementary equation a(n) = a(n-1)*b(n-2) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 7, 29, 146, 877, 7017, 63154, 631541, 6946952, 83363425, 1083724526, 15172143365
Offset: 0

Views

Author

Clark Kimberling, Oct 29 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294381 for a guide to related sequences.

Examples

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

Links

Crossrefs

Cf. A293076, A293765, A294381.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1]*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}]  (* A294383 *)
Table[b[n], {n, 0, 10}]

A294384 Solution of the complementary equation a(n) = a(n-1)*b(n-2) - n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 4, 13, 61, 361, 2521, 20161, 181441, 1814401, 19958401, 239500801, 3353011202, 50295168017
Offset: 0

Views

Author

Clark Kimberling, Oct 29 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294381 for a guide to related sequences.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1)*b(0) - 2 = 4
Complement: (b(n)) = (2, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, ...)

Links

Crossrefs

Cf. A293076, A293765, A294381.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[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}]  (* A294384 *)
Table[b[n], {n, 0, 10}]

A294385 Solution of the complementary equation a(n) = a(n-1)*b(n-2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 8, 35, 179, 1079, 7559, 68038, 680388, 7484277, 89811334, 1167547353, 16345662954
Offset: 0

Views

Author

Clark Kimberling, Oct 29 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294381 for a guide to related sequences.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1)*b(0) + 2 = 8
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 12, 14, 15, 16, ...)

Links

Crossrefs

Cf. A293076, A293765, A294381.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[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}]  (* A294385 *)
Table[b[n], {n, 0, 10}]

A294397 Solution of the complementary equation a(n) = a(n-1) + b(n-2) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 34, 44, 55, 68, 82, 97, 113, 130, 149, 169, 190, 212, 235, 259, 284, 311, 339, 368, 398, 429, 461, 494, 528, 564, 601, 639, 678, 718, 759, 801, 844, 888, 934, 981, 1029, 1078, 1128, 1179, 1231, 1284, 1338, 1393, 1450, 1508, 1567, 1627
Offset: 0

Views

Author

Clark Kimberling, Oct 30 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A022940 for a guide to related sequences.
Apart from the first two entries this is the same as A081689. - R. J. Mathar, Oct 31 2017

Examples

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

Links

Crossrefs

Cf. A081689, A293076, A293765, A022940.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + 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}]  (* A294397 *)
Table[b[n], {n, 0, 10}]

A294398 Solution of the complementary equation a(n) = a(n-1) + b(n-2) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 7, 13, 20, 28, 38, 49, 61, 74, 88, 104, 121, 139, 158, 178, 199, 222, 246, 271, 297, 324, 352, 381, 412, 444, 477, 511, 546, 582, 619, 657, 696, 737, 779, 822, 866, 911, 957, 1004, 1052, 1101, 1151, 1203, 1256, 1310, 1365, 1421, 1478, 1536, 1595, 1655
Offset: 0

Views

Author

Clark Kimberling, Oct 30 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A022940 for a guide to related sequences.

Examples

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

Links

Crossrefs

Cf. A293076, A293765, A022940.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + 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}]  (* A294398 *)
Table[b[n], {n, 0, 10}]

A294399 Solution of the complementary equation a(n) = a(n-1) + b(n-2) + 3, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 8, 15, 23, 32, 42, 54, 67, 81, 96, 112, 129, 148, 168, 189, 211, 234, 258, 283, 310, 338, 367, 397, 428, 460, 493, 527, 563, 600, 638, 677, 717, 758, 800, 843, 887, 933, 980, 1028, 1077, 1127, 1178, 1230, 1283, 1337, 1392, 1448, 1506, 1565, 1625, 1686
Offset: 0

Views

Author

Clark Kimberling, Oct 30 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A022940 for a guide to related sequences.

Examples

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

Links

Crossrefs

Cf. A293076, A293765, A022940.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + 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}]  (* A294399 *)
Table[b[n], {n, 0, 10}]

A294400 Solution of the complementary equation a(n) = a(n-1) + b(n-2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 7, 14, 23, 34, 48, 64, 82, 102, 124, 148, 175, 204, 235, 268, 303, 340, 379, 420, 464, 510, 558, 608, 660, 714, 770, 828, 888, 950, 1015, 1082, 1151, 1222, 1295, 1370, 1447, 1526, 1607, 1690, 1775, 1862, 1951, 2043, 2137, 2233, 2331, 2431, 2533, 2637
Offset: 0

Views

Author

Clark Kimberling, Oct 31 2017

Keywords

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A022940 for a guide to related sequences.

Examples

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

Links

Crossrefs

Cf. A293076, A293765, A022940.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[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}]  (* A294400 *)
Table[b[n], {n, 0, 10}]
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