A294381 -id:A294381 - cp's OEIS frontend

A022940 a(n) = a(n-1) + b(n-2) for n >= 3, a( ) increasing, given a(1) = 1, a(2) = 3; where b( ) is complement of a( ).

1, 3, 5, 9, 15, 22, 30, 40, 51, 63, 76, 90, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 354, 383, 414, 446, 479, 513, 548, 584, 621, 659, 698, 739, 781, 824, 868, 913, 959, 1006, 1054, 1103, 1153, 1205, 1258, 1312, 1367, 1423, 1480

Offset: 1

Clark Kimberling

From Clark Kimberling, Oct 30 2017: (Start)
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:
here: a(n) = a(n-1) + b(n-2) [with a different offset]
A294397: a(n) = a(n-1) + b(n-2) + 1;
A294398: a(n) = a(n-1) + b(n-2) + 2;
A294399: a(n) = a(n-1) + b(n-2) + 3;
A294400: a(n) = a(n-1) + b(n-2) + n;
A294401: a(n) = a(n-1) + b(n-2) + 2*n.
(End)

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

Ivan Neretin, Table of n, a(n) for n = 1..10000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A005228 and references therein.

Cf. A293076, A293765, A294381.

Fold[Append[#1, #1[[-1]] + Complement[Range[Max@#1 + 1], #1][[#2]]] &, {1, 3}, Range[50]] (* Ivan Neretin, Apr 04 2016 *)

A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.

Original entry on oeis.org

1, 1, 2, 2, 0, 7, 1, 2, 9, 4, 7, 8, 7, 2, 0, 1, 9, 1, 3, 1, 3, 5, 6, 3, 9, 9, 3, 2, 1, 2, 0, 7, 4, 4, 8, 2, 2, 3, 5, 2, 3, 0, 1, 4, 9, 2, 6, 1, 9, 0, 4, 2, 5, 0, 7, 7, 3, 3, 5, 9, 0, 7, 6, 1, 3, 8, 9, 6, 1, 1, 3, 4, 2, 2, 3, 5, 4, 8, 8, 0, 1, 0, 7, 9, 7, 0

Offset: 2

Clark Kimberling, Dec 20 2017

Suppose that A = (a(n)), for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A294170, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 11.22071294787201913135639932120744822352...

Cf. A001622, A294381, A296284, A296491.

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] + 2 n;
j = 1; While[j < 16, 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}];  (* A294170 *)
z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120]   (* A296492 *)

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

Clark Kimberling, Oct 29 2017

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

			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, ...)

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

Cf. A293076, A293765, A294381.

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

Clark Kimberling, Oct 29 2017

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

			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, ...)

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

Cf. A293076, A293765, A294381.

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

Clark Kimberling, Oct 29 2017

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

			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, ...)

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

Cf. A293076, A293765, A294381.

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

Clark Kimberling, Oct 29 2017

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

			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, ...)

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

Cf. A293076, A293765, A294381.

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}]

cp's OEIS Frontend

A022940 a(n) = a(n-1) + b(n-2) for n >= 3, a( ) increasing, given a(1) = 1, a(2) = 3; where b( ) is complement of a( ).

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A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.

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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.

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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.

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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.

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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.

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