A293316 -id:A293316 - cp's OEIS frontend

A293076 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 6, 13, 24, 44, 76, 129, 215, 355, 582, 951, 1548, 2515, 4080, 6613, 10712, 17345, 28078, 45445, 73546, 119016, 192588, 311631, 504247, 815907, 1320184, 2136122, 3456338, 5592493, 9048865, 14641393, 23690294, 38331724, 62022056, 100353819, 162375915

Offset: 0

Clark Kimberling, Oct 28 2017

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values, which for each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
A293076:  a(n) = a(n-1) + a(n-2) + b(n-2)
A293316:  a(n) = a(n-1) + a(n-2) + b(n-2) + 1
A293057:  a(n) = a(n-1) + a(n-2) + b(n-2) + 2
A293058:  a(n) = a(n-1) + a(n-2) + b(n-2) + 3
A293317:  a(n) = a(n-1) + a(n-2) + b(n-2) - 1
A293349:  a(n) = a(n-1) + a(n-2) + b(n-2) + n
A293350:  a(n) = a(n-1) + a(n-2) + b(n-2) + 2*n
A293351:  a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1
A293357:  a(n) = a(n-1) + a(n-2) + b(n-2) + n + 1
Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

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

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

Cf. A001622 (golden ratio), A293358.

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 - 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}]  (* A293076 *)
Table[b[n], {n, 0, 10}]

Comments corrected by Georg Fischer, Sep 23 2020

A293321 The integer k that minimizes |k/2^n - tau^2|, where tau = (1+sqrt(5))/2 = golden ratio.

Original entry on oeis.org

3, 5, 10, 21, 42, 84, 168, 335, 670, 1340, 2681, 5362, 10723, 21447, 42894, 85788, 171575, 343151, 686302, 1372604, 2745208, 5490415, 10980830, 21961661, 43923322, 87846643, 175693287, 351386574, 702773148, 1405546295, 2811092590, 5622185181, 11244370361

Offset: 0

Clark Kimberling, Oct 07 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A293313, A293319, A293320.

z = 120; r = 1+GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293319 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293320 *)
Table[Round[r*2^n], {n, 0, z}]; (* A293321 *)

a(n) = (2^n*(3+sqrt(5))+1)\2; \\ Altug Alkan, Oct 08 2017

a(n) = floor(1/2 + r*2^n), where r = (3+sqrt(5))/2.

a(n) = A293319(n) if (fractional part of r*2^n) < 1/2, else a(n) = A293316(n).

A293057 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 8, 17, 32, 57, 98, 166, 276, 455, 745, 1215, 1976, 3208, 5202, 8430, 13653, 22105, 35781, 57910, 93716, 151652, 245395, 397075, 642499, 1039604, 1682134, 2721770, 4403937, 7125742, 11529715, 18655494, 30185247, 48840780, 79026067, 127866888, 206892997

Offset: 0

Clark Kimberling, Oct 28 2017

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A293076 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

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

Cf. A001622 (golden ratio), A293076.

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 - 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}]  (* A293316 *)
Table[b[n], {n, 0, 10}]

A293058 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

Original entry on oeis.org

1, 3, 9, 19, 36, 64, 110, 185, 308, 507, 830, 1353, 2200, 3571, 5790, 9381, 15192, 24596, 39812, 64433, 104271, 168731, 273030, 441790, 714850, 1156671, 1871553, 3028257, 4899844, 7928136, 12828016, 20756189, 33584243, 54340472, 87924756, 142265270

Offset: 0

Clark Kimberling, Oct 28 2017

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A293076 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + a(0) + b(0) + 3 = 9;
a(3) = a(2) + a(1) + b(1) + 1 = 19.
Complement: (b(n)) = (2,4,5,6,7,8,10,11,12,13,14,...)

Cf. A001622 (golden ratio), A293076.

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 - 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}]  (* A293316 *)
Table[b[n], {n, 0, 10}]

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A293076 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

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A293321 The integer k that minimizes |k/2^n - tau^2|, where tau = (1+sqrt(5))/2 = golden ratio.

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A293057 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

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A293058 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

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