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A293406 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 2
1, 3, 9, 18, 34, 60, 103, 174, 289, 476, 779, 1270, 2065, 3352, 5435, 8807, 14263, 23092, 37378, 60494, 97897, 158417, 256341, 414786, 671156, 1085972, 1757159, 2843163, 4600355, 7443552, 12043943, 19487532, 31531513, 51019084, 82550637, 133569762 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) + 1 = 8;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, ...)
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] + 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}] (* A293406 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A001622 (golden ratio), A293076.
Sequence in context: A093446 A256524 A210970 * A295862 A246695 A132920
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 29 2017
STATUS
approved

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Last modified July 13 11:56 EDT 2024. Contains 374282 sequences. (Running on oeis4.)