A296846 Solution of the complementary equation a(n) = a(n-1) + a(n-2) - b(n-2), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

3, 5, 7, 10, 13, 17, 22, 30, 41, 59, 86, 130, 200, 312, 493, 785, 1257, 2019, 3252, 5246, 8472, 13691, 22135, 35797, 57901, 93666, 151534, 245166, 396665, 641795, 1038423, 1680180, 2718564, 4398704, 7117226, 11515887, 18633069, 30148911, 48781934, 78930798

Offset

0,1

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..999

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

Example

a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4
a(2) = a(0) + a(1) - b(0) = 7
Complement: (b(n)) = (1, 2, 4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, ...)

Mathematica

a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] - b[n - 2];
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}];  (* A296846 *)
Table[b[n], {n, 0, 20}] (* complement *)

Crossrefs

Cf. A001622, A296245, A296847.

Sequence in context: A184013 A057672 A112509 * A173137 A310020 A280433

Adjacent sequences: A296843 A296844 A296845 * A296847 A296848 A296849

Keyword

nonn,easy

Author

Clark Kimberling, Jan 12 2018

Status

approved