A295961 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 10, 19, 35, 61, 104, 175, 290, 477, 780, 1271, 2066, 3353, 5436, 8808, 14264, 23093, 37379, 60495, 97898, 158418, 256342, 414787, 671157, 1085973, 1757160, 2843164, 4600356, 7443553, 12043944, 19487533, 31531514, 51019085, 82550638, 133569763

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 A295862 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..2000

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

Example

a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 10
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, ...)

Mathematica

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A295961 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A228705 A253772 A043330 * A219555 A263738 A011963

Adjacent sequences: A295958 A295959 A295960 * A295962 A295963 A295964

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved