A295964 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 4, 9, 18, 33, 58, 100, 168, 279, 459, 751, 1224, 1990, 3230, 5238, 8487, 13745, 22253, 36020, 58296, 94340, 152661, 247027, 399715, 646770, 1046514, 1693314, 2739859, 4433206, 7173099, 11606340, 18779475, 30385852, 49165365, 79551256, 128716661, 208267958

Offset

0,2

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) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 9
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, ...)

Mathematica

a[0] = 1; a[1] = 4; b[0] = 2; 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}];  (* A295964 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A266340 A266339 A192760 * A292765 A357282 A301157

Adjacent sequences: A295961 A295962 A295963 * A295965 A295966 A295967

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved