A295954 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, 12, 23, 43, 75, 128, 214, 354, 582, 951, 1549, 2517, 4084, 6620, 10724, 17365, 28111, 45499, 73635, 119160, 192822, 312010, 504861, 816901, 1321793, 2138726, 3460552, 5599312, 9059899, 14659247, 23719183, 38378468, 62097690, 100476198, 162573929

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 = 12
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 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}];  (* A295954 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A173650 A303030 A291404 * A364753 A052416 A059322

Adjacent sequences: A295951 A295952 A295953 * A295955 A295956 A295957

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved