A297011 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) - b(n), 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, 9, 17, 36, 81, 188, 446, 1068, 2569, 6192, 14938, 36052, 87024, 210081, 507166, 1224392, 2955928, 7136225, 17228354, 41592908, 100414144, 242421169, 585256454, 1412934048, 3411124520, 8235183057, 19881490602, 47998164228, 115877819024, 279753802241

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(2). See A296245 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

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) = 2*a(1) + a(0) - b(2) = 9
Complement: (b(n)) = (1, 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 19, ...)

Mathematica

a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n];
j = 1; While[j < 9, 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}]; (* A297011 *)
Table[b[n], {n, 0, 25}] (* complement *)
Take[u, 30]

Crossrefs

Cf. A297012, A297013.

Sequence in context: A351938 A018095 A003217 * A337780 A178717 A006723

Adjacent sequences: A297008 A297009 A297010 * A297012 A297013 A297014

Keyword

nonn,easy

Author

Clark Kimberling, Jan 13 2018

Status

approved