A296256 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

3, 4, 11, 40, 87, 176, 327, 584, 1011, 1739, 2919, 4854, 7998, 13108, 21395, 34827, 56583, 91810, 148834, 241128, 390491, 632195, 1023311, 1656182, 2680222, 4337188, 7018251, 11356339, 18375551, 29732914, 48109554, 77843624, 125954403, 203799323, 329755095

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..1000

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

Formula

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.

Example

a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2;
a(2) = a(0) + a(1) + b(1)^2 = 11;
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)

Mathematica

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
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}]   (* A296256 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A259845 A037185 A299047 * A351510 A101982 A041947

Adjacent sequences: A296253 A296254 A296255 * A296257 A296258 A296259

Keyword

nonn,easy

Author

Clark Kimberling, Dec 11 2017

Status

approved