A296249 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, 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, 31, 71, 151, 286, 518, 904, 1543, 2591, 4303, 7090, 11618, 18964, 30871, 50159, 81391, 131950, 213782, 346216, 560527, 907319, 1468471, 2376466, 3845666, 6222916, 10069423, 16293239, 26363686, 42658014, 69022856, 111682095, 180706247, 292389711

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(2)^2 + f(n-2)*b(3)^2 + ... + f(2)*b(n-1)^2 + f(1)*b(n)^2, where f(n) = A000045(n), the n-th Fibonacci number.

Example

a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5;
a(2) = a(0) + a(1) + b(2)^2 = 31;
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)

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]^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}]     (* A296249 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A053037 A018294 A334488 * A181620 A220283 A339603

Adjacent sequences: A296246 A296247 A296248 * A296250 A296251 A296252

Keyword

nonn,easy

Author

Clark Kimberling, Dec 10 2017

Status

approved