A296265 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), 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, 9, 23, 62, 127, 245, 452, 807, 1391, 2354, 3927, 6491, 10658, 17421, 28385, 46148, 74913, 121481, 196856, 318865, 516321, 835836, 1352859, 2189451, 3543122, 5733443, 9277495, 15011930, 24290481, 39303533, 63595204, 102899997, 166496533, 269397936

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..999

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

Example

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

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] b[n - 2];
j = 1; While[j < 10, 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}];  (* A296265 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A032789 A299123 A245455 * A034921 A038222 A038629

Adjacent sequences: A296262 A296263 A296264 * A296266 A296267 A296268

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved