A296266 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 18, 44, 97, 189, 349, 618, 1066, 1804, 3013, 4985, 8193, 13402, 21850, 35556, 57746, 93701, 151887, 246071, 398486, 645132, 1044242, 1690049, 2735019, 4425851, 7161710, 11588460, 18751130, 30340613, 49092831, 79434599, 128528654, 207964548, 336494570

Offset

0,2

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) = 1, a(1) = 2, b(0) = 3, b(1) = 1, b(2) = 4, b(3) = 5;
a(2) = a(0) + a(1) + b(0)*b(2) = 18;
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, ...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] b[n];
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}];  (* A296266 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A098857 A189333 A072278 * A365493 A280316 A131538

Adjacent sequences: A296263 A296264 A296265 * A296267 A296268 A296269

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved