A296270 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), 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, 11, 33, 79, 160, 302, 542, 952, 1624, 2744, 4563, 7531, 12349, 20168, 32840, 53368, 86607, 140415, 227505, 368448, 596528, 965600, 1562803, 2529131, 4092717, 6622688, 10716304, 17339952, 28057310, 45398382, 73456916, 118856593, 192314877, 311172913

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.

Example

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

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] 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}];  (* A296270 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A178925 A298445 A294224 * A123439 A026164 A025191

Adjacent sequences: A296267 A296268 A296269 * A296271 A296272 A296273

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved