A296267 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 14, 41, 90, 179, 332, 591, 1022, 1733, 2898, 4811, 7917, 12983, 21188, 34494, 56042, 90935, 147417, 238835, 386780, 626190, 1013594, 1640459, 2654781, 4296023, 6951644, 11248566, 18201170, 29450759, 47653017, 77104931, 124759172, 201865398, 326625938

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..1000

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

Example

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

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; 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}];  (* A296267 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A117662 A196236 A213482 * A104905 A055650 A367985

Adjacent sequences: A296264 A296265 A296266 * A296268 A296269 A296270

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved