A296272 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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, 23, 55, 120, 231, 423, 744, 1277, 2153, 3586, 5921, 9717, 15878, 25867, 42051, 68260, 110691, 179371, 290524, 470423, 761547, 1232620, 1994869, 3228245, 5223926, 8453041, 13677897, 22131930, 35810883, 57943935, 93756008, 151701203, 245458543, 397161152

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) = 2, b(0) = 3, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(1)*b(2) = 23;
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

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 - 1] 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}];  (* A296272 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A084237 A106928 A070934 * A031915 A247603 A102385

Adjacent sequences: A296269 A296270 A296271 * A296273 A296274 A296275

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved