A296268 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 4, 15, 37, 87, 172, 322, 574, 995, 1689, 2827, 4684, 7719, 12641, 20648, 33612, 54620, 88631, 143691, 232805, 377024, 610404, 988052, 1599131, 2587911, 4187825, 6776576, 10965300, 17742836, 28709159, 46453083, 75163397, 121617704, 196782431, 318401539

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

Mathematica

a[0] = 1; a[1] = 4; b[0] = 2; 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}];  (* A296268 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A212974 A033813 A296295 * A209409 A241302 A112666

Adjacent sequences: A296265 A296266 A296267 * A296269 A296270 A296271

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved