A294552 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 12, 26, 53, 97, 171, 292, 490, 812, 1336, 2186, 3563, 5793, 9403, 15246, 24702, 40004, 64765, 104831, 169661, 274560, 444292, 718927, 1163298, 1882307, 3045690, 4928085, 7973866, 12902045, 20876008, 33778153, 54654264, 88432523, 143086896, 231519531

Offset: 0

Author

Clark Kimberling, Nov 04 2017

Keywords

• nonn
• easy

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences, and a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

The sequence is also the solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2n with initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5.

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number");
a(2) = a(1) + a(0) + b(1) + b(0) + 2 = 12.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...).
		

Links

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

Crossrefs

Cf. A001622, A294532.

Programs

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + n;
j = 1; While[j < 5, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]  (* A294552 *)
Table[b[n], {n, 0, 20}] (* complement *)

Extensions

Edited by Clark Kimberling, Dec 21 2017