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

0,2

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.

Links

Table of n, a(n) for n=0..35.

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

Example

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, ...).

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 *)

Crossrefs

Cf. A001622, A294532.

Sequence in context: A294554 A098707 A152811 * A294170 A102960 A166151

Adjacent sequences: A294549 A294550 A294551 * A294553 A294554 A294555

Keyword

nonn,easy

Author

Clark Kimberling, Nov 04 2017

Extensions

Edited by Clark Kimberling, Dec 21 2017

Status

approved