A294170 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2*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, 12, 26, 53, 97, 171, 292, 490, 813, 1337, 2187, 3564, 5794, 9404, 15247, 24703, 40005, 64766, 104832, 169662, 274561, 444294, 718929, 1163300, 1882309, 3045692, 4928087, 7973868, 12902047, 20876010, 33778155, 54654266, 88432525, 143086898, 231519533

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..2000

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(2) + 4 = 12
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 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] + 2 n;
j = 1; While[j < 16, 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}];  (* A294170 *)

Crossrefs

Cf. A296245, A296491, A296492.

Sequence in context: A098707 A152811 A294552 * A102960 A166151 A154149

Adjacent sequences: A294167 A294168 A294169 * A294171 A294172 A294173

Keyword

nonn,easy

Author

Clark Kimberling, Feb 10 2018

Status

approved