A296556 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 11, 23, 45, 81, 141, 239, 400, 661, 1085, 1772, 2885, 4687, 7604, 12325, 19965, 32328, 52333, 84704, 137082, 221833, 358964, 580848, 939865, 1520768, 2460690, 3981517, 6442268, 10423848, 16866181, 27290096, 44156346, 71446513, 115602932, 187049520

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

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 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++];
u = Table[a[n], {n, 0, k}];  (* A296556 *)
Table[b[n], {n, 0, 20}] (* complement *)

Crossrefs

Cf. A001622, A296245, A296493, A296565, A296566.

Sequence in context: A159791 A078723 A294368 * A141187 A107138 A145473

Adjacent sequences: A296553 A296554 A296555 * A296557 A296558 A296559

Keyword

nonn,easy

Author

Clark Kimberling, Dec 20 2017

Status

approved