A296557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 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, 6, 11, 21, 36, 61, 102, 168, 275, 448, 728, 1181, 1914, 3100, 5019, 8125, 13150, 21281, 34437, 55724, 90167, 145897, 236070, 381973, 618049, 1000028, 1618083, 2618117, 4236206, 6854330, 11090543, 17944880, 29035430, 46980317, 76015754, 122996078

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

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] - 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}];  (* A296557 *)
Table[b[n], {n, 0, 20}] (* complement *)

Crossrefs

Cf. A001622, A296245, A296493, A296567, A296568.

Sequence in context: A081691 A085571 A007684 * A241564 A135348 A305101

Adjacent sequences: A296554 A296555 A296556 * A296558 A296559 A296560

Keyword

nonn,easy

Author

Clark Kimberling, Dec 20 2017

Status

approved