A296274 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 4, 20, 54, 116, 226, 414, 730, 1254, 2116, 3526, 5824, 9560, 15624, 25456, 41386, 67184, 108969, 176615, 286090, 463257, 749947, 1213854, 1964503, 3179113, 5144428, 8324411, 13469769, 21795172, 35265997, 57062291, 92329478, 149393029, 241723839, 391118274

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.

a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5

a(2) = a(0) + a(1) + b(1)*b(2) = 20

Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)

a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; b[2] = 5;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];

j = 1; While[j < 10, 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}]; (* A296274 *)

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

Cf. A001622, A296245.

easy

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) = 4, b(0) = 2, b(1) = 3, b(2) = 5;
a(2) = a(0) + a(1) + b(1)*b(2) = 20;
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)

Mathematica

a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];
j = 1; While[j < 10, 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}];  (* A296274 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A023667 A213485 A366151 * A035007 A047810 A002492

Adjacent sequences: A296271 A296272 A296273 * A296275 A296276 A296277

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved