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A296277 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
3, 4, 17, 51, 110, 217, 399, 706, 1215, 2053, 3424, 5659, 9293, 15192, 24773, 40307, 65460, 106187, 172109, 278802, 451463, 730865, 1182978, 1914545, 3098279, 5013636, 8112785, 13127351, 21241128, 34369535, 55611785, 89982510, 145595555, 235579397, 381176358 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

a(2) = a(0) + a(1) + b(1)*b(2) = 17

Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, ...)

MATHEMATICA

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; 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}]; (* A296277 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A257330 A115388 A187995 * A303480 A254200 A009208

Adjacent sequences:  A296274 A296275 A296276 * A296278 A296279 A296280

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 13 2017

STATUS

approved

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Last modified December 1 04:32 EST 2021. Contains 349426 sequences. (Running on oeis4.)