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A296265 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences. 2
3, 4, 9, 23, 62, 127, 245, 452, 807, 1391, 2354, 3927, 6491, 10658, 17421, 28385, 46148, 74913, 121481, 196856, 318865, 516321, 835836, 1352859, 2189451, 3543122, 5733443, 9277495, 15011930, 24290481, 39303533, 63595204, 102899997, 166496533, 269397936 (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..999

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

EXAMPLE

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

a(2) = a(0) + a(1) + b(0)*b(1) = 9;

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

MATHEMATICA

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

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

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}];  (* A296265 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A089243 A299123 A245455 * A034921 A038222 A038629

Adjacent sequences:  A296262 A296263 A296264 * A296266 A296267 A296268

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 12 2017

STATUS

approved

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Last modified May 25 19:44 EDT 2019. Contains 323576 sequences. (Running on oeis4.)