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A296259 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2, where a(0) = 2, a(1) = 3, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences. 2
2, 3, 6, 25, 56, 130, 250, 461, 811, 1393, 2348, 3910, 6454, 10589, 17299, 28177, 45800, 74338, 120538, 195317, 316339, 512185, 829100, 1341961, 2171790, 3514535, 5687166, 9202601, 14890728, 24094353, 38986170, 63081679, 102069074, 165152049, 267222492 (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.

FORMULA

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(0)^2 + f(n-2)*b(1)^2 + ... + f(2)*b(n-3)^2 + f(1)*b(n-2)^2, where f(n) = A000045(n), the n-th Fibonacci number.

EXAMPLE

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

a(2) = a(0) + a(1) + b(0)^2 = 6;

Complement: (b(n)) = (1, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)

MATHEMATICA

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

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

j = 1; While[j < 6 , 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}]   (* A296259 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A099000 A032540 A063728 * A000341 A144857 A090445

Adjacent sequences:  A296256 A296257 A296258 * A296260 A296261 A296262

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 11 2017

STATUS

approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)