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A296246 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 3
1, 3, 29, 68, 146, 278, 505, 883, 1509, 2536, 4214, 6946, 11385, 18587, 30261, 49172, 79794, 129366, 209601, 339451, 549581, 889608, 1439814, 2330098, 3770641, 6101523, 9873064, 15975548, 25849636, 41826273, 67677065, 109504563, 177182924, 286688856 (list; graph; refs; listen; history; text; internal format)
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.

FORMULA

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

EXAMPLE

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

a(2) = a(0) + a(1) + b(2) = 29.

Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...).

MATHEMATICA

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

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]^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}]   (* A296246 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A220953 A031912 A119951 * A257293 A221745 A087210

Adjacent sequences:  A296243 A296244 A296245 * A296247 A296248 A296249

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 06 2018

STATUS

approved

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)