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A295965 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
2, 3, 9, 17, 32, 56, 97, 163, 271, 446, 730, 1190, 1935, 3142, 5095, 8256, 13371, 21648, 35041, 56712, 91777, 148514, 240317, 388858, 629203, 1018090, 1647323, 2665445, 4312801, 6978280, 11291116, 18269432, 29560585, 47830055, 77390679, 125220774, 202611494 (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 A295862 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..2000

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

EXAMPLE

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

b(3) = 6 (least "new number")

a(2) = a(1) + a(0) + b(2) - 1 = 9

Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, ...)

MATHEMATICA

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

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

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}];  (* A295965 *)

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A272057 A056658 A302164 * A034467 A234646 A065965

Adjacent sequences:  A295962 A295963 A295964 * A295966 A295967 A295968

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified March 24 06:56 EDT 2019. Contains 321444 sequences. (Running on oeis4.)