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A294562 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 5, 10, 17, 29, 48, 80, 130, 212, 344, 558, 904, 1465, 2371, 3838, 6211, 10051, 16264, 26317, 42583, 68902, 111487, 180391, 291881, 472274, 764157, 1236433, 2000592, 3237027, 5237621, 8474650, 13712273, 22186925, 35899200, 58086127 (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. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

LINKS

Table of n, a(n) for n=0..35.

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

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, so that

b(1) = 4 (least "new number")

a(2) = a(1) + a(0) + b(1) - b(0) + 1 = 5

Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, ...)

MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

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

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

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

Table[a[n], {n, 0, 40}] (* A294562 *)

Table[b[n], {n, 0, 10}]

CROSSREFS

Cf. A001622, A294532.

Sequence in context: A178137 A071602 A046485 * A109377 A109472 A172167

Adjacent sequences: A294559 A294560 A294561 * A294563 A294564 A294565

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 15 2017

STATUS

approved

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Last modified January 31 04:27 EST 2023. Contains 359947 sequences. (Running on oeis4.)