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A294558 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 4
1, 2, 14, 31, 64, 118, 209, 358, 602, 999, 1644, 2690, 4386, 7133, 11580, 18778, 30427, 49278, 79782, 129141, 209008, 338238, 547339, 885674, 1433114, 2318893, 3752116, 6071122, 9823356, 15894601, 25718084, 41612816, 67331035, 108943990, 176275168, 285219305 (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 = 11

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, ...)

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] + 2*n;

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

Table[a[n], {n, 0, 18}] (* A294558 *)

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

CROSSREFS

Cf. A001622, A294532.

Sequence in context: A075490 A031301 A335200 * A202638 A226565 A231050

Adjacent sequences: A294555 A294556 A294557 * A294559 A294560 A294561

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 16 2017

EXTENSIONS

Definition corrected by Georg Fischer, Sep 27 2020

STATUS

approved

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Last modified February 4 13:27 EST 2023. Contains 360055 sequences. (Running on oeis4.)