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A294555 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 13, 27, 54, 97, 169, 286, 477, 787, 1290, 2106, 3428, 5568, 9032, 14638, 23710, 38390, 62144, 100580, 162772, 263402, 426226, 689682, 1115965, 1805707, 2921734, 4727505, 7649305, 12376878, 20026253, 32403203, 52429530, 84832809, 137262417, 222095306 (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) + 3 = 13

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 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] + 3;

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

Table[a[n], {n, 0, 40}] (* A294555 *)

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

CROSSREFS

Cf. A001622, A294532.

Sequence in context: A274869 A018400 A091052 * A031090 A358296 A294556

Adjacent sequences: A294552 A294553 A294554 * A294556 A294557 A294558

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 15 2017

EXTENSIONS

Definition corrected by Georg Fischer, Sep 27 2020

STATUS

approved

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Last modified December 9 14:59 EST 2022. Contains 358700 sequences. (Running on oeis4.)