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A294564 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 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, 7, 14, 27, 50, 86, 146, 243, 401, 657, 1074, 1747, 2838, 4603, 7460, 12083, 19564, 31669, 51256, 82949, 134230, 217205, 351464, 568698, 920192, 1488921, 2409145, 3898099, 6307278, 10205412, 16512726, 26718175, 43230939, 69949153, 113180132, 183129326 (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..36.

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) + 2*b(1) - b(0) - 1 = 7

Complement: (b(n)) = (3, 4, 5, 6, 8, 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] + 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}]  (* A294564 *)

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

CROSSREFS

Cf. A001622, A294532.

Sequence in context: A210728 A294533 A294541 * A068040 A200084 A184704

Adjacent sequences:  A294561 A294562 A294563 * A294565 A294566 A294567

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 15 2017

STATUS

approved

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Last modified January 22 04:27 EST 2020. Contains 331133 sequences. (Running on oeis4.)