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A294560 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 17, 37, 76, 139, 245, 418, 701, 1161, 1908, 3119, 5081, 8258, 13401, 21727, 35202, 57007, 92291, 149384, 241765, 391243, 633106, 1024451, 1657663, 2682224, 4340001, 7022343, 11362466, 18384935, 29747531, 48132600, 77880269, 126013011, 203893428 (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..34.

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(0) + a(1) + 2*b(0) + 2*b(1) = 17

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 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] + 2 b[n - 1] + 2 b[n - 2];

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

Table[a[n], {n, 0, 40}]  (* A294560 *)

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

CROSSREFS

Cf. A001622, A294532.

Sequence in context: A041965 A095075 A307161 * A258977 A069042 A121923

Adjacent sequences:  A294557 A294558 A294559 * A294561 A294562 A294563

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 15 2017

STATUS

approved

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Last modified June 19 21:25 EDT 2021. Contains 345151 sequences. (Running on oeis4.)