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A294425 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) = 3, b(0) = 2, b(1) = 4. 2
1, 3, 9, 17, 32, 56, 96, 163, 270, 445, 728, 1187, 1930, 3133, 5082, 8234, 13336, 21591, 34949, 56563, 91536, 148124, 239686, 387837, 627551, 1015417, 1642998, 2658446, 4301478, 6959958, 11261471, 18221465, 29482973, 47704476, 77187488, 124892004, 202079533 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A294414 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

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) = 3, b(0) = 2, b(1) = 4, so that

a(2)  = a(1) + a(0) + 2*b(1) - b(0) - 1 = 9

Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14,...)

MATHEMATICA

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

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;

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}]  (* A294425 *)

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

CROSSREFS

Cf. A293076, A293765, A294414.

Sequence in context: A349489 A049778 A270105 * A123325 A239206 A116688

Adjacent sequences:  A294422 A294423 A294424 * A294426 A294427 A294428

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 01 2017

STATUS

approved

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Last modified May 28 07:30 EDT 2022. Contains 354112 sequences. (Running on oeis4.)