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A294367 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 2
1, 3, 9, 19, 37, 67, 117, 200, 335, 555, 912, 1491, 2429, 3948, 6407, 10387, 16829, 27253, 44121, 71415, 115579, 187039, 302665, 489753, 792469, 1282275, 2074799, 3357131, 5431989, 8789181, 14221233, 23010479, 37231779, 60242328, 97474179, 157716581 (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. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. See A293358 for a guide to related sequences.

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

a(2) = a(1) + a(0) + b(1) + 1 = 12;

Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...)

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] + b[n - 1] + n - 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}] (* A294367 *)

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

CROSSREFS

Cf. A001622 (golden ratio), A293765.

Sequence in context: A147158 A014540 A293058 * A339495 A146694 A146050

Adjacent sequences: A294364 A294365 A294366 * A294368 A294369 A294370

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Oct 29 2017

STATUS

approved

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Last modified February 8 08:49 EST 2023. Contains 360138 sequences. (Running on oeis4.)