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A295140 Solution of the complementary equation a(n) = 3*a(n-2) - b(n-2) + 4, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 3, 5, 9, 13, 24, 35, 66, 98, 190, 284, 559, 840, 1664, 2506, 4977, 7502, 14914, 22488, 44723, 67443, 134147, 202306, 402417, 606893, 1207225, 1820652, 3621647, 5461927, 10864911, 16385749, 32594700, 49157213, 97784065, 147471603, 293352158 (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 A295053 for a guide to related sequences.

The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.40..., 2.13...

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

a(2) =3*a(0) - b(0) + 4 = 5

Complement: (b(n)) = (2, 4, 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; b[1]=4;

a[n_] := a[n] = 3 a[n - 2] + b[n - 2] + 4;

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

Table[a[n], {n, 0, 18}]  (* A295140 *)

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

CROSSREFS

Cf. A295053.

Sequence in context: A001993 A284829 A153263 * A144933 A133033 A145365

Adjacent sequences:  A295137 A295138 A295139 * A295141 A295142 A295143

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 19 2017

STATUS

approved

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Last modified March 18 12:10 EDT 2019. Contains 321283 sequences. (Running on oeis4.)