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A295139 Solution of the complementary equation a(n) = 3*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 6, 10, 23, 37, 77, 120, 242, 372, 739, 1130, 2232, 3406, 6713, 10236, 20158, 30728, 60495, 92206, 181509, 276643, 544553, 829956, 1633687, 2489897, 4901091, 7469722, 14703305, 22409199, 44109949, 67227632, 132329883 (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 1.52..., 1.96...
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) =3*a(0) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 15, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1]=4;
a[n_] := a[n] = 3 a[n - 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, 18}] (* A295139 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A295053.
Sequence in context: A200572 A342136 A049750 * A134016 A072297 A358908
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 19 2017
STATUS
approved

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Last modified April 14 16:48 EDT 2024. Contains 371666 sequences. (Running on oeis4.)