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A293766
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 3, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
2
1, 3, 11, 22, 42, 74, 127, 213, 353, 581, 950, 1548, 2516, 4083, 6619, 10723, 17364, 28110, 45498, 73634, 119159, 192821, 312009, 504860, 816900, 1321792, 2138725, 3460551, 5599311, 9059898, 14659246, 23719182, 38378467, 62097689, 100476197, 162573928
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
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) + 3 = 11;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 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] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A293766 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A001622 (golden ratio), A293765.
Sequence in context: A129215 A139593 A121471 * A178946 A087078 A177789
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 29 2017
STATUS
approved