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A294418 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 2
1, 3, 12, 28, 56, 103, 181, 309, 518, 858, 1411, 2309, 3763, 6118, 9930, 16100, 26085, 42243, 68389, 110696, 179152, 289918, 469143, 759137, 1228359, 1987579, 3216026, 5203696, 8419816, 13623609, 22043525, 35667237, 57710868, 93378214, 151089194, 244467523 (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
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) + 2*b(0) = 12
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 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] + b[n - 1] + 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, 40}] (* A294418 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A237426 A066643 A140065 * A308669 A115549 A005995
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 31 2017
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)