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A294543 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 9, 18, 35, 62, 107, 181, 301, 496, 812, 1324, 2153, 3495, 5667, 9183, 14872, 24078, 38974, 63077, 102077, 165181, 267286, 432496, 699812, 1132339, 1832183, 2964555, 4796772, 7761362, 12558170, 20319570, 32877779, 53197389, 86075209, 139272640 (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 A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number");
a(2) = a(1) + a(0) + b(1) + 2 = 9.
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...).
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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}] (* A294543 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A103256 A028881 A294535 * A295956 A296843 A200085
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 04 2017
STATUS
approved

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Last modified February 28 16:48 EST 2024. Contains 370400 sequences. (Running on oeis4.)