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A294556
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
4
1, 2, 13, 28, 57, 104, 183, 312, 523, 866, 1423, 2327, 3792, 6164, 10004, 16219, 26277, 42553, 68890, 111506, 180462, 292037, 472571, 764683, 1237332, 2002097, 3239515, 5241701, 8481308, 13723104, 22204510, 35927715, 58132329, 94060151, 152192590, 246252854
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) + b(0) + 3 = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + n + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294556 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A294555 A031090 A358296 * A294559 A041017 A033837
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 15 2017
EXTENSIONS
Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018
Definition corrected by Georg Fischer, Sep 27 2020
STATUS
approved