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A294561
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 2, 14, 30, 61, 111, 195, 332, 556, 920, 1511, 2469, 4023, 6539, 10612, 17204, 27872, 45135, 73069, 118269, 191406, 309746, 501226, 811049, 1312355, 2123487, 3435928, 5559506, 8995529, 14555133, 23550763, 38106000, 61656870, 99762980, 161419963, 261183059
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) + 2*b(1) + b(0) = 14
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, ...)
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] + 2 b[n - 1] + 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}] (* A294561 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A075718 A259710 A093794 * A075490 A031301 A335200
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 15 2017
STATUS
approved