login
A293765
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
41
1, 3, 10, 20, 38, 67, 115, 193, 321, 528, 864, 1408, 2289, 3715, 6023, 9758, 15802, 25583, 41409, 67017, 108452, 175496, 283976, 459501, 743507, 1203039, 1946578, 3149650, 5096262, 8245947, 13342245, 21588229, 34930512, 56518780, 91449333, 147968155
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) + 2 = 10;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 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] + 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}] (* A293765 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A001622 (golden ratio), A293765.
Sequence in context: A246520 A338631 A272764 * A295953 A005997 A213850
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 29 2017
STATUS
approved