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A295364
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2), where a(0) = 1, a(1) = 3, a[2] = 5, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 3, 5, 32, 79, 167, 318, 575, 1003, 1710, 2869, 4761, 7840, 12841, 20953, 34100, 55395, 89875, 145690, 236027, 382223, 618802, 1001625, 1621077, 2623404, 4245237, 6869453, 11115560, 17985943, 29102526, 47089591
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295357 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
FORMULA
a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
EXAMPLE
a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, so that
b(2) = 6 (least "new number")
a(3) = a(2) + a(1) + b(2)*b(1) = 32
Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 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}]]];
z = 32; u = Table[a[n], {n, 0, z}] (* A295364 *)
v = Table[b[n], {n, 0, 10}] (* complement *)
CROSSREFS
Sequence in context: A318554 A109173 A179387 * A199774 A235267 A284379
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 21 2017
STATUS
approved