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A296277
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
2
3, 4, 17, 51, 110, 217, 399, 706, 1215, 2053, 3424, 5659, 9293, 15192, 24773, 40307, 65460, 106187, 172109, 278802, 451463, 730865, 1182978, 1914545, 3098279, 5013636, 8112785, 13127351, 21241128, 34369535, 55611785, 89982510, 145595555, 235579397, 381176358
OFFSET
0,1
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5
a(2) = a(0) + a(1) + b(1)*b(2) = 17
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, ...)
MATHEMATICA
a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];
j = 1; While[j < 10, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]; (* A296277 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A257330 A115388 A187995 * A303480 A254200 A009208
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 13 2017
STATUS
approved