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A296273
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 3, 24, 57, 123, 236, 431, 757, 1298, 2187, 3641, 6010, 9861, 16111, 26244, 42661, 69247, 112288, 181955, 294705, 477166, 772446, 1250262, 2023410, 3274428, 5298650, 8573948, 13873528, 22448468, 36323052, 58772642, 95096884, 153870786, 248969002, 402841194
OFFSET
0,2
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) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(1)*b(2) = 24;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)
MATHEMATICA
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; 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}]; (* A296273 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A160665 A211034 A220868 * A101008 A070734 A009113
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 12 2017
STATUS
approved