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A296291
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
2
2, 3, 13, 31, 68, 134, 250, 447, 777, 1323, 2220, 3697, 6097, 10002, 16337, 26609, 43250, 70199, 113827, 184444, 298731, 483679, 782960, 1267237, 2050845, 3318782, 5370381, 8689973, 14061250, 22752180, 36814450, 59567715, 96383317, 155952253, 252336862
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) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
a(2) = a(0) + a(1) + 2*b(1) = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, ...)
MATHEMATICA
a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
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}]; (* A296291 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A092175 A317898 A317187 * A072997 A037428 A073688
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 14 2017
STATUS
approved