login
A296284
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
33
1, 2, 9, 23, 52, 105, 199, 360, 639, 1098, 1857, 3098, 5123, 8416, 13763, 22434, 36485, 59242, 96087, 155728, 252255, 408487, 661292, 1070377, 1732317, 2803394, 4536465, 7340669, 11878002, 19219599, 31098591, 50319244, 81418955, 131739387, 213159600
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) = 2, b(0) = 3, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + 2*b(0) = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-2];
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}]; (* A296284 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A062445 A009304 A154118 * A376750 A115185 A349189
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 13 2017
STATUS
approved