login
A296290
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 4, 11, 30, 65, 130, 243, 436, 759, 1303, 2192, 3649, 6021, 9878, 16137, 26285, 42726, 69351, 112455, 182224, 295139, 477867, 773556, 1252021, 2026225, 3278946, 5305925, 8585708, 13892529, 22479194, 36372743, 58853022, 95226917, 154081160, 249309369
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) = 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] = 1; a[1] = 4; b[0] = 2; b[1] = 3;
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}]; (* A296290 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A110579 A361218 A024829 * A224215 A308082 A099065
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 14 2017
STATUS
approved