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A296289
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 3, 12, 30, 66, 131, 245, 439, 764, 1302, 2196, 3652, 6028, 9888, 16154, 26312, 42770, 69422, 112570, 182410, 295440, 478354, 774344, 1253296, 2028288, 3282284, 5311326, 8594447, 13906669, 22502073, 36409762, 58912920, 95323834, 154237975, 249563101
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) + 2*b(1) = 12
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, ...)
MATHEMATICA
a[0] = 1; a[1] = 3; b[0] = 2; 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}]; (* A296289 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A048088 A064181 A281434 * A089143 A073952 A107231
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 14 2017
STATUS
approved