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A305746 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n), where a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7. See Comments. 1
1, 2, 3, 12, 30, 66, 130, 241, 429, 742, 1258, 2103, 3481, 5722, 9360, 15259, 24817, 40296, 65356, 105919, 171567, 277804, 449716, 727893, 1178011, 1906337, 3084813, 4991648, 8076993, 13069208, 21146804, 34216652, 55364134, 89581503, 144946394, 234528695 (list; graph; refs; listen; history; text; internal format)
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, the golden ratio (A001622).
LINKS
EXAMPLE
a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7, and a(3) = 2*3 - 1 + 7 = 12.
MATHEMATICA
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6; b[3] = 7;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + b[n];
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 60}] (* A305746 *)
CROSSREFS
Sequence in context: A025560 A073618 A247651 * A221510 A109489 A169636
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 10 2018
STATUS
approved

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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)