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A296293 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 13, 33, 74, 147, 275, 492, 855, 1455, 2450, 4070, 6712, 11003, 17967, 29255, 47542, 77154, 125092, 202683, 328255, 531463, 860290, 1392374, 2253336, 3646435, 5900551, 9547823, 15449270, 24998079, 40448399, 65447594, 105897177, 171346025, 277244528 (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 = golden ratio (A001622). See A296245 for a guide to related sequences.
LINKS
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + 2*b(2) = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, ...)
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n];
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}]; (* A296293 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A100512 A051474 A062708 * A190816 A084910 A124024
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 14 2017
STATUS
approved

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Last modified April 14 05:31 EDT 2024. Contains 371655 sequences. (Running on oeis4.)