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A296281 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*b(n), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
2, 3, 25, 148, 383, 867, 1754, 3341, 6085, 10746, 18547, 31477, 52754, 87591, 144425, 236912, 387151, 630903, 1026034, 1666177, 2702837, 4381158, 7098347, 11496353, 18614356, 30132633, 48771349, 78930952, 127732061, 206695749, 334463714, 541198733, 875705287 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + b(0)*b(1)*b(2) = 25
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)
MATHEMATICA
a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-2] b[n - 1] 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}]; (* A296281 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A296275 A295328 A281169 * A226018 A094998 A208203
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 13 2017
STATUS
approved

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)