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A296262 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*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, 71, 143, 270, 485, 845, 1450, 2451, 4083, 6744, 11067, 18083, 29456, 47881, 77717, 126018, 204197, 330721, 535470, 866791, 1402911, 2270404, 3674071, 5945287, 9620257, 15566536, 25187849, 40755507, 65944546, 106701313, 172647191, 279349910 (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
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3,
a(2) = a(0) + a(1) + b(0)*b(1) = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)
MATHEMATICA
a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n - 2];
j = 1; While[j < 10, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296262 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A224215 A308082 A099065 * A102281 A269083 A026583
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 11 2017
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)