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A296263
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
2
2, 3, 9, 32, 71, 145, 272, 497, 879, 1508, 2543, 4233, 6986, 11459, 18717, 30482, 49541, 80403, 130364, 211229, 342099, 553880, 896579, 1451109, 2348390, 3800255, 6149457, 9950582, 16100969, 26052574, 42154665, 68208429, 110364354, 178574115, 288939875
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) = 2, b(1) = 1, b(2) = 4
a(2) = a(0) + a(1) + b(0)*b(1) = 9
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)
MATHEMATICA
a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
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}]; (* A296263 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A322752 A277345 A259943 * A064020 A204442 A306523
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 11 2017
STATUS
approved