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A296250
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
2
3, 4, 32, 72, 153, 289, 523, 912, 1556, 2612, 4337, 7145, 11707, 19108, 31104, 50536, 82001, 132937, 215379, 348800, 564708, 914084, 1479417, 2394177, 3874323, 6269284, 10144448, 16414632, 26560041, 42975762, 69536959, 112513946, 182052201, 294567516
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.
FORMULA
a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2)^2 + f(n-2)*b(3)^2 + ... + f(2)*b(n-1)^2 + f(1)*b(n)^2, where f(n) = A000045(n), the n-th Fibonacci number.
EXAMPLE
a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5;
a(2) = a(0) + a(1) + b(2)^2 = 32;
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)
MATHEMATICA
a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]^2;
j = 1; While[j < 6 , 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}] (* A296250 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A219679 A042431 A269724 * A032834 A025140 A246015
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 10 2017
STATUS
approved