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A296251
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
4
1, 2, 19, 46, 101, 196, 361, 638, 1099, 1858, 3101, 5128, 8425, 13778, 22459, 36526, 59309, 96235, 155985, 252704, 409218, 662498, 1072341, 1735515, 2808585, 4544884, 7354310, 11900094, 19255365, 31156483, 50412937, 81570576, 131984738, 213556610, 345542717
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.
FORMULA
a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
a(2) = a(0) + a(1) + b(1)^2 = 19;
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, ...)
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^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}] (* A296251 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A062587 A109946 A141067 * A307554 A031911 A136685
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 10 2017
STATUS
approved