A120866 a(n) is the number j for which there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 20*n^2.

1, 4, 9, 16, 4, 11, 20, 31, 5, 16, 29, 44, 4, 19, 36, 55, 1, 20, 41, 64, 89, 19, 44, 71, 100, 16, 45, 76, 109, 11, 44, 79, 116, 4, 41, 80, 121, 164, 36, 79, 124, 171, 29, 76, 125, 176, 20, 71, 124, 179, 9, 64, 121, 180, 241, 55, 116, 179, 244, 44, 109, 176, 245, 31, 100

Offset

1,2

Comments

The k's that match these j's comprise A120867.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Clark Kimberling, The equation (j+k+1)^2-4*k=Q*n^2 and related dispersions, Journal of Integer Sequences, 10 (2007), Article #07.2.7.

Formula

a(n) = 5*n^2 - floor(n*sqrt(5))^2.

Example

1 = 5*1 - floor(sqrt(5))^2,
4 = 5*4 - floor(2*sqrt(5))^2,
9 = 5*9 - floor(3*sqrt(5))^2, etc.
Moreover,
for n = 1, the unique (j,k) is (1,4): (1 + 4 + 1)^2 - 4*4 = 20*1;
for n = 2, the unique (j,k) is (4,5): (4 + 5 + 1)^2 - 4*5 = 20*4;
for n = 3, the unique (j,k) is (9,4): (9 + 4 + 1)^2 - 4*4 = 20*9.

Prog

(Magma) [5*n^2-Floor(n*Sqrt(5))^2: n in [1..70]]; // Vincenzo Librandi, Sep 13 2011

Crossrefs

Cf. A120867.

Sequence in context: A258682 A070445 A070444 * A070443 A279403 A330377

Adjacent sequences: A120863 A120864 A120865 * A120867 A120868 A120869

Keyword

nonn

Author

Clark Kimberling, Jul 09 2006

Status

approved