A120865 a(n) is the number k for which there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 12*n^2.

1, 4, 9, 1, 6, 13, 22, 4, 13, 24, 37, 9, 22, 37, 1, 16, 33, 52, 6, 25, 46, 69, 13, 36, 61, 88, 22, 49, 78, 4, 33, 64, 97, 13, 46, 81, 118, 24, 61, 100, 141, 37, 78, 121, 9, 52, 97, 144, 22, 69, 118, 169, 37, 88, 141, 1, 54, 109, 166, 16, 73, 132, 193, 33, 94, 157, 222, 52

Offset

1,2

Comments

The j's that match these k's comprise A120864.

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) = -3*n^2 + floor(1 + n*sqrt(3))^2.

Example

1 = -3*1 + floor(1 + sqrt(3))^2,
4 = -3*4 + floor(1 + 2*sqrt(3))^2,
9 = -3*9 + floor(1 + 3*sqrt(3))^2, etc.
Moreover,
for n = 1, the unique (j,k) is (2,1): (2+1+1)^2 - 4*1 = 12*1;
for n = 2, the unique (j,k) is (3,4): (3+4+1)^2 - 4*4 = 12*4;
for n = 3, the unique (j,k) is (2,9): (2+9+1)^2 - 4*9 = 12*9.

Prog

(Magma) [-3*n^2+Floor(1+n*Sqrt(3))^2: n in [1..70]]; // Vincenzo Librandi, Sep 13 2011

Crossrefs

Cf. A120864.

Sequence in context: A007892 A010297 A001191 * A219031 A243452 A133868

Adjacent sequences: A120862 A120863 A120864 * A120866 A120867 A120868

Keyword

nonn

Author

Clark Kimberling, Jul 09 2006

Status

approved