A120864 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 = 12*n^2.

2, 3, 2, 12, 11, 8, 3, 23, 18, 11, 2, 32, 23, 12, 50, 39, 26, 11, 59, 44, 27, 8, 66, 47, 26, 3, 71, 48, 23, 99, 74, 47, 18, 104, 75, 44, 11, 107, 74, 39, 2, 108, 71, 32, 146, 107, 66, 23, 147, 104, 59, 12, 146, 99, 50, 192, 143, 92, 39, 191, 138, 83, 26, 188, 131, 72, 11, 183

Offset

1,1

Comments

The k's that match these j's comprise A120865.

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(n*sqrt(3))^2.

(a(n) + A120865(n) + 1)^2 - 4*A120865(n) = 12*n^2. - Petros Hadjicostas, Jul 10 2020

Example

a(1) = 3*1^2 - floor(1*sqrt(3))^2 = 2;
a(2) = 3*2^2 - floor(2*sqrt(3))^2 = 3;
a(3) = 3*3^2 - floor(3*sqrt(3))^2 = 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.

Mathematica

Table[3n^2-Floor[n*Sqrt[3]]^2, {n, 68}] (* James C. McMahon, Oct 12 2024 *)

Prog

(Magma) [3*n^2-Floor(n*Sqrt(3))^2: n in [1..70]]; // Vincenzo Librandi, Sep 13 2011
(PARI) a(n) = 3*n^2 - sqrtint(3*n^2)^2; \\ Michel Marcus, Jul 10 2020

Crossrefs

Cf. A120865.

Sequence in context: A225787 A012960 A013117 * A173463 A023641 A083775

Adjacent sequences: A120861 A120862 A120863 * A120865 A120866 A120867

Keyword

nonn

Author

Clark Kimberling, Jul 09 2006

Status

approved