A120867 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 = 20*n^2.

4, 5, 4, 1, 19, 16, 11, 4, 36, 29, 20, 9, 55, 44, 31, 16, 76, 61, 44, 25, 4, 80, 59, 36, 11, 101, 76, 49, 20, 124, 95, 64, 31, 149, 116, 81, 44, 5, 139, 100, 59, 16, 164, 121, 76, 29, 191, 144, 95, 44, 220, 169, 116, 61, 4, 196, 139, 80, 19, 225, 164, 101, 36, 256, 191

Offset

1,1

Comments

The j's that match these k's comprise A120866.

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(1 + n*sqrt(5))^2.

Example

4 = -5*1 + floor(1 + sqrt(5))^2,
5 = -5*4 + floor(1 + 2*sqrt(5))^2,
4 = -5*9 + floor(1 + 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(1+n*Sqrt(5))^2: n in [1..70]]; // Vincenzo Librandi, Sep 13 2011

Crossrefs

Cf. A120866.

Sequence in context: A322193 A174984 A092141 * A011427 A366610 A175622

Adjacent sequences: A120864 A120865 A120866 * A120868 A120869 A120870

Keyword

nonn

Author

Clark Kimberling, Jul 09 2006

Status

approved