A162155 Integer quartets a(4k)= 2, a(4k+1) = 32*k^2-24*k+3, a(4k+2) = 32*k^2-24*k+2, a(4k+3) = 8*k-3, k>=1.

2, 11, 10, 5, 2, 83, 82, 13, 2, 219, 218, 21, 2, 419, 418, 29, 2, 683, 682, 37, 2, 1011, 1010, 45, 2, 1403, 1402, 53, 2, 1859, 1858, 61, 2, 2379, 2378, 69, 2, 2963, 2962, 77, 2, 3611, 3610, 85, 2, 4323, 4322, 93, 2, 5099, 5098, 101, 2, 5939, 5938, 109, 2, 6843, 6842, 117

Offset

4,1

Comments

The members of each quartet are related by the Diophantine formula,

( (4*k-3)^2+(4*k-2)^2 ) * ( (4*k-1)^2+(4*k)^2 ) = a(4*k)^2 + a(4*k+1)^2 = a(4*k+2)^2 + a(4*k+3)^2 .

Links

Table of n, a(n) for n=4..63.

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,2,-2,2,-2,-1,1,-1,1).

Formula

a(n)= a(n-1) -a(n-2) +a(n-3) +2*a(n-4) -2*a(n-5) +2*a(n-6) -2*a(n-7) -a(n-8) +a(n-9) -a(n-10) +a(n-11).

G.f.: x^4* (-2-9*x-x^2-4*x^3+8*x^4-58*x^5+6*x^6+2*x^8-5*x^9+3*x^10-4*x^7)/( (1+x)^2 * (x-1)^3 * (x^2+1)^3 ).

Example

k=1 contributes the quartet (2,11,10,5). k=2 contributes (2,83,82,13) etc.

Crossrefs

Sequence in context: A009204 A023642 A025536 * A163344 A064743 A109868

Adjacent sequences: A162152 A162153 A162154 * A162156 A162157 A162158

Keyword

nonn,less,easy

Author

Juri-Stepan Gerasimov, Jun 26 2009

Extensions

Edited and corrected by R. J. Mathar, Oct 04 2009

Status

approved