login
A158443
a(n) = 16*n^2 - 4.
3
12, 60, 140, 252, 396, 572, 780, 1020, 1292, 1596, 1932, 2300, 2700, 3132, 3596, 4092, 4620, 5180, 5772, 6396, 7052, 7740, 8460, 9212, 9996, 10812, 11660, 12540, 13452, 14396, 15372, 16380, 17420, 18492, 19596, 20732, 21900, 23100, 24332, 25596, 26892, 28220
OFFSET
1,1
COMMENTS
The identity (8*n^2-1)^2-(16*n^2-4) *(2*n)^2=1 can be written as A157914(n)^2-a(n)*A005843(n)^2=1.
Sequence found by reading the line from 12, in the direction 12, 60,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 4*x*(3+6*x-x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi-2)/16. (End)
MATHEMATICA
16Range[60]^2-4 (* Harvey P. Dale, Mar 18 2011 *)
PROG
(Magma) I:=[12, 60, 140]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
(PARI) a(n) = 16*n^2 - 4.
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 19 2009
STATUS
approved