OFFSET
0,1
COMMENTS
From Klaus Purath, Aug 18 2022: (Start)
This is A028560(8*n+1), and thus a(n) + 9 is a square. (See formulas.)
7 is the only prime number of this sequence in which all odd prime factors occur.
Each prime factor p appears exactly twice in any interval of p consecutive terms. If a(m) and a(n) are within such an interval containing p, then m + n == -1 (mod p). (End)
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 4*A001539(n) - 5.
a(n) = 128*n + a(n-1) with a(0)=7. - Vincenzo Librandi, Nov 12 2010
Sum_{n>=0} 1/a(n) = (Psi(7/8)-Psi(1/8))/48 = 0.1580099..., see A250129. - R. J. Mathar, May 30 2022 [ = (sqrt(2)+1)*Pi/48. - Amiram Eldar, Sep 08 2022]
From Klaus Purath, Aug 18 2022: (Start)
a(n) = A028560(8*n+1).
a(n) + 9 = ((a(n+1) - a(n-1))/32)^2 = A017113(n)^2.
a(2*n) = (a(n+1) - a(n-1))*n + 7. (End)
From Amiram Eldar, Feb 19 2023: (Start)
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(cot(Pi/16)) + sin(Pi/8) * log(cot(3*Pi/16)))/12.
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/8)*cos(sqrt(5/2)*Pi/4).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/8)*cos(sqrt(2)*Pi/4). (End)
G.f.: -(7+114*x+7*x^2)/(x-1)^3 . - R. J. Mathar, Apr 23 2024
MATHEMATICA
a[n_] := (8 n + 1)*(8 n + 7); Array[a, 40, 0] (* Amiram Eldar, Sep 08 2022 *)
LinearRecurrence[{3, -3, 1}, {7, 135, 391}, 40] (* Harvey P. Dale, Jan 07 2023 *)
PROG
(PARI) a(n)=(8*n+1)*(8*n+7) \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved