%I #55 Oct 25 2024 07:40:35
%S 7,135,391,775,1287,1927,2695,3591,4615,5767,7047,8455,9991,11655,
%T 13447,15367,17415,19591,21895,24327,26887,29575,32391,35335,38407,
%U 41607,44935,48391,51975,55687,59527,63495,67591,71815,76167,80647,85255,89991,94855,99847,104967
%N a(n) = (8*n+1)*(8*n+7).
%C From _Klaus Purath_, Aug 18 2022: (Start)
%C This is A028560(8*n+1), and thus a(n) + 9 is a square. (See formulas)
%C 7 is the only prime number of this sequence in which all odd prime factors occur.
%C 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)
%H T. D. Noe, <a href="/A001533/b001533.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 4*A001539(n) - 5.
%F a(n) = 128*n + a(n-1) with a(0)=7. - _Vincenzo Librandi_, Nov 12 2010
%F 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]
%F From _Klaus Purath_, Aug 18 2022: (Start)
%F a(n) = A028560(8*n+1).
%F a(n) + 9 = ((a(n+1) - a(n-1))/32)^2 = A017113(n)^2.
%F a(2*n) = (a(n+1) - a(n-1))*n + 7. (End)
%F From _Amiram Eldar_, Feb 19 2023: (Start)
%F a(n) = A017077(n)*A004771(n).
%F Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(cot(Pi/16)) + sin(Pi/8) * log(cot(3*Pi/16)))/12.
%F Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/8)*cos(sqrt(5/2)*Pi/4).
%F Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/8)*cos(sqrt(2)*Pi/4). (End)
%F G.f.: -(7+114*x+7*x^2)/(x-1)^3. - _R. J. Mathar_, Apr 23 2024
%F From _Elmo R. Oliveira_, Oct 25 2024: (Start)
%F E.g.f.: exp(x)*(7 + 64*x*(2 + x)).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t a[n_] := (8 n + 1)*(8 n + 7); Array[a, 40, 0] (* _Amiram Eldar_, Sep 08 2022 *)
%t LinearRecurrence[{3,-3,1},{7,135,391},40] (* _Harvey P. Dale_, Jan 07 2023 *)
%o (PARI) a(n)=(8*n+1)*(8*n+7) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A001539, A004771, A017077, A017113, A028560, A250129.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_