OFFSET
0,3
COMMENTS
The sequence of n such that n is prime and (2*n+1) is prime is the sequence of Sophie Germain primes A005384; the subsequence of those for which in addition (3*n+2) is prime is A067256; and the subsequence of those for which in addition (4*n+3) is prime is A067257. - Jonathan Vos Post, Dec 15 2004
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: (1 -5*x +115*x^2 +345*x^3 +120*x^4)/(1-x)^5. - R. J. Mathar, Jan 30 2011
From G. C. Greubel, Mar 05 2020: (Start)
a(n) = n^4* Pochhammer((n-1)/n, 4).
E.g.f.: (1 - x + 53*x^2 + 94*x^3 + 24*x^4)*exp(x). (End)
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=2} 1/a(n) = 29/36 + (4/3 - 3*sqrt(3)/4)*Pi - 12*log(2) + 27*log(3)/4.
Sum_{n>=2} (-1)^n/a(n) = (1 + 4*sqrt(2)/3 - 3*sqrt(3)/2)*Pi + 14*log(2)/3 - 4*sqrt(2)*log(2)/3 + 8*sqrt(2)*log(2-sqrt(2))/3 - 29/36. (End)
MAPLE
1, seq( n^4*pochhammer((n-1)/n, 4), n=1..40); # G. C. Greubel, Mar 05 2020
MATHEMATICA
Table[1-10 n+35 n^2-50 n^3+24 n^4, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 0, 105, 880, 3465}, 40] (* Harvey P. Dale, Jan 29 2011 & Apr 26 2011 *)
PROG
(Magma) [ 24*n^4-50*n^3+35*n^2-10*n+1: n in [0..40]]; // Vincenzo Librandi, Jan 30 2011
(Magma) [&*[s*n-1: s in [1..4]]: n in [0..40]]; // Bruno Berselli, May 23 2011
(PARI) a(n)=24*n^4-50*n^3+35*n^2-10*n+1 \\ Charles R Greathouse IV, May 23 2011
(Sage) [1]+[n^4*rising_factorial((n-1)/n, 4) for n in (1..40)] # G. C. Greubel, Mar 05 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved