OFFSET
0,2
COMMENTS
If gcd(n,7) = gcd(n+1,7) = gcd(2*n+1,7) = 1 then a(n) == 0 (mod 7) (E. Picutti, see References).
REFERENCES
Ettore Picutti, Sul numero e la sua storia, Feltrinelli Economica, 1977, p. 208.
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..10000.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: (1 + 2*x + 66*x^2 + 2*x^3 + x^4)/(1-x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 6*12.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6*A008594(n-1).
a(n) = 2*a(n-1) - a(n-2) + 6*A003154(n).
a(n) = a(n-1) + 6*A007588(n).
a(n) = 1 + 6*A062392(n).
Sum_{i=0..n} a(i) = (3*n^5 + 15*n^4 + 20*n^3 - 3*n + 5)/5.
a(n) = 7*(3*n^2 + 3*n - 1)*(Sum_{k=1..n} k^6)/(5*Sum_{k=1..n} k^4), n > 0. - Gary Detlefs, Oct 18 2011
MATHEMATICA
Table[3 n^4 + 6 n^3 - 3 n + 1, {n, 0, 40}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 7, 91, 397, 1141}, 40] (* Harvey P. Dale, Jul 12 2022 *)
PROG
(Magma) [3*n^4+6*n^3-3*n+1: n in [0..31]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Oct 25 2010 - Oct 29 2010
EXTENSIONS
Formula, program and crossref added by Bruno Berselli, Aug 22 2011
STATUS
approved