OFFSET
0,2
COMMENTS
Consider all sequences of numbers of the form m^2+h*m+1 (with h natural number and m = 0,1,2,3,4,5,...) which contain exactly 7 squares; the present sequence has the smallest value of h. Note that for 6 squares the smallest h is 23 and for 8 squares the smallest h is 37.
For n < 365^2, the squares of the form n^2+731*n+1 are 1, 239121, 2653641, 24413481, 220255281, 1982831841, 17846020921; for n > 365^2-1 we have (n+365)^2 < n^2+731*n+1 < (n+366)^2 and therefore n^2+731*n+1 cannot be a square.
a(A155095(k)) == 0 (mod 17).
LINKS
B. Berselli, Table of n, a(n) for n = 0..10000.
Berselli et al., SeqFan mailing list, Jan 18 2011 ff.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (1+730*x-729*x^2)/(1-x)^3.
a(2*n-1) - a(n) - a(n-1) = A142463(n-1) for n>0.
a(0)=1, a(1)=733, a(2)=1467; for n>2, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 14 2012
MATHEMATICA
Table[n^2 + 731 n + 1, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 733, 1467}, 40] (* Harvey P. Dale, Oct 14 2012 *)
PROG
(Magma) [n^2+731*n+1: n in [0..40]]; // Vincenzo Librandi, Jan 26 2011
(Sage) [n^2+731*n+1 for n in (0..40)] # Bruno Berselli, May 13 2014
(PARI) a(n)=n^2+731*n+1 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Sep 25 2010 - Jan 26 2011
STATUS
approved