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%I #52 Jan 02 2023 12:30:47
%S 1,733,1467,2203,2941,3681,4423,5167,5913,6661,7411,8163,8917,9673,
%T 10431,11191,11953,12717,13483,14251,15021,15793,16567,17343,18121,
%U 18901,19683,20467,21253,22041,22831,23623,24417,25213,26011,26811
%N a(n) = n^2 + 731*n + 1.
%C 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.
%C 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.
%C a(A155095(k)) == 0 (mod 17).
%H B. Berselli, <a href="/A180919/b180919.txt">Table of n, a(n) for n = 0..10000</a>.
%H Berselli et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2011-January/006866.html">SeqFan mailing list</a>, Jan 18 2011 ff.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1+730*x-729*x^2)/(1-x)^3.
%F a(2*n-1) - a(n) - a(n-1) = A142463(n-1) for n>0.
%F 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
%t Table[n^2 + 731 n + 1, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 733, 1467}, 40] (* _Harvey P. Dale_, Oct 14 2012 *)
%o (Magma) [n^2+731*n+1: n in [0..40]]; // _Vincenzo Librandi_, Jan 26 2011
%o (Sage) [n^2+731*n+1 for n in (0..40)] # _Bruno Berselli_, May 13 2014
%o (PARI) a(n)=n^2+731*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Sep 25 2010 - Jan 26 2011
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