A180919 a(n) = n^2 + 731*n + 1.

1, 733, 1467, 2203, 2941, 3681, 4423, 5167, 5913, 6661, 7411, 8163, 8917, 9673, 10431, 11191, 11953, 12717, 13483, 14251, 15021, 15793, 16567, 17343, 18121, 18901, 19683, 20467, 21253, 22041, 22831, 23623, 24417, 25213, 26011, 26811

Offset: 0

Author

Bruno Berselli, Sep 25 2010 - Jan 26 2011

Keywords

• nonn
• easy

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).

Programs

Magma

[n^2+731*n+1: n in [0..40]];  // Vincenzo Librandi, Jan 26 2011

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 *)

PARI

a(n)=n^2+731*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[n^2+731*n+1 for n in (0..40)] # Bruno Berselli, May 13 2014

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