%I #52 Sep 08 2022 08:45:19

%S 17,65,145,257,401,577,785,1025,1297,1601,1937,2305,2705,3137,3601,

%T 4097,4625,5185,5777,6401,7057,7745,8465,9217,10001,10817,11665,12545,

%U 13457,14401,15377,16385,17425,18497,19601,20737,21905,23105,24337,25601

%N a(n) = 16*n^2 + 1.

%C Area of a Maltese cross conventionally inscribed in a 5n X 5n-grid.

%C Areas of some other crosses, each made from unit squares, as shown in Weisstein's illustrations: Greek Cross = x-pentomino = 5. Latin Cross = 6. Saint Andrew's cross = crux decussata = 9. Saint Anthony's Cross = tau cross = crux commissa = 10. Gaullist Cross = cross of Lorraine or patriarchal cross = 13. Papal Cross = 22. - _Jonathan Vos Post_, Jun 18 2005

%C The identity (16*n^2+1)^2-(64*n^2+8)*(2*n)^2 = 1 can be written as a(n)^2-A158488(n)*A005843(n)^2 = 1. - _Vincenzo Librandi_, Feb 08 2012

%C Sequence found by reading the line from 17, in the direction 17, 65,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Nov 02 2012

%C Conjecture: a(n) = floor(1/((4n) - log(2) + 1/(n+1) + 1/(n+2) + ... + 1/(2n)). - _Clark Kimberling_, Sep 09 2014

%H Vincenzo Librandi, <a href="/A108211/b108211.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MalteseCross.html">Maltese Cross</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GaullistCross.html">Gaullist Cross</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreekCross.html">Greek Cross</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatinCross.html">Latin Cross</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A002522(4*n) = A016802(n) + 1.

%F G.f.: x*(17+14*x+x^2)/(1-x)^3. - _Bruno Berselli_, Feb 08 2012

%F From _Amiram Eldar_, Jul 13 2020: (Start)

%F Sum_{n>=1} 1/a(n) = Pi*coth(Pi/4)/8 - 1/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - Pi*csch(Pi/4)/8. (End)

%F From _Amiram Eldar_, Feb 05 2021: (Start)

%F Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/4)*sinh(Pi/sqrt(8)).

%F Product_{n>=1} (1 - 1/a(n)) = (Pi/4)*csch(Pi/4). (End)

%p A108211:=n->16*n^2+1: seq(A108211(n), n=1..50); # _Wesley Ivan Hurt_, Sep 24 2014

%t LinearRecurrence[{3, -3, 1}, {17, 65, 145}, 40] (* _Vincenzo Librandi_, Feb 08 2012 *)

%o (PARI) a(n)= 16*n^2+1 \\ _Charles R Greathouse IV_, Dec 23 2011

%o (Magma) I:=[17, 65, 145]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Feb 08 2012

%Y Cf. A002522, A005843, A016802, A158488.

%K nonn,easy

%O 1,1

%A _Reinhard Zumkeller_, Jun 15 2005