%I A033991
%S A033991 0,3,14,33,60,95,138,189,248,315,390,473,564,663,770,885,1008,1139,
%T A033991 1278,1425,1580,1743,1914,2093,2280,2475,2678,2889,3108,3335,3570,3813,
%U A033991 4064,4323,4590,4865,5148,5439,5738,6045,6360,6683,7014,7353,7700,8055
%N A033991 n(4n-1).
%C A033991 Write 0,1,2,... in clockwise spiral; sequence gives numbers on negative 
               x axis.
%C A033991 This sequence is the number of expressions x generated for a given modulus 
               n in finite arithmetic. For example, n=1 (modulus 1) generates 3 
               expressions: 0+0=0(mod 1), 0-0=0(mod 1), 0*0=0(mod 1). By subtracting 
               n from 4n^2, we eliminate the counting of those expressions that 
               would include division by zero, which would be, of course, undefined. 
               - David Quentin Dauthier (d_dauthier(AT)yahoo.com), Nov 04 2007
%C A033991 If A=[A033991] n(4n-1) (for n>0, 3,14,33,60,95,...,); Y=[A157330] 64*n-8 
               (56,120,184,...,); X=[A157331] 128*n^2-32*n+1 (97,449,1057,...,) 
               ; , we have for all terms, Pell's equation X^2-A*Y^2=1. Example: 
               97^2-3*56^2=1; 449^2-14*120^2=1; 1057^2-33*184^2=1; 1921^2-60*248^2=1. 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 27 2009]
%D A033991 S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 
               1998) 188; 30 (#4, 1999-2000), 246-250.
%D A033991 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 2nd ed., 1994, p. 99.
%H A033991 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A033991 Emilio Apricena, <a href="a035608.png">A version of the Ulam spiral</
               a>
%H A033991 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5773864&tstart=0">
               X^2-AY^2=1</a> [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 27 2009]
%F A033991 G.f.: x(3+5x)/(1-x)^3. - Michael Somos, Mar 03 2003
%F A033991 a(n)=A014635/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 16 
               2007
%F A033991 a(n)= A000326(n)+A005476(n), example:95=60+35, etc... a(n)=A049452(n)-A001105(n), 
               example:33=51-18, etc. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 12 2007
%F A033991 a(n)=8*n+a(n-1)-13 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 12 2009]
%e A033991 16 17 18 19 ...
%e A033991 15 4 5 6 ...
%e A033991 14 3 0 7 ...
%e A033991 13 2 1 8 ...
%e A033991 For n=2, a(2)=8*2+0-13=3; n=3, a(3)=8*3+3-13=14; n=4, a(4)=8*4+14-13=33 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
%p A033991 [seq(binomial(4*n, 2)/2, n=0..45)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 16 2007
%t A033991 (Mathematica 6.0) Manipulate[4 x^2 - x, {x, 1, 100, 1}] - David Quentin 
               Dauthier (d_dauthier(AT)yahoo.com), Nov 04 2007
%o A033991 (PARI) a(n)=4*n^2-n.
%o A033991 (Other) sage: [lucas_number1(3,2*n,n) for n in xrange(0, 46)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 20 2009]
%Y A033991 Sequences from spirals: A001107, A002939, A007742, A033951, A033952, 
               A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y A033991 Cf. A016742.
%Y A033991 a(n)=A007742(-n)=A074378(2n-1)=A014848(2n).
%Y A033991 Cf. A014635.
%Y A033991 Cf. A157330, A157331 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 27 2009]
%Y A033991 Sequence in context: A071396 A032525 A130697 this_sequence A155154 A081269 
               A140064
%Y A033991 Adjacent sequences: A033988 A033989 A033990 this_sequence A033992 A033993 
               A033994
%K A033991 nonn,easy,nice,new
%O A033991 0,2
%A A033991 N. J. A. Sloane (njas(AT)research.att.com).