Search: id:A007742
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%I A007742
%S A007742 0,5,18,39,68,105,150,203,264,333,410,495,588,689,798,915,1040,1173,
%T A007742 1314,1463,1620,1785,1958,2139,2328,2525,2730,2943,3164,3393,3630,3875,
%U A007742 4128,4389,4658,4935,5220,5513,5814,6123,6440,6765,7098,7439,7788,8145
%N A007742 n(4n+1).
%C A007742 Write 0,1,2,... in clockwise spiral; sequence gives numbers on positive 
               y axis.
%C A007742 Central terms of the triangle in A126890. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 30 2006
%C A007742 If A=[A007742] n(4n+1) (for n>0, 5,18,39,68,...,); Y=[A157336] 64*n+8 
               (72,136,200,...,); X=[A157337] 128*n^2+32*n+1 (161,577,1249,...,) 
               ; , we have for all terms, Pell's equation X^2-A*Y^2=1. Example: 
               161^2-5*72^2=1; 577^2-18*136^2=1; 1249^2-39*200^2=1; 2177^2-68*264^2=1. 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 27 2009]
%D A007742 S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 
               1998) 188; 30 (#4, 1999-2000), 246-250.
%D A007742 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 2nd ed., 1994, p. 99.
%H A007742 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A007742 Emilio Apricena, <a href="a035608.png">A version of the Ulam spiral</
               a>
%H A007742 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 A007742 G.f.: x(5+3x)/(1-x)^3. - Michael Somos, Mar 03 2003
%F A007742 a(n)=8*n+a(n-1)-11 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 12 2009]
%e A007742 Part of the spiral:
%e A007742 16 17 18 19 ...
%e A007742 15 4 5 6 ...
%e A007742 14 3 0 7 ...
%e A007742 13 2 1 8 ...
%e A007742 For n=2, a(2)=8*2+0-11=5; n=3, a(3)=8*3+5-11=18; n=4, a(4)=8*4+18-11=39 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
%t A007742 s=0;lst={s};Do[s+=n++ +5;AppendTo[lst, s], {n, 0, 7!, 8}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 16 2008]
%o A007742 (PARI) a(n)=4*n^2+n
%Y A007742 a(n)=A033991(-n)=A074378(2n).
%Y A007742 Sequences from spirals: A001107, A002939, A007742, A033951, A033952, 
               A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y A007742 Cf. A157336, A157337 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 27 2009]
%Y A007742 Sequence in context: A038346 A065007 A031428 this_sequence A000338 A056640 
               A160969
%Y A007742 Adjacent sequences: A007739 A007740 A007741 this_sequence A007743 A007744 
               A007745
%K A007742 nonn,easy,nice,new
%O A007742 0,2
%A A007742 N. J. A. Sloane (njas(AT)research.att.com).

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