%I A004767
%S A004767 3,7,11,15,19,23,27,31,35,39,43,47,51,55,59,63,67,71,75,
%T A004767 79,83,87,91,95,99,103,107,111,115,119,123,127,131,135,
%U A004767 139,143,147,151,155,159,163,167,171,175,179,183,187,191
%N A004767 4n+3.
%C A004767 Apart from initial term(s), dimension of the space of weight 2n cusp 
               forms for Gamma_0( 12 ).
%C A004767 These are the numbers which end in 11 in their binary expansion; also 
               the numbers for which zeta(2*x+1) needs just 2 terms to be evaluated. 
               - Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Dec 16 2004
%C A004767 a(n) = smallest k such that for every r from 0 to 2n-1 there exist j 
               and i, k >= j > i > 2n-1, such that j - i == r ( mod (2n-1)), with 
               (k,(2n-1))=(j,(2n-1))=(i,(2n-1)) = 1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Sep 24 2003
%C A004767 Apart from initial terms, same as 4n-9.
%C A004767 Campbell reference shows: "A graph on n vertices with at least 4n-9 edges 
               is intrinsically linked. A graph on n vertices with at least 5n-14 
               edges is intrinsically knotted." - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Jan 18 2007
%C A004767 Any (4n+3)-dimensional manifold endowed with a mixed 3-Sasakian structure 
               is an Einstein space with Einstein constant lambda = 4n+2 [Theorem 
               3, p.10 of Ianus et al]. [From Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Nov 24 2008]
%C A004767 For all n, a(n) is not a^2+b^2. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 27 2009]
%C A004767 If X=(n^3+9)/4, Y=(n^3+9)/2, Z=(n^4+9*n)/4, then X^3+Y^3+Z^3=(X*Y)^2 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jun 30 2009]
%H A004767 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A004767 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0n.gp">
               Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A004767 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The 
               modular forms database</a>
%H A004767 J. Campbell, T.W. Mattman, R. Ottman, J. Pyzer, M. Rodrigues and S. Williams, 
               <a href="http://arXiv.org/abs/math.GT/0701422">Intrinsic knotting 
               and linking of almost complete graphs</a>, 15 Jan 2007.
%H A004767 Stere Ianus, Mihai Visinescu and Gabriel-Eduard Vilcu, <a href="http:/
               /arxiv.org/abs/0811.3478">Hidden symmetries and Killing tensors on 
               curved spaces</a>, Nov 21, 2008. [From Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Nov 24 2008]
%F A004767 Binary expansion ends 11.
%F A004767 G.f.: (3+x)/(1-x)^2 - Paul Barry (pbarry(AT)wit.ie), Feb 27 2003
%F A004767 Complement of A004773. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 29 2005
%F A004767 a(n)=2*a(n-1)-a(n-2); a(0)=3, a(1)=7. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2008]
%F A004767 a(n)=8*n-a(n-1)-6 (with a(1)=3) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 24 2009]
%e A004767 For n=3, X=9, Y=18, Z=27, and 9^3+18^3+27^3=(9*18)^2; n=7, X=88, Y=176, 
               Z=616, and 88^3+176^3+616^3=(88*176)^2; [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Jun 30 2009]
%e A004767 For n=2, a(2)=8*2-3-6=7; n=3, a(3)=8*3-7-6=11; n=4, a(4)=8*4-11-6=15 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 24 2009]
%p A004767 seq( (3+4*x),x=0..100 );
%p A004767 a:=n->sum(sum(binomial(2,j), j=0..k),k=1..n): seq(a(n), n=1..48); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2007
%p A004767 a[1]:=-1:for n from 2 to 100 do a[n]:=a[n-1]+4 od: seq(a[n], n=2..49); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
%p A004767 with(finance):seq(add(cashflows([0,0,4], 0 ),k=1..n)-1,n=1..48); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008
%t A004767 f[n_]:=4*n+3; lst={};Do[a=f[n];AppendTo[lst,a],{n,0,6!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]
%o A004767 (Other) sage: [i+3 for i in range(190) if gcd(i,4) == 4] # [From Zerinvary 
               Lajosz (zerinvarylajos(AT)yahoo.com), May 20 2009]
%o A004767 (Other) sage: [crt(3, n, 4,3 ) for n in xrange(3, 51)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 07 2009]
%Y A004767 Sequence in context: A124981 A059554 A103543 this_sequence A131098 A118894 
               A039957
%Y A004767 Adjacent sequences: A004764 A004765 A004766 this_sequence A004768 A004769 
               A004770
%K A004767 nonn,easy,new
%O A004767 0,1
%A A004767 N. J. A. Sloane (njas(AT)research.att.com).