%I A007583 M2895
%S A007583 1,3,11,43,171,683,2731,10923,43691,174763,699051,2796203,11184811,
%T A007583 44739243,178956971,715827883,2863311531,11453246123,45812984491,
%U A007583 183251937963,733007751851,2932031007403,11728124029611,46912496118443
%N A007583 (2^(2n+1) + 1)/3.
%C A007583 Let u(k), v(k), w(k) be the 3 sequences defined by u(1)=1, v(1)=0, w(1)=0 
               and u(k+1)=u(k)+v(k)-w(k), v(k+1)=u(k)-v(k)+w(k), w(k+1)=-u(k)+v(k)+w(k); 
               let M(k)=Max(u(k),v(k),w(k)); then a(n)=M(2n)=M(2n-1) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Mar 25 2002
%C A007583 Also the number of words of length 2n generated by the two letters s 
               and t that reduce to the identity 1 by using the relations ssssss=1, 
               tt=1 and stst=1. The generators s and t along with the three relations 
               generate the dihedral group D6=C2xD3. - Jamaine Paddyfoot and John 
               Layman (jay_paddyfoot(AT)hotmail.com/layman(AT)math.vt.edu), Jul 
               08 2002
%C A007583 Binomial transform of A025192. - Paul Barry (pbarry(AT)wit.ie), Apr 11 
               2003
%C A007583 a(n) = A020988(n-1)+1 = A039301(n+1)-1 = A083584(n-1)+2. - Ralf Stephan 
               (ralf(AT)ark.in-berlin.de), Jun 14 2003
%C A007583 Number of walks of length 2n+1 between two adjacent vertices in the cycle 
               graph C_6. Example: a(1)=3 because in the cycle ABCDEF we have three 
               walks of length 3 between A and B: ABAB, ABCB and AFAB. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%C A007583 a(n) = A072197(n) - A020988(n). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Dec 31 2004
%C A007583 Numbers of the form 1+Sum_{i=1..m} [2^(2i-1)]. - Artur Jasinski (grafix(AT)csl.pl), 
               Feb 09 2007
%C A007583 Prime numbers of the form 1+Sum[2^(2n-1)] are in A000979. Numbers x such 
               1+Sum[2^(2n-1)] is prime for n=1,2,...,x is A127936. - Artur Jasinski 
               (grafix(AT)csl.pl), Feb 09 2007
%C A007583 Related to A024493(6n+1), A131708(6n+3), A024495(6n+5). - Paul Curtz 
               (bpcrtz(AT)free.fr), Mar 27 2008
%D A007583 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007583 H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 
               10.
%D A007583 S. Hong and J. H. Kwak, Regular fourfold covering with respect to the 
               identity automorphism, J. Graph Theory, 17 (1993), 621-627.
%H A007583 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A007583 C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="http:/
               /arXiv.org/abs/math.CO/0506334">On the x-rays of permutations</a>
%H A007583 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=893">
               Encyclopedia of Combinatorial Structures 893</a>
%H A007583 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Repunit.html">Repunit</a>
%F A007583 a(n) = sum(A060920(n, m), m = 0..n) = A002450(n+1)-2*A002450(n). G.f.: 
               (1-2*x)/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Apr 24 2001
%F A007583 a(n)=sum(binomial(n+k, 2*k)/2^(k-n), k=0..n). a(n)=4a(n-1)-1, n>0.
%F A007583 a(n)=1 + 2*sum{k=0..n-1, 4^k} a(n)=A001045(2n+1). - Paul Barry (pbarry(AT)wit.ie), 
               Mar 17 2003
%F A007583 u(0) = 0; u(n+1) = 4*u(n) - 1 - Regis Decamps (decamps(AT)users.sf.net), 
               Feb 04 2004
%F A007583 a(n)=sum(i+j+k=n, (n+k)!/i!/j!/(2*k)!) 0<=i, j, k<=n - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Mar 25 2004
%F A007583 a(n)=5a(n-1)-4a(n-2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 
               01 2004
%F A007583 a(n)=4^n-A001045(2n) - Paul Barry (pbarry(AT)wit.ie), Apr 17 2004
%F A007583 a(n)=2*(A001045(n))^2+(A001045(n+1))^2. - Paul Barry (pbarry(AT)wit.ie), 
               Jul 15 2004
%F A007583 a(n) = left and right terms in M^n * [1 1 1] where M = the 3X3 matrix 
               [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A002450(n+1) a(n)] 
               E.g. a(3) = 43 since M^n * [1 1 1] = [43 85 43] = [a(3) A002450(4) 
               a(3)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004
%F A007583 a(n)=4*a(n-1)-1 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Oct 29 2009]
%e A007583 for n=2, a(2)=4*1-1=3; n=3, a(3)=4*3-1=11; n=4, a(4)=4*11-1=43 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 29 2009]
%p A007583 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=4*a[n-1]-1 od: seq(a[n], 
               n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 
               2008
%t A007583 a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; AppendTo[a, c], {x, 0, 
               30}]; a - Artur Jasinski (grafix(AT)csl.pl), Feb 09 2007
%t A007583 a = {}; ZZ = 1; Do[ZZ = ZZ + 4^(x); AppendTo[a, ZZ], {x, 0, 24}]; a/2 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2007
%o A007583 (PARI) a(n)=sum(k=-n\3,n\3,binomial(2*n+1,n+1+3*k))
%Y A007583 a(n) = (2*A002450(n))+1. Cf. also A006054, A006356, A005578.
%Y A007583 Partial sums of A081294.
%Y A007583 Cf. A002450.
%Y A007583 Cf. A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, 
               A127936.
%Y A007583 Sequence in context: A034477 A140803 A084643 this_sequence A026671 A026876 
               A151090
%Y A007583 Adjacent sequences: A007580 A007581 A007582 this_sequence A007584 A007585 
               A007586
%K A007583 nonn,easy
%O A007583 0,2
%A A007583 Simon Plouffe (simon.plouffe(AT)gmail.com)