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A007583 (2^(2n+1) + 1)/3.
(Formerly M2895)
55

%I M2895

%S 1,3,11,43,171,683,2731,10923,43691,174763,699051,2796203,11184811,

%T 44739243,178956971,715827883,2863311531,11453246123,45812984491,

%U 183251937963,733007751851,2932031007403,11728124029611,46912496118443

%N (2^(2n+1) + 1)/3.

%C 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_, Mar 25 2002

%C 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 (jay_paddyfoot(AT)hotmail.com) and _John W. Layman_, Jul 08 2002

%C Binomial transform of A025192. - _Paul Barry_, Apr 11 2003

%C 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_, Apr 01 2004

%C Numbers of the form 1+Sum_{i=1..m} [2^(2i-1)]. - _Artur Jasinski_, Feb 09 2007

%C 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_, Feb 09 2007

%C Related to A024493(6n+1), A131708(6n+3), A024495(6n+5). - _Paul Curtz_, Mar 27 2008

%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - _Milan Janjic_, Feb 21 2010

%C Number of toothpicks in the toothpick structure of A139250 after 2^n stages. - _Omar E. Pol_, Feb 28 2011

%C Numbers whose binary representation is 10, n-1 times, together with 11, n >= 1. For example 171 = 10101011 (2). - _Omar E. Pol_, Nov 22 2012

%C a(n) is the smallest number for which A072219(a(n)) = 2*n+1. - _Ramasamy Chandramouli_, Dec 22 2012.

%C An Engel expansion of 2 to the base b := 4/3 as defined in A181565, with the associated series expansion 2 = b + b^2/3 + b^3/(3*11) + b^4/(3*11*43) + .... Cf. A007051. - _Peter Bala_, Oct 29 2013

%C The positive integer solution (x,y) of 3*x - 2^n*y = 1, n>=0, with smallest x is (a(n/2), 2) if n is even and (a((n-1)/2), 1) if n is odd. - _Wolfdieter Lang_, Feb 15 2014

%D H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 10.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A007583/b007583.txt">Table of n, a(n) for n = 0..170</a>

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://neilsloane.com/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>

%H 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 E. Estrada and J. A. de la Pena, <a href="http://arxiv.org/abs/1302.1176">From Integer Sequences to Block Designs via Counting Walks in Graphs</a>, arXiv preprint arXiv:1302.1176, 2013. - From _N. J. A. Sloane_, Feb 28 2013

%H E. Estrada and J. A. de la Pena, <a href="http://www.nntdm.net/papers/nntdm-19/NNTDM-19-3-78-84.pdf">Integer sequences from walks in graphs</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84

%H S. Hong and J. H. Kwak, <a href="http://dx.doi.org/10.1002/jgt.3190170509">Regular fourfold coverings with respect to the identity automorphism</a>, J. Graph Theory, 17 (1993), 621-627.

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=893">Encyclopedia of Combinatorial Structures 893</a>

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Note on Cosine Power Sums</a> J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.

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

%H <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>

%F 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_, Apr 24 2001

%F a(n)=sum(binomial(n+k, 2*k)/2^(k-n), k=0..n). a(n)=4a(n-1)-1, n>0.

%F a(n)=1 + 2*sum{k=0..n-1, 4^k} a(n)=A001045(2n+1). - _Paul Barry_, Mar 17 2003

%F a(n) = A020988(n-1)+1 = A039301(n+1)-1 = A083584(n-1)+2. - _Ralf Stephan_, Jun 14 2003

%F u(0) = 0; u(n+1) = 4*u(n) - 1. - Regis Decamps (decamps(AT)users.sf.net), Feb 04 2004

%F a(n)=sum(i+j+k=n, (n+k)!/i!/j!/(2*k)!) 0<=i, j, k<=n. - _Benoit Cloitre_, Mar 25 2004

%F a(n)=5a(n-1)-4a(n-2). - _Emeric Deutsch_, Apr 01 2004

%F a(n)=4^n-A001045(2n). - _Paul Barry_, Apr 17 2004

%F a(n)=2*(A001045(n))^2+(A001045(n+1))^2. - _Paul Barry_, Jul 15 2004

%F 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_, Dec 18 2004

%F a(n) = A072197(n) - A020988(n). - _Creighton Dement_, Dec 31 2004

%F a(n) = A139250(2^n). - _Omar E. Pol_, Feb 28 2011

%F a(n) = A193652(2*n+1). - _Reinhard Zumkeller_, Aug 08 2011

%F a(n) = sum(binomial(2*n,n+3*k)/2, k=-floor(n/3)..floor(n/3)). - _Mircea Merca_, Jan 28 2012

%F a(n) = 2^(2(n+1)) - A072197(n). - _Vladimir Pletser_, Apr 12 2014

%p 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_, Feb 22 2008

%t a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; AppendTo[a, c], {x, 0, 30}]; a (* _Artur Jasinski_, Feb 09 2007 *)

%t a = {}; ZZ = 1; Do[ZZ = ZZ + 4^(x); AppendTo[a, ZZ], {x, 0, 24}]; a/2 (* _Zerinvary Lajos_, Apr 03 2007 *)

%o (PARI) a(n)=sum(k=-n\3,n\3,binomial(2*n+1,n+1+3*k))

%o (MAGMA) [(2^(2*n+1) + 1)/3: n in [0..40] ]; // _Vincenzo Librandi_, Apr 28 2011

%o (Haskell)

%o a007583 = (`div` 3) . (+ 1) . a004171

%o -- _Reinhard Zumkeller_, Jan 09 2013

%Y a(n) = (2*A002450(n))+1. Cf. also A006054, A006356, A005578.

%Y Partial sums of A081294.

%Y Cf. A002450.

%Y Cf. A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936.

%Y Cf. A004171. A007051, A083065, A083066.

%K nonn,easy,changed

%O 0,2

%A _Simon Plouffe_

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Last modified April 20 05:52 EDT 2014. Contains 240779 sequences.