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A007583
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(2^(2n+1) + 1)/3.
(Formerly M2895)
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48
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1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203, 11184811, 44739243, 178956971, 715827883, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 2932031007403, 11728124029611, 46912496118443
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OFFSET
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0,2
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COMMENTS
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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
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
Binomial transform of A025192. - Paul Barry, Apr 11 2003
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
Numbers of the form 1+Sum_{i=1..m} [2^(2i-1)]. - Artur Jasinski, Feb 09 2007
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
Related to A024493(6n+1), A131708(6n+3), A024495(6n+5). - Paul Curtz, Mar 27 2008
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). [From Milan Janjic, Feb 21 2010]
Number of toothpicks in the toothpick structure of A139250 after 2^n stages. - Omar E. Pol, Feb 28 2011
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
a(n) is the smallest number for which A072219(a(n)) = 2*n+1. - Ramasamy Chandramouli, Dec 22 2012.
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REFERENCES
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E. Estrada and J. A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176, 2013. - From N. J. A. Sloane, Feb 28 2013
H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 10.
S. Hong and J. H. Kwak, Regular fourfold covering with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..170
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the x-rays of permutations
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 893
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Eric Weisstein's World of Mathematics, Repunit
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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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
a(n)=sum(binomial(n+k, 2*k)/2^(k-n), k=0..n). a(n)=4a(n-1)-1, n>0.
a(n)=1 + 2*sum{k=0..n-1, 4^k} a(n)=A001045(2n+1). - Paul Barry, Mar 17 2003
a(n) = A020988(n-1)+1 = A039301(n+1)-1 = A083584(n-1)+2. - Ralf Stephan, Jun 14 2003
u(0) = 0; u(n+1) = 4*u(n) - 1 - Regis Decamps (decamps(AT)users.sf.net), Feb 04 2004
a(n)=sum(i+j+k=n, (n+k)!/i!/j!/(2*k)!) 0<=i, j, k<=n - Benoit Cloitre, Mar 25 2004
a(n)=5a(n-1)-4a(n-2). - Emeric Deutsch, Apr 01 2004
a(n)=4^n-A001045(2n) - Paul Barry, Apr 17 2004
a(n)=2*(A001045(n))^2+(A001045(n+1))^2. - Paul Barry, Jul 15 2004
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
a(n) = A072197(n) - A020988(n). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 31 2004
a(n) = A139250(2^n). - Omar E. Pol, Feb 28 2011
a(n) = A193652(2*n+1). [Reinhard Zumkeller, Aug 08 2011]
a(n) = sum(binomial(2*n,n+3*k)/2, k=-floor(n/3)..floor(n/3)). [Mircea Merca, Jan 28 2012]
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MAPLE
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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
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MATHEMATICA
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a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; AppendTo[a, c], {x, 0, 30}]; a - Artur Jasinski, Feb 09 2007
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
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PROG
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(PARI) a(n)=sum(k=-n\3, n\3, binomial(2*n+1, n+1+3*k))
(MAGMA) [(2^(2*n+1) + 1)/3: n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011
(Haskell)
a007583 = (`div` 3) . (+ 1) . a004171
-- Reinhard Zumkeller, Jan 09 2013
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CROSSREFS
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a(n) = (2*A002450(n))+1. Cf. also A006054, A006356, A005578.
Partial sums of A081294.
Cf. A002450.
Cf. A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936.
Cf. A004171.
Sequence in context: A034477 A140803 A084643 * A026671 A026876 A151090
Adjacent sequences: A007580 A007581 A007582 * A007584 A007585 A007586
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KEYWORD
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nonn,easy
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AUTHOR
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Simon Plouffe
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STATUS
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approved
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