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A002450
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(4^n - 1)/3.
(Formerly M3914 N1608)
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130
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0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101, 5592405, 22369621, 89478485, 357913941, 1431655765, 5726623061, 22906492245, 91625968981, 366503875925, 1466015503701, 5864062014805, 23456248059221, 93824992236885
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| For n>0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3)=21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman (layman(AT)math.vt.edu), Dec 18 2001
a(n) is composite for all n > 2 and has factors x, (3*x+2*(-1)^n) where x belongs to A001045. In binary the terms>0 are 1, 101, 10101, 1010101, etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002
Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - R. H. Hardin (rhhardin(AT)att.net), Mar 16 2002
Collatz-function iteration started at with a(n) will surely end at 1 in 2*n steps. - Labos E. (labos(AT)ana.sote.hu), Sep 30 2002
Also sum of squares of divisors of 2^n: a(n)=A001157(A000079(n)) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast (ghodges14(AT)comcast.net), Apr 10 2003
Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
Number of walks of length 2n between two vertices at distance 2 in the cycle graph C_6. For n=2 we have for example 5 walks of length 4 from vertex A to C: ABABC, ABCBC, ABCDC, AFABC and AFEDC. - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004
Also number of walks of length 2n+1 between two vertices at distance 3 in the cycle graph C_12. - Herbert Kociemba (kociemba(AT)t-online.de), Jul 05 2004
a(n+1) is the number of steps which are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to mark one vertex on the lattice (compare A080674). - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 13 2005
a(n+1)=sum of square divisors of 4^n. - Paul Barry (pbarry(AT)wit.ie), Oct 13 2005
a(n+1) is the decimal number generated by the binary bits in the n-th generation of the Rule 250 elementary cellular automaton. - Eric Weisstein (eric(AT)weisstein.com), Apr 08 2006
a(k)=[M^k]_2,1, where M is the 3 by 3 matrix defined as follows: M = [1,1,1;1,3,1;1,1,1]. - Simone Severini (simoseve(AT)gmail.com), Jun 11 2006
a(n-1) / a(n) = percentage of wasted storage if a single image is stored as a pyramid with a each subsequent higher resolution layer containing four times as many pixels as the previous layer. n is the number of layers. - Victor Brodsky (victorbrodsky(AT)gmail.com), Jun 15 2006
n is in the sequence if and only if C(4n+1,n) (A052203) is odd; - Paul Barry (pbarry(AT)wit.ie), Mar 26 2007
This sequence also gives the number of distinct 3-colorings of the odd cycle C(2*n-1). - Keith Briggs (keith.briggs(AT)bt.com), Jun 19 2007
All numbers of the form n*4^n+(4^n-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity n*4^n+(4^n-1)/3=4(4(..4(4n+1)+1)+1)+1..)+1. - Artur Jasinski (grafix(AT)csl.pl), Nov 12 2007
Successive numbers contain only the digit 1 in base 4 positional system: 1, 11, 111, 1111 etc. [From Artur Jasinski (grafix(AT)csl.pl), Sep 30 2008]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=charpoly(A,1). [From Milan R. Janjic (agnus(AT)blic.net), Jan 27 2010]
This is the sequence A(0,1;3,4;2) = A(0,1;4,0;1) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 18 2010]
6*a(n)+1 = every second Mersenne number >=M3 hence all Mersenne primes greater than M2 must be a 6*a(n)+1 of this sequence. [From Roderick(Roddy) MacPhee (science_man_88(AT)yahoo.com), Nov 01 2010]
Let N(n) be the number of distinct prime knots with n crossings, counting chiral versions of the same knot separately. Then (((2^(n-2)-1)/3 =< N(n) =< e^n, according to Ernst. Note that (2^n)/3 is an integer only when n is even, hence this relates to (4^n - 1)/3. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 21 2010]
Smallest number having alternating bit sum n. Cf. A065359.
For n=1,2,..., the last digit of a(n) is 1,5,1,5,.... [From W. Bomfim (webonfim(AT)gmail.com) Jan 21 2011]
Rule 50 elementary cellular automaton generates this sequence. [Paul Muljadi, Jan 27 2011]
This sequence also appears in the second column of array in A173588. [Paul Muljadi, Jan 27 2011]
Subsequence of A003714.
Sequence found by reading the line from 0, in the direction 0, 5,... and the line from 1, in the direction 1, 21,..., in the square spiral whose edges are the Jacobsthal numbers A001045 and whose vertices are the numbers A000975. These parallel lines are two semi-diagonals in the spiral. - Omar E. Pol, Sep 10 2011
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REFERENCES
| A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.1.
J. V. Leyendekkers and A.G. Shannon, Modular Rings and the Integer 3, Notes on Number Theory & Discrete Mathematics, 17 (2011), 47-51; http://www.nntdm.net/papers/nntdm-17/NNTDM-17-2-47-51.pdf
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
H. Bottomley, Illustration of initial terms
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Ernst, C. and Sumners, D. W. The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 [Jonathan Vos Post, Nov 21 2010]
W. Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 18 2010]
Eric Weisstein's World of Mathematics, Repunit, Rule 250, Prime Knot.
Wikipedia, Elementary cellular automaton
Index entries for sequences related to linear recurrences with constant coefficients, signature (5,-4).
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FORMULA
| a(n+1)= sum(A060921(n, m), m=0..n). G.f.: x/((1-x)*(1-4*x)). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
a(n)= sum{k=0..n-1, 4^k}; a(n)= A001045(2*n). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
E.g.f.: (exp(4*x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
a(n+1) = 4*a(n) + 1, with a(0) = 0. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004
a(n) = sum(i=0..n-1, C(2*n-1-i, i)*2^i). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
a(n+1) = sum{k=0..n, binomial(n+1, k+1)*3^k} - Paul Barry (pbarry(AT)wit.ie), Aug 20 2004
a(n) = center term in M^n * [1 0 0], where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 0 0] = [A007583(n-1) a(n) A007583(n-1)]. E.g. a(4) = 85 since M^4 * [1 0 0] = [43 85 43] = [A007583(3) a(4) A007583(3)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004
a(n) = sum{k=0..n, sum{j=0..n, C(n, j)*C(j, k)A001045(j-k)}}; - Paul Barry (pbarry(AT)wit.ie), Feb 15 2005
a(n) = sum{k=0..n, C(n, k)*A001045(n-k)*2^k} = sum{k=0..n, C(n, k)*A001045(k)*2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), Apr 22 2005
a(n) = A125118(n,3) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2006
a(n) = sum(k=0..n, 2^(n-k)*A128908(n,k) ), n>=1 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 19 2008]
a(n) = sum(k=0..n, A106566(n,k)*A100335(k) ). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]
If we define f(m,j,x)=sum(binomial(m,k)*stirling2(k,j)*x^(m-k),k=j..m) then a(n-1)=f(2*n,4,-2), (n>=2). [From Milan R. Janjic (agnus(AT)blic.net), Apr 26 2009]
a(n) = A014551(n)*A001045(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 08 2009]
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3) = 5*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1, a(2)=5. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 18 2010]
a(0) = 0, a(n+1) = a(n)+2^(2*n) - W. Bomfim Jan 21, 2011
A036555(a(n)) = 2*n. - Reinhard Zumkeller, Jan 28 2011
a(n) = sum(k=1..floor((p+2)/3),C(2*p+1,p+2-3*k)). [Mircea Merca, Jun 25 2011]
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MAPLE
| [seq((4^n-1)/3, n=0..40)];
A002450:=1/(4*z-1)/(z-1); [S. Plouffe in his 1992 dissertation, dropping the initial zero.]
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MATHEMATICA
| Table[(4^n-1)/3, {n, 0, 100}] [From Vladimir Orlovsky, Sep 29 2008]
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PROG
| (MAGMA) [ (4^n-1)/3: n in [0..25] ]; [From Klaus Brockhaus, Oct 28 2008]
(PARI) a(n) = (4^n-1)/3;
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CROSSREFS
| a(n) = (A007583(n)-1)/2.
Partial sums of powers of 4, A000302.
a(n)=A000975(2*n)/2.
A084160(n) = 2*a(n).
When converted to binary, produces A094028.
Cf. A002446, A024036, A020988, A080674, A047849, A007583, A080355, A112627, A113860, A129735, A018215.
Cf. A160967, A139391. [From Reinhard Zumkeller, May 31 2009]
Sequence in context: A002054 A028948 * A084241 A187063 A026855 A097113
Adjacent sequences: A002447 A002448 A002449 * A002451 A002452 A002453
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Artur Jasinski (grafix(AT)csl.pl), Jan 27 2006
More terms from Artur Jasinski (grafix(AT)csl.pl), Nov 12 2007
Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009
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