%I A002450 M3914 N1608
%S A002450 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A002450 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A002450 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%N A002450 (4^n - 1)/3.
%C A002450 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
%C A002450 a(n) is composite for all n > 2 and has factors x, (3x+2(-1)^n) where 
               x belongs to A001045. In binary the terms are 1, 101, 10101, 1010101, 
               etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002
%C A002450 Number of n X 2 binary arrays with path of adjacent 1's from upper left 
               corner to right column. - Ron Hardin (rhhardin(AT)att.net), Mar 16 
               2002
%C A002450 Collatz-function iteration started at with a[n] will surely ended by 
               1 in exactly 2n steps. - Labos E. (labos(AT)ana.sote.hu), Sep 30 
               2002
%C A002450 Also sum of squares of divisors of 2^n: a(n)=A001157[A000079(n)] - Paul 
               Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A002450 All members of sequence are also generalized octagonal numbers (A001082). 
               - Matthew Vandermast (ghodges14(AT)comcast.net), Apr 10 2003
%C A002450 Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), 
               Apr 11 2003
%C A002450 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
%C A002450 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
%C A002450 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
%C A002450 a(n+1)=sum of square divisors of 4^n. - Paul Barry (pbarry(AT)wit.ie), 
               Oct 13 2005
%C A002450 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
%C A002450 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 (ss54(AT)york.ac.uk), Jun 
               11 2006
%C A002450 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
%C A002450 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
%C A002450 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
%C A002450 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
%C A002450 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]
%C A002450 a(n) for n >= 1 written in base 2: 1,101,10101,1010101,101010101, ...(see 
               A094028(n-1)). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Aug 02 2009]
%C A002450 Except for the first term, a(n)=4*a(n-1)+1 (with a(1)=1) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Oct 29 2009]
%D A002450 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002450 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002450 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.
%D A002450 A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial 
               Interpretations, Journal of Integer Sequences, Vol. 9 (2006), Article 
               06.3.1.
%D A002450 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002450 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.
%H A002450 T. D. Noe, <a href="b002450.txt">Table of n, a(n) for n=0..200</a>
%H A002450 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002450 H. Bottomley, <a href="a060919.gif">Illustration of initial terms</a>
%H A002450 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=373">
               Encyclopedia of Combinatorial Structures 373</a>
%H A002450 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002450 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002450 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Repunit.html">Link to a section of The World of Mathematics.</a>
%H A002450 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rule250.html">Rule 250</a>
%F A002450 a(n+1)= sum(A060921(n, m), m=0..n). G.f.: x/(1-5*x+4*x^2). - Wolfdieter 
               Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
%F A002450 a(n)= sum{k=0..n-1, 4^k} a(n)= A001045(2n). - Paul Barry (pbarry(AT)wit.ie), 
               Mar 17 2003
%F A002450 Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), 
               Mar 28 2003
%F A002450 E.g.f. (exp(4x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 a(0) = 0, a(n+1) = 4*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Feb 25 2004
%F A002450 a(n)=Sum(C(2n-1-i, i)2^i, i=0, .., n-1). - Mario Catalani (mario.catalani(AT)unito.it), 
               Jul 23 2004
%F A002450 a(n+1)=sum{k=0..n, binomial(n+1, k+1)3^k} - Paul Barry (pbarry(AT)wit.ie), 
               Aug 20 2004
%F A002450 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
%F A002450 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
%F A002450 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
%F A002450 Coefficients of series expansion of (1+4x)/(1-x-16x^2+16x^3) at point 
               x=0. - Artur Jasinski (grafix(AT)csl.pl), Jan 27 2006
%F A002450 a(n) = A125118(n,3) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 21 2006
%F A002450 a(n)=Sum_{k, 0<=k<=n}2^(n-k)*A128908(n,k), n>=1 . [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 19 2008]
%F A002450 a(n)=Sum_{k, 0<=k<=n} A106566(n,k)*A100335(k). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A002450 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]
%F A002450 a(n)=A014551(n)*A001045(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jul 08 2009]
%p A002450 [seq((4^n-1)/3,n=0..40)];
%p A002450 a:=n->sum(4^(n-j),j=1..n): seq(a(n), n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 04 2007
%p A002450 A002450:=1/(4*z-1)/(z-1); [S. Plouffe in his 1992 dissertation, dropping 
               the initial zero.]
%p A002450 with(finance):seq(add(futurevalue( 2, 3, k),k=0..n)/2,n=-1..23); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008
%t A002450 lst={};Do[p=(4^n-1)/3;AppendTo[lst, p], {n, 0, 5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008]
%t A002450 b = {}; a = {1}; Do[s = FromDigits[a, 4]; AppendTo[b, s]; AppendTo[a, 
               1], {n, 1, 50}]; b [From Artur Jasinski (grafix(AT)csl.pl), Sep 30 
               2008]
%o A002450 (MAGMA) [ (4^n-1)/3: n in [0..25] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), 
               Oct 28 2008]
%o A002450 (Other) sage: [lucas_number1(n,5,4) for n in xrange(0, 25)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A002450 (Other) sage: [gaussian_binomial(n,1,4) for n in xrange(0,25)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
%Y A002450 a(n) = (A007583(n)-1)/2.
%Y A002450 Partial sums of powers of 4, A000302.
%Y A002450 a(n)=A000975(2n)/2.
%Y A002450 A084160(n) = 2*a(n).
%Y A002450 Cf. A002446, A024036, A020988, A080674, A047849, A007583.
%Y A002450 Cf. A080355, A112627, A113860, A129735.
%Y A002450 Cf. A018215.
%Y A002450 A160967, A139391. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 31 2009]
%Y A002450 Sequence in context: A026027 A002054 A028948 this_sequence A084241 A026855 
               A097113
%Y A002450 Adjacent sequences: A002447 A002448 A002449 this_sequence A002451 A002452 
               A002453
%K A002450 nonn,easy,nice
%O A002450 0,3
%A A002450 N. J. A. Sloane (njas(AT)research.att.com).
%E A002450 More terms from Artur Jasinski (grafix(AT)csl.pl), Jan 27 2006
%E A002450 More terms from Artur Jasinski (grafix(AT)csl.pl), Nov 12 2007
%E A002450 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009