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%I M3914 N1608 #559 Mar 24 2024 13:22:54
%S 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,
%T 22369621,89478485,357913941,1431655765,5726623061,22906492245,
%U 91625968981,366503875925,1466015503701,5864062014805,23456248059221,93824992236885,375299968947541
%N a(n) = (4^n - 1)/3.
%C 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_, Dec 18 2001
%C 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 greater than 0 are 1, 101, 10101, 1010101, etc. - _John McNamara_, Jan 16 2002
%C Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - _R. H. Hardin_, Mar 16 2002
%C The Collatz-function iteration started at a(n), for n >= 1, will end at 1 after 2*n+1 steps. - _Labos Elemer_, Sep 30 2002 [corrected by _Wolfdieter Lang_, Aug 16 2021]
%C Second binomial transform of A001045. - _Paul Barry_, Mar 28 2003
%C All members of sequence are also generalized octagonal numbers (A001082). - _Matthew Vandermast_, Apr 10 2003
%C Also sum of squares of divisors of 2^(n-1): a(n) = A001157(A000079(n-1)), for n > 0. - _Paul Barry_, Apr 11 2003
%C Binomial transform of A000244 (with leading zero). - _Paul Barry_, Apr 11 2003
%C 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_, May 31 2004
%C Also number of walks of length 2n + 1 between two vertices at distance 3 in the cycle graph C_12. - _Herbert Kociemba_, Jul 05 2004
%C a(n+1) is the number of steps that 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 a(n+1) is the sum of square divisors of 4^n. - _Paul Barry_, Oct 13 2005
%C 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 W. Weisstein_, Apr 08 2006
%C a(k) = [M^k]_2,1, where M is the 3 X 3 matrix defined as follows: M = [1, 1, 1; 1, 3, 1; 1, 1, 1]. - _Simone Severini_, Jun 11 2006
%C 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 k is in the sequence if and only if C(4k + 1, k) (A052203) is odd. - _Paul Barry_, Mar 26 2007
%C This sequence also gives the number of distinct 3-colorings of the odd cycle C(2*n - 1). - _Keith Briggs_, Jun 19 2007
%C All numbers of the form m*4^m + (4^m-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 m*4^m + (4^m-1)/3 = 4(4(..4(4m + 1) + 1) + 1) + 1 ..) + 1. - _Artur Jasinski_, Nov 12 2007
%C For n > 0, terms are the numbers that, in base 4, are repunits: 1_4, 11_4, 111_4, 1111_4, etc. - _Artur Jasinski_, Sep 30 2008
%C 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). - _Milan Janjic_, Jan 27 2010
%C 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. - _Wolfdieter Lang_, Oct 18 2010
%C 6*a(n) + 1 is every second Mersenne number greater than or equal to M3, hence all Mersenne primes greater than M2 must be a 6*a(n) + 1 of this sequence. - _Roderick MacPhee_, Nov 01 2010
%C Smallest number having alternating bit sum n. Cf. A065359.
%C For n = 1, 2, ..., the last digit of a(n) is 1, 5, 1, 5, ... . - _Washington Bomfim_, Jan 21 2011
%C Rule 50 elementary cellular automaton generates this sequence. This sequence also appears in the second column of array in A173588. - _Paul Muljadi_, Jan 27 2011
%C 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
%C a(n), n >= 1, is also the inverse of 3, denoted by 3^(-1), Modd(2^(2*n - 1)). For Modd n see a comment on A203571. E.g., a(2) = 5, 3 * 5 = 15 == 1 (Modd 8), because floor(15/8) = 1 is odd and -15 == 1 (mod 8). For n = 1 note that 3 * 1 = 3 == 1 (Modd 2) because floor(3/2) = 1 and -3 == 1 (mod 2). The inverse of 3 taken Modd 2^(2*n) coincides with 3^(-1) (mod 2^(2*n)) given in A007583(n), n >= 1. - _Wolfdieter Lang_, Mar 12 2012
%C If an AVL tree has a leaf at depth n, then the tree can contain no more than a(n+1) nodes total. - _Mike Rosulek_, Nov 20 2012
%C Also, this is the Lucas sequence V(5, 4). - _Bruno Berselli_, Jan 10 2013
%C Also, for n > 0, a(n) is an odd number whose Collatz trajectory contains no odd number other than n and 1. - _Jayanta Basu_, Mar 24 2013
%C Sum_{n >= 1} 1/a(n) converges to (3*(log(4/3) - QPolyGamma[0, 1, 1/4]))/log(4) = 1.263293058100271... = A321873. - _K. G. Stier_, Jun 23 2014
%C Consider n spheres in R^n: the i-th one (i=1, ..., n) has radius r(i) = 2^(1-i) and the coordinates of its center are (0, 0, ..., 0, r(i), 0, ..., 0) where r(i) is in position i. The coordinates of the intersection point in the positive orthant of these spheres are (2/a(n), 4/a(n), 8/a(n), 16/a(n), ...). For example in R^2, circles centered at (1, 0) and (0, 1/2), and with radii 1 and 1/2, meet at (2/5, 4/5). - _Jean M. Morales_, May 19 2015
%C From _Peter Bala_, Oct 11 2015: (Start)
%C a(n) gives the values of m such that binomial(4*m + 1,m) is odd. Cf. A003714, A048716, A263132.
%C 2*a(n) = A020988(n) gives the values of m such that binomial(4*m + 2, m) is odd.
%C 4*a(n) = A080674(n) gives the values of m such that binomial(4*m + 4, m) is odd. (End)
%C Collatz Conjecture Corollary: Except for powers of 2, the Collatz iteration of any positive integer must eventually reach a(n) and hence terminate at 1. - _Gregory L. Simay_, May 09 2016
%C Number of active (ON, black) cells at stage 2^n - 1 of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 16 2016
%C From _Luca Mariot_ and _Enrico Formenti_, Sep 26 2016: (Start)
%C a(n) is also the number of coprime pairs of polynomials (f, g) over GF(2) where both f and g have degree n + 1 and nonzero constant term.
%C a(n) is also the number of pairs of one-dimensional binary cellular automata with linear and bipermutive local rule of neighborhood size n+1 giving rise to orthogonal Latin squares of order 2^m, where m is a multiple of n. (End)
%C Except for 0, 1 and 5, all terms are Brazilian repunits numbers in base 4, and so belong to A125134. For n >= 3, all these terms are composite because a(n) = {(2^n-1) * (2^n + 1)}/3 and either (2^n - 1) or (2^n + 1) is a multiple of 3. - _Bernard Schott_, Apr 29 2017
%C Given the 3 X 3 matrix A = [2, 1, 1; 1, 2, 1; 1, 1, 2] and the 3 X 3 unit matrix I_3, A^n = a(n)(A - I_3) + I_3. - _Nicolas Patrois_, Jul 05 2017
%C The binary expansion of a(n) (n >= 1) consists of n 1's alternating with n - 1 0's. Example: a(4) = 85 = 1010101_2. - _Emeric Deutsch_, Aug 30 2017
%C a(n) (n >= 1) is the viabin number of the integer partition [n, n - 1, n - 2, ..., 2, 1] (for the definition of viabin number see comment in A290253). Example: a(4) = 85 = 1010101_2; consequently, the southeast border of the Ferrers board of the corresponding integer partition is ENENENEN, where E = (1, 0), N = (0, 1); this leads to the integer partition [4, 3, 2, 1]. - _Emeric Deutsch_, Aug 30 2017
%C Numbers whose binary and Gray-code representations are both palindromes (i.e., intersection of A006995 and A281379). - _Amiram Eldar_, May 17 2021
%C Starting with n = 1 the sequence satisfies {a(n) mod 6} = repeat{1, 5, 3}. - _Wolfdieter Lang_, Jan 14 2022
%C Terms >= 5 are those q for which the multiplicative order of 2 mod q is floor(log_2(q)) + 2 (and which is 1 more than the smallest possible order for any q). - _Tim Seuré_, Mar 09 2024
%C The order of 2 modulo a(n) is 2*n for n >= 2. - _Joerg Arndt_, Mar 09 2024
%D 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 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002450/b002450.txt">Table of n, a(n) for n = 0..200</a>
%H Peter Bala, <a href="/A002450/a002450.txt">A characterization of A002450, A020988 and A080674</a>.
%H Henry Bottomley, <a href="/A060919/a060919.gif">Illustration of initial terms</a>
%H Sung-Hyuk Cha, <a href="http://naun.org/multimedia/UPress/ami/16-125.pdf">On Complete and Size Balanced k-ary Tree Integer Sequences</a>, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012
%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 8.
%H D. Dumont, <a href="/A001469/a001469_3.pdf">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
%H David Eppstein, <a href="https://arxiv.org/abs/1804.07396">Making Change in 2048</a>, arXiv:1804.07396 [cs.DM], 2018.
%H C. Ernst and D. W. Sumners, <a href="http://dx.doi.org/10.1017/S0305004100067323">The Growth of the Number of Prime Knots</a>, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987
%H Ernesto Estrada and José A. de la Peña, <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 Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
%H Enrico Formenti and Luca Mariot, <a href="https://lucamariot.org/files/slides/talk_automata_2023_poly.pdf">Exhaustive Generation of Linear Orthogonal CA</a>, Automata 2023, Univ. Twente (Netherlands), see p. 22 of 38.
%H Mattia Fregola, <a href="https://docs.google.com/spreadsheets/d/1629UXZ07lVK1-LVR0T7u1RDVKeW4f55K688CAtS5maw/edit?usp=sharing">Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450</a>
%H A. Frosini and S. Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Frosini/fros2.html">On the Sequence A079500 and Its Combinatorial Interpretations</a>, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
%H Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=373">Encyclopedia of Combinatorial Structures 373</a>
%H Petro Kosobutskyy, Anastasiia Yedyharova, and Taras Slobodzyan, <a href="https://doi.org/10.23939/cds2023.01.121">From Newton's binomial and Pascal's triangle to Collatz's problem</a>, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, 121-127.
%H Wolfdieter Lang, <a href="/A002450/a002450.pdf">Notes on certain inhomogeneous three term recurrences.</a>
%H J. V. Leyendekkers and A.G. Shannon, <a href="http://www.nntdm.net/papers/nntdm-17/NNTDM-17-2-47-51.pdf">Modular Rings and the Integer 3</a>, Notes on Number Theory & Discrete Mathematics, 17 (2011), 47-51.
%H Luca Mariot, <a href="https://lucamariot.org/files/slides/talk_zagreb_2017.pdf">Cryptography by Cellular Automata</a>, 2017.
%H Luca Mariot, <a href="https://lucamariot.org/files/slides/talk_iwoca_2020.pdf">Orthogonal labelings in de Bruijn graphs</a>, IWOCA 2020 - Open Problems Session, Delft University of Technology (Netherlands).
%H Luca Mariot, <a href="https://lucamariot.org/files/slides/talk_algebra_seminars_2023.pdf">Connections between Latin squares, Cellular Automata and Coprime Polynomials</a>, Univ. Twente (Netherlands, 2023). See p. 16/37.
%H Luca Mariot and Enrico Formenti, <a href="http://openit.disco.unimib.it/~mariot/mf_counting_coprime_polynomials_2016.pdf">The number of coprime/non-coprime pairs of polynomials over F_2 with degree n and nonzero constant term</a>.
%H Luca Mariot, Enrico Formenti and Alberto Leporati, <a href="http://openit.disco.unimib.it/~mariot/mfl_short_paper_automata_2016.pdf">Constructing Orthogonal Latin Squares from Linear Cellular Automata</a>. In: Exploratory papers of AUTOMATA 2016.
%H Luca Mariot, Maximilien Gadouleau, Enrico Formenti, and Alberto Leporati, <a href="https://arxiv.org/abs/1906.08249">Mutually Orthogonal Latin Squares based on Cellular Automata</a>, arXiv:1906.08249 [cs.DM], 2019.
%H Luca Mariot, <a href="https://lucamariot.org/files/slides/talk_matapp_2023.pdf">Counting Coprime Polynomials over Finite Fields with Formal Languages and Compositions of Natural Numbers</a>, Univ. Twente (Netherlands 2023). See p. 11.
%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 Mező István, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mezo/mezo5.html">Several Generating Functions for Second-Order Recurrence Sequences </a>, JIS 12 (2009) 09.3.7.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, <a href="https://arxiv.org/abs/2307.08073">The sequence of higher order Mersenne numbers and associated binomial transforms</a>, arXiv:2307.08073 [math.NT], 2023.
%H A. G. Shannon, <a href="http://www.nntdm.net/papers/nntdm-17/NNTDM-17-4-09-13.pdf">Some recurrence relations for binary sequence matrices</a>, NNTDM 17 (2011), 4, 913. - From _N. J. A. Sloane_, Jun 13 2012
%H T. N. Thiele, <a href="https://archive.org/details/interpolationsre00thieuoft">Interpolationsrechnung</a>, Teubner, Leipzig, 1909, p. 35.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>, <a href="http://mathworld.wolfram.com/Rule250.html">Rule 250</a>, <a href="http://mathworld.wolfram.com/PrimeKnot.html">Prime Knot</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Elementary_cellular_automaton">Elementary cellular automaton</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.
%H Michael Williams, <a href="https://doi.org/10.13140/RG.2.2.29146.31686">Collatz conjecture: an order isomorphic recursive machine</a>, ResearchGate (2024). See pp. 8, 13.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F From _Wolfdieter Lang_, Apr 24 2001: (Start)
%F a(n+1) = Sum_{m = 0..n} A060921(n, m).
%F G.f.: x/((1-x)*(1-4*x)). (End)
%F a(n) = Sum_{k = 0..n-1} 4^k; a(n) = A001045(2*n). - _Paul Barry_, Mar 17 2003
%F E.g.f.: (exp(4*x) - exp(x))/3. - _Paul Barry_, Mar 28 2003
%F a(n) = (A007583(n) - 1)/2. - _N. J. A. Sloane_, May 16 2003
%F a(n) = A000975(2*n)/2. - _N. J. A. Sloane_, Sep 13 2003
%F a(n) = A084160(n)/2. - _N. J. A. Sloane_, Sep 13 2003
%F a(n+1) = 4*a(n) + 1, with a(0) = 0. - _Philippe Deléham_, Feb 25 2004
%F 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
%F a(n+1) = Sum_{k = 0..n} binomial(n+1, k+1)*3^k. - _Paul Barry_, Aug 20 2004
%F a(n) = center term in M^n * [1 0 0], where M is the 3 X 3 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_, Dec 18 2004
%F a(n) = Sum_{k = 0..n, j = 0..n} C(n, j)*C(j, k)*A001045(j - k). - _Paul Barry_, Feb 15 2005
%F 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_, Apr 22 2005
%F a(n) = A125118(n, 3) for n > 2. - _Reinhard Zumkeller_, Nov 21 2006
%F a(n) = Sum_{k = 0..n} 2^(n - k)*A128908(n, k), n >= 1. - _Philippe Deléham_, Oct 19 2008
%F a(n) = Sum_{k = 0..n} A106566(n, k)*A100335(k). - _Philippe Deléham_, Oct 30 2008
%F If we define f(m, j, x) = Sum_{k = j..m} binomial(m, k)*stirling2(k, j)*x^(m - k) then a(n-1) = f(2*n, 4, -2), n >= 2. - _Milan Janjic_, Apr 26 2009
%F a(n) = A014551(n) * A001045(n). - _R. J. Mathar_, Jul 08 2009
%F 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. - _Wolfdieter Lang_, Oct 18 2010
%F a(0) = 0, a(n+1) = a(n) + 2^(2*n). - _Washington Bomfim_, Jan 21 2011
%F A036555(a(n)) = 2*n. - _Reinhard Zumkeller_, Jan 28 2011
%F a(n) = Sum_{k = 1..floor((n+2)/3)} C(2*n + 1, n + 2 - 3*k). - _Mircea Merca_, Jun 25 2011
%F a(n) = Sum_{i = 1..n} binomial(2*n + 1, 2*i)/3. - _Wesley Ivan Hurt_, Mar 14 2015
%F a(n+1) = 2^(2*n) + a(n), a(0) = 0. - _Ben Paul Thurston_, Dec 27 2015
%F a(k*n)/a(n) = 1 + 4^n + ... + 4^((k-1)*n). - _Gregory L. Simay_, Jun 09 2016
%F Dirichlet g.f.: (PolyLog(s, 4) - zeta(s))/3. - _Ilya Gutkovskiy_, Jun 26 2016
%F A000120(a(n)) = n. - _André Dalwigk_, Mar 26 2018
%F a(m) divides a(m*n), in particular: a(2*n) == 0 (mod 5), a(3*n) == 0 (mod 3*7), a(5*n) == 0 (mod 11*31), etc. - _M. F. Hasler_, Oct 19 2018
%F a(n) = 4^(n-1) + a(n-1). - _Bob Selcoe_, Jan 01 2020
%F a(n) = A178415(1, n) = A347834(1, n-1), arrays, for n >= 1. - _Wolfdieter Lang_, Nov 29 2021
%F a(n) = A000225(2*n)/3. - _John Keith_, Jan 22 2022
%F a(n) = A080674(n) + 1 = A047849(n) - 1 = A163834(n) - 2 = A155701(n) - 3 = A163868(n) - 4 = A156605(n) - 7. - _Ray Chandler_, Jun 16 2023
%e Apply Collatz iteration to 9: 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1.
%e Apply Collatz iteration to 27: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1. [Corrected by _Sean A. Irvine_ at the suggestion of Stephen Cornelius, Mar 04 2024]
%e a(5) = (4^5 - 1)/3 = 341 = 11111_4 = {(2^5 - 1) * (2^5 + 1)}/3 = 31 * 33/3 = 31 * 11. - _Bernard Schott_, Apr 29 2017
%p [seq((4^n-1)/3,n=0..40)];
%p A002450:=1/(4*z-1)/(z-1); # _Simon Plouffe_ in his 1992 dissertation, dropping the initial zero
%t Table[(4^n - 1)/3, {n, 0, 127}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 29 2008 *)
%t LinearRecurrence[{5, -4}, {0, 1}, 30] (* _Harvey P. Dale_, Jun 23 2013 *)
%o (Magma) [ (4^n-1)/3: n in [0..25] ]; // _Klaus Brockhaus_, Oct 28 2008
%o (Magma) [n le 2 select n-1 else 5*Self(n-1)-4*Self(n-2): n in [1..70]]; // _Vincenzo Librandi_, Jun 13 2015
%o (PARI) a(n) = (4^n-1)/3;
%o (PARI) my(z='z+O('z^40)); Vec(z/((1-z)*(1-4*z))) \\ _Altug Alkan_, Oct 11 2015
%o (Haskell)
%o a002450 = (`div` 3) . a024036
%o a002450_list = iterate ((+ 1) . (* 4)) 0
%o -- _Reinhard Zumkeller_, Oct 03 2012
%o (Maxima) makelist((4^n-1)/3, n, 0, 30); /* _Martin Ettl_, Nov 05 2012 */
%o (GAP) List([0..25], n -> (4^n-1)/3); # _Muniru A Asiru_, Feb 18 2018
%o (Scala) ((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)(_ * _)).scanLeft(0: BigInt)(_ + _) // _Alonso del Arte_, Sep 17 2019
%o (Python)
%o def A002450(n): return ((1<<(n<<1))-1)//3 # _Chai Wah Wu_, Jan 29 2023
%Y Partial sums of powers of 4, A000302.
%Y When converted to binary, this gives A094028.
%Y Subsequence of A003714.
%Y Primitive factors: A129735.
%Y Cf. A000225, A002446, A006995, A007583, A018215, A020988, A024036, A047849, A048716, A080355, A080674, A112627, A113860, A139391, A155701, A156605, A160967, A163834, A163868, A178415, A263132, A281379, A347834.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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