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%I #72 Dec 19 2023 15:18:37
%S 0,4,20,84,340,1364,5460,21844,87380,349524,1398100,5592404,22369620,
%T 89478484,357913940,1431655764,5726623060,22906492244,91625968980,
%U 366503875924,1466015503700,5864062014804,23456248059220,93824992236884,375299968947540,1501199875790164
%N a(n) = (4/3)*(4^n - 1).
%C a(n) 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 move along one edge on the lattice. - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 10 2005
%C Conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4 and 5 as a digit. - _Alexandre Wajnberg_, Apr 25 2005
%C Gives the values of m such that binomial(4*m + 4,m) is odd. Cf. A002450, A020988 and A263132. - _Peter Bala_, Oct 11 2015
%C Also the partial sums of 4^n for n>0, cf. A000302. - _Robert G. Wilson v_, Sep 18 2016
%H Vincenzo Librandi, <a href="/A080674/b080674.txt">Table of n, a(n) for n = 0..170</a>
%H Peter Bala, <a href="/A002450/a002450.txt">A characterization of A002450, A020988 and A080674.</a>
%H Mattia Fregola, <a href="https://docs.google.com/spreadsheets/d/1xApi5KNDGye7I2lxsdlbvng_r7On6G2AyPxUduD3ANw/edit?usp=sharing">Cellular Automata RULE13 generating OEIS sequence A080674</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F a(n) = 2*A020988(n) = A002450(n+1) - 1 = 4*A002450(n).
%F a(n) = Sum_{i = 1..n} 4^i. - Adam McDougall (mcdougal(AT)stolaf.edu), Sep 29 2004
%F a(n) = 4*a(n-1) + 4. - _Alexandre Wajnberg_, Apr 25 2005
%F a(n) = 4^n + a(n-1) (with a(0) = 0). - _Vincenzo Librandi_, Aug 08 2010
%F From _Colin Barker_, Oct 12 2015: (Start)
%F a(n) = 5*a(n-1) - 4*a(n-2).
%F G.f.: 4*x / ((x-1)*(4*x-1)). (End)
%F E.g.f.: 4*(exp(x)*(exp(3*x) - 1)/3 - _Elmo R. Oliveira_, Dec 17 2023
%t Table[4*(4^n-1)/3,{n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 30 2012 *)
%t LinearRecurrence[{5,-4},{0,4},40] (* _Harvey P. Dale_, May 05 2018 *)
%o (Magma) [(4/3)*(4^n-1): n in [0..40] ]; // _Vincenzo Librandi_, Apr 28 2011
%o (PARI) vector(100, n, n--; (4/3)*(4^n-1)) \\ _Altug Alkan_, Oct 11 2015
%o (PARI) Vec(4*x/((x-1)*(4*x-1)) + O(x^40)) \\ _Colin Barker_, Oct 12 2015
%Y Row n = 4 of A228275.
%Y Cf. A000301, A000302, A002450, A020988, A024036, A080674, A263132.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Mar 02 2003
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