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%I M4193 N1748 #53 Sep 08 2022 08:44:30
%S 0,6,30,126,510,2046,8190,32766,131070,524286,2097150,8388606,
%T 33554430,134217726,536870910,2147483646,8589934590,34359738366,
%U 137438953470,549755813886,2199023255550,8796093022206,35184372088830
%N a(n) = 2^(2*n+1) - 2.
%D H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
%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 S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
%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 Vincenzo Librandi, <a href="/A002446/b002446.txt">Table of n, a(n) for n = 0..300</a>
%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F G.f.: 6*x/((1-x)*(1-4*x)). - _Simon Plouffe_, see MAPLE line
%F E.g.f.: (cos(i*x)-1)^2. - _Vladimir Kruchinin_, Oct 28 2012
%p A002446:=6*z/((4*z-1)*(z-1)); # [Generating function. _Simon Plouffe_ in his 1992 dissertation.]
%t f[n_] := Det[{{1, 1}, {1, 4}}^(n - 1) {{1, 2}, {1, 2}}]; Array[f, 30] (* _Robert G. Wilson v_, Jul 13 2011 *)
%t 2^(2*Range[0,30]+1)-2 (* or *) LinearRecurrence[{5,-4},{0,6},30] (* _Harvey P. Dale_, Sep 01 2016 *)
%o (Magma) [2^(2*n+1) - 2: n in [0..30]]; // _Vincenzo Librandi_, Jun 01 2011
%o (PARI) a(n) = 2*(4^n - 1); \\ _G. C. Greubel_, Jul 04 2019
%o (Sage) [2*(4^n -1) for n in (0..30)] # _G. C. Greubel_, Jul 04 2019
%o (GAP) List([0..30], n-> 2*(4^n - 1)) # _G. C. Greubel_, Jul 04 2019
%Y Equals 6 * A002450(n).
%Y A diagonal of the triangle in A241171.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vincenzo Librandi_, Jun 01 2011
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