login
a(n) = 2^(2*n+1) - 2.
(Formerly M4193 N1748)
7

%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