A002446 a(n) = 2^(2*n+1) - 2.

0, 6, 30, 126, 510, 2046, 8190, 32766, 131070, 524286, 2097150, 8388606, 33554430, 134217726, 536870910, 2147483646, 8589934590, 34359738366, 137438953470, 549755813886, 2199023255550, 8796093022206, 35184372088830

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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.
• 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.
• 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.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Crossrefs

Equals 6 * A002450(n).

A diagonal of the triangle in A241171.

Programs

GAP

List([0..30], n-> 2*(4^n - 1)) # G. C. Greubel, Jul 04 2019

Magma

[2^(2*n+1) - 2: n in [0..30]]; // Vincenzo Librandi, Jun 01 2011

Maple

A002446:=6*z/((4*z-1)*(z-1)); # [Generating function. Simon Plouffe in his 1992 dissertation.]

Mathematica

f[n_] := Det[{{1, 1}, {1, 4}}^(n - 1) {{1, 2}, {1, 2}}]; Array[f, 30] (* Robert G. Wilson v, Jul 13 2011 *)
2^(2*Range[0,30]+1)-2 (* or *) LinearRecurrence[{5,-4},{0,6},30] (* Harvey P. Dale, Sep 01 2016 *)

PARI

a(n) = 2*(4^n - 1); \\ G. C. Greubel, Jul 04 2019

Sage

[2*(4^n -1) for n in (0..30)] # G. C. Greubel, Jul 04 2019

Formula

G.f.: 6*x/((1-x)*(1-4*x)). - Simon Plouffe, see MAPLE line

E.g.f.: (cos(i*x)-1)^2. - Vladimir Kruchinin, Oct 28 2012

Extensions

More terms from Vincenzo Librandi, Jun 01 2011