A130824 a(n) = 2*A004273(n).

0, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 0

Author

Paul Curtz, Jul 17 2007

Keywords

• nonn
• easy

Comments

Equals A111284 from the 2nd term on. - R. J. Mathar, Jun 13 2008

Besides the first term, this sequence gives the denominators of the alternating series Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - Mohammad K. Azarian, Oct 14 2011 [edited by Jon E. Schoenfield, Mar 07 2015]

Numbers that cannot be a side of a primitive Pythagorean triangle. - Torlach Rush, Nov 07 2019

Simple continued fraction expansion of tanh(1/2) = (e - 1)/(e + 1) = 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))). - Peter Bala, Oct 01 2023

References

• Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A016825, A111284, A160327.

Programs

GAP

Concatenation([0], List([1..60], n-> 4*n-2 )); # G. C. Greubel, Dec 30 2019

Magma

[4*n-2*Floor((n+2) mod (n+1)):n in [0..60]]; // Vincenzo Librandi, Sep 22 2011

Maple

A130827 := proc(n) if n =0 then 0 ; else 4*n-2 ; fi ; end: seq(A130827(n),n=0..120) ; # R. J. Mathar, Oct 28 2007

Mathematica

2 Join[{0}, Range[1, 200, 2]] (* Michael De Vlieger, Mar 07 2015 *)

PARI

vector(61, n, if(n==1, 0, 4*(n-1) -2) ) \\ G. C. Greubel, Dec 30 2019

Sage

[0]+[4*n-2 for n in (1..60)] # G. C. Greubel, Dec 30 2019

Formula

From Stefano Spezia, Dec 09 2019: (Start)

G.f.: 2*x*(1+x)/(1-x)^2.

a(n) = 2*a(n-1) - a(n-2) for n > 0.

a(n) = 4*n - 1 - (-1)^(2^n). (End)

E.g.f: 2*(1 - (1-2*x)*exp(x)). - G. C. Greubel, Dec 30 2019

Extensions

More terms from R. J. Mathar, Oct 28 2007