

A145979


a(n) = (2*n + 4)/gcd(n,4).


23



1, 6, 4, 10, 3, 14, 8, 18, 5, 22, 12, 26, 7, 30, 16, 34, 9, 38, 20, 42, 11, 46, 24, 50, 13, 54, 28, 58, 15, 62, 32, 66, 17, 70, 36, 74, 19, 78, 40, 82, 21, 86, 44, 90, 23, 94, 48, 98, 25, 102, 52, 106, 27, 110, 56, 114, 29, 118, 60, 122
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OFFSET

0,2


COMMENTS

Previous name was: Square root of A061038(n+2).
a(n) = denominator(Sum_{k=1..n} 1/((k+1)*(k+2))), n > 0. This summation has a closed form of 1/2  1/(n+2) and numerator of A060819(n).  Gary Detlefs, Sep 16 2011
Prefixing this sequence with 2 makes it a shift of the involution b defined on positive integers by b(n) = n if 4n, b(n) = 2n if n is odd, b(n) = n/2 if n mod 4 = 2. This sequence b, when n > 2, occurs as the number of congruent regular ngons in various ways of making cycles of them by sticking them together along edges with constant rotation angle between the two stickings on any one ngon. For example, it is well known that only the triangle, square and hexagon can make cycles going once around a common point. But allowing the ngons to keep going any number of times around the common corner, they will eventually close up into a cycle for any n (since their corner interior angle is a rational multiple of Pi), and the number of ngons in that cycle is b(n).  David Pasino, Nov 12 2017
Here is another example of b(n) in the behavior of regular polygons as said in the comment of Nov 12 2017. For integers n and k, both exceeding 2, consider congruent regular kgon tiles arranged as a ring going once around a central region, each tile adjacent to two others by sharing an exact edge, such that, if possible for n and k, the centers of the kgons are the vertices of a regular ngon. Then for any given n, the numbers k for which this arrangement is possible are exactly the multiples of b(n). (In the cases where (n, k) is (3, 6) or (4, 4) or (6, 3), the central region is only a point.)  David Pasino, Feb 20 2018
The generating function of the rationals A060819(n)/a(n) = 1/2  1/(n+2), n >= 0, with A060819(0) = 0, mentioned in the comment on a sum by Gary Detlefs above is (1/2)*(1hypergeom([1, 1], [3], x/(1x)))/(1x) = (x*(2  x) + 2*(1  x)*log(1x) )/(2*(1x)*x^2). Thanks to him for leading me to Jolley's general remark (201) on p. 38 on such sums.  Wolfdieter Lang, Mar 08 2018


REFERENCES

L. B. W. Jolley, Summation of Series, Second revised ed., Dover, 1961, p. 38, (201). For the sum given in the comment by Gary Detlefs.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,1).


FORMULA

a(2n) = A022998(n+1). a(2n+1) = A016825(n+1) = 2*A005408(n+1).
a(4n) = 2n+1 = A005408(n). a(4n+2) = A008586(n+1) = 4*A000027(n+1).
From R. J. Mathar, Dec 08 2008: (Start)
a(n) = 2*a(n4)  a(n8).
G.f.: (1 + 6x + 4x^2 + 10x^3 + x^4 + 2x^5  2x^7) / ((x1)^2*(1+x)^2*(x^2+1)^2). (End)
a(n) = (n+2)*(11  5*(1)^n  i^n  (i)^n)/8, where i is the imaginary unit.  Bruno Berselli, Feb 25 2011
a(n) = A060819(n) + A060819(n+4).  Paul Curtz, Mar 13 2011
a(n) = (2*n + 4)/gcd(n,4).  Joerg Arndt, Jan 17 2015
E.g.f.: (1/4)*(2*(4*x+3)*cosh(x) + (3*x+16)*sinh(x) + x*sin(x)  2*cos(x)).  G. C. Greubel, Jan 29 2016
a(n) = b(n+2), for b as in comment of Nov 12 2017. Same b is b(n) = (2n)/gcd(2n, n+2).  David Pasino, Feb 20 2018


MAPLE

seq(denom(1/21/(n+2)), n=0..25); # Gary Detlefs, Sep 16 2011


MATHEMATICA

Table[(2*n + 4)/GCD[n, 4], {n, 0, 50}] (* G. C. Greubel, Jan 29 2016 *)
LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, 1}, {1, 6, 4, 10, 3, 14, 8, 18}, 70] (* Vincenzo Librandi, Jan 29 2016 *)
CoefficientList[ Series[(2x^7 + 2x^5 + x^4 + 10x^3 + 4x^2 + 6x + 1)/(x^4  1)^2, {x, 0, 60}], x] (* Robert G. Wilson v, Nov 25 2016 *)


PROG

(Sage)
a = lambda n: (2 + n) / (2  (n % 2) / 2  (n % 4 != 0))
[a(n) for n in range(60)] # Peter Luschny, Jan 17 2015
(MAGMA) [(2*n+4)/GCD(n, 4): n in [0..70]]; // Vincenzo Librandi, Jan 29 2016
(PARI) a(n) = (2*n + 4)/gcd(n, 4); \\ Michel Marcus, Jan 29 2016
(GAP) List([1..70], n>(2*n+4)/Gcd(n, 4)); # Muniru A Asiru, Apr 08 2018


CROSSREFS

Sequence in context: A264770 A143520 A075450 * A182164 A294093 A257926
Adjacent sequences: A145976 A145977 A145978 * A145980 A145981 A145982


KEYWORD

nonn


AUTHOR

Paul Curtz, Oct 26 2008


EXTENSIONS

Edited by R. J. Mathar, Dec 08 2008
New name from Joerg Arndt, Jan 17 2015


STATUS

approved



