A162330 Blocks of 4 numbers of the form 2k, 2k-1, 2k, 2k+1, k=1,2,3,4,...

2, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 7, 8, 9, 10, 9, 10, 11, 12, 11, 12, 13, 14, 13, 14, 15, 16, 15, 16, 17, 18, 17, 18, 19, 20, 19, 20, 21, 22, 21, 22, 23, 24, 23, 24, 25, 26, 25, 26, 27, 28, 27, 28, 29, 30, 29, 30, 31, 32, 31, 32, 33, 34, 33, 34, 35, 36, 35, 36, 37, 38, 37

Offset: 1

Author

Juri-Stepan Gerasimov, Jul 01 2009

Keywords

• nonn
• easy

Comments

This illustrates the infinite product Pi/2 = Product_{k>=1} ((2*k)/(2k-1))*((2k)/(2k+1)): read the 4 terms of numerator and denominator of the factor in the product in that order shown.

Number of roots of the polynomial 1+x+x^2+...+x^(n+1) = (x^(n+2)-1)/(x-1) in the left half plane. - Michel Lagneau, Oct 30 2012

Links

• Eric Weisstein's World of Mathematics, Wallis Formula.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A000796, A019669, A134967, A183041.

Programs

Mathematica

Flatten[#+{0,-1,0,1}&/@Range[2,40,2]] (* Harvey P. Dale, Aug 12 2014 *)

PARI

A162330(n)=n+1-(n+2)\4*2  \\ M. F. Hasler, Nov 01 2012

Formula

a(n) = a(n-1) + a(n-4) - a(n-5).

G.f.: x*(2-x+x^2+x^3-x^4)/((1+x)*(1+x^2)*(1-x)^2).

a(n) = n + 1 - 2*floor( (n+2)/4 ). - M. F. Hasler, Nov 01 2012

a(n) = (2*n + 3 - (-1)^n + 2*(-1)^((2*n - 1 + (-1)^n)/4))/4. - Luce ETIENNE, Mar 08 2016

Sum_{n>=1} (-1)^n/a(n) = 2*log(2) - 1. - Amiram Eldar, Sep 10 2023

Extensions

Edited by R. J. Mathar, Sep 16 2009