A047451 Numbers that are congruent to {0, 6} mod 8.

0, 6, 8, 14, 16, 22, 24, 30, 32, 38, 40, 46, 48, 54, 56, 62, 64, 70, 72, 78, 80, 86, 88, 94, 96, 102, 104, 110, 112, 118, 120, 126, 128, 134, 136, 142, 144, 150, 152, 158, 160, 166, 168, 174, 176, 182, 184, 190, 192, 198, 200, 206, 208, 214, 216, 222, 224, 230

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

All even numbers m such that Integral_{x=0..2*Pi} Product_{i=1..m/2} cos(2*i*x) dx is nonzero. - William Boyles, Oct 12 2019

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Union of A008590 and A017137.

Cf. A030308, A047504 (complement).

Programs

Mathematica

Array[8 # + {0, 6} &, 29, 0] // Flatten (* or *)
Rest@ CoefficientList[Series[2 x^2*(3 + x)/((1 + x) (x - 1)^2), {x, 0, 58}], x] (* Michael De Vlieger, Nov 18 2019 *)
LinearRecurrence[{1,1,-1},{0,6,8},80] (* Harvey P. Dale, Apr 09 2022 *)

PARI

forstep(n=0,200,[6,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

PARI

a(n) = 4*n - 3 + (-1)^n; \\ David Lovler, Jul 25 2022

Formula

a(n) = 8*n - a(n-1) - 10 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010

From R. J. Mathar, Oct 08 2011: (Start)

a(n) = 4*n - 3 + (-1)^n.

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

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=6 and b(k)=2^(k+2) for k > 0. - Philippe Deléham, Oct 17 2011

a(n) = ceiling((8/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012

Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - Pi/16. - Amiram Eldar, Dec 18 2021

E.g.f.: (4*x + 1)*exp(x) - exp(-x) = 4*x*exp(x) + 2*sinh(x). - David Lovler, Aug 02 2022