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

1,2

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).

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

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 *)

Prog

(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

Crossrefs

Union of A008590 and A017137.

Cf. A030308, A047504 (complement).

Sequence in context: A359827 A315884 A178950 * A194399 A315885 A315886

Adjacent sequences: A047448 A047449 A047450 * A047452 A047453 A047454

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved