A047566 Numbers that are congruent to {4, 5, 6, 7} mod 8.

4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 68, 69, 70, 71, 76, 77, 78, 79, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers having a 1 in position 2 of their binary expansion. One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner, Jan 22 2006

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Maths Magic, Mystery Calculator.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A003628 (primes).

Mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421.

Programs

Haskell

a047566 n = a047566_list !! (n-1)
a047566_list = [n | n <- [1..], mod n 8 > 3]
-- Reinhard Zumkeller, Dec 29 2012

Maple

A047566:= n-> n+3 + 4*iquo(n-1, 4):
seq(A047566(n), n=1..100);  # Alois P. Heinz, Aug 22 2011

Mathematica

Flatten[# + {4, 5, 6, 7}&/@(8Range[0, 14])] (* Harvey P. Dale, Feb 02 2011 *)

Formula

G.f.: x*(4+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, May 19 2016: (Start)

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

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

E.g.f.: 1 + sin(x) - cos(x) + sinh(x) + 2*x*exp(x). - Ilya Gutkovskiy, May 20 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 - 3*log(2)/8. - Amiram Eldar, Dec 26 2021