A047409 Numbers that are congruent to {0, 1, 4, 6} mod 8.

0, 1, 4, 6, 8, 9, 12, 14, 16, 17, 20, 22, 24, 25, 28, 30, 32, 33, 36, 38, 40, 41, 44, 46, 48, 49, 52, 54, 56, 57, 60, 62, 64, 65, 68, 70, 72, 73, 76, 78, 80, 81, 84, 86, 88, 89, 92, 94, 96, 97, 100, 102, 104, 105, 108, 110, 112, 113, 116, 118, 120, 121, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

All squares and the products of any terms belong to the sequence. This sequence (n > 1) is closed under multiplication. - Klaus Purath, Feb 13 2023

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A008586, A047452.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 1, 4, 6]]; // Wesley Ivan Hurt, May 24 2016

Maple

A047409:=n->(8*n-9-I^(2*n)+I^(-n)+I^n)/4: seq(A047409(n), n=1..100); # Wesley Ivan Hurt, May 24 2016

Mathematica

Select[Range[0,110], MemberQ[{0,1,4,6}, Mod[#,8]]&] (* Harvey P. Dale, Sep 28 2011 *)

Formula

G.f.: x^2*(1 + 3*x + 2*x^2 + 2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011

From Wesley Ivan Hurt, May 24 2016: (Start)

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

a(n) = (8*n - 9 - i^(2n) + i^(-n) + i^n)/4 where i=sqrt(-1).

a(2k) = A047452(k), a(2k-1) = A008586(k-1) for k > 0. (End)

E.g.f.: (4 + cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016

Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*Pi/16 + (10 - sqrt(2))*log(2)/16 + sqrt(2)*log(2 + sqrt(2))/8. - Amiram Eldar, Dec 20 2021

a(n) = 2*(n-1) + floor((n+1)/4) - floor((n+2)/4). - Ridouane Oudra, Aug 19 2024