A047454 Numbers that are congruent to {1, 2, 3, 4} mod 8.

1, 2, 3, 4, 9, 10, 11, 12, 17, 18, 19, 20, 25, 26, 27, 28, 33, 34, 35, 36, 41, 42, 43, 44, 49, 50, 51, 52, 57, 58, 59, 60, 65, 66, 67, 68, 73, 74, 75, 76, 81, 82, 83, 84, 89, 90, 91, 92, 97, 98, 99, 100, 105, 106, 107, 108, 113, 114, 115, 116, 121, 122, 123

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A047463, A047471.

Programs

Magma

I:=[1, 2, 3, 4, 9]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 15 2012

Maple

A047454:=n->(-5-I^(2*n)-(1-I)*(-I)^n-(1+I)*I^n+4*n)/2: seq(A047454(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Mathematica

Select[Range[0,300], MemberQ[{1,2,3,4}, Mod[#,8]]&] (* Vincenzo Librandi, May 15 2012 *)

PARI

my(x='x+O('x^100)); Vec(x*(1+x+x^2+x^3+4*x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

Formula

From Colin Barker, May 14 2012: (Start)

a(n) = (-5-(-1)^n-(1-i)*(-i)^n-(1+i)*i^n+4*n)/2 where i=sqrt(-1).

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

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 15 2012

a(2k) = A047463(k), a(2k-1) = A047471(k). - Wesley Ivan Hurt, Jun 01 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 23 2021