A047541 Numbers that are congruent to {1, 2, 4, 7} mod 8.

1, 2, 4, 7, 9, 10, 12, 15, 17, 18, 20, 23, 25, 26, 28, 31, 33, 34, 36, 39, 41, 42, 44, 47, 49, 50, 52, 55, 57, 58, 60, 63, 65, 66, 68, 71, 73, 74, 76, 79, 81, 82, 84, 87, 89, 90, 92, 95, 97, 98, 100, 103, 105, 106, 108, 111, 113, 114, 116, 119, 121, 122, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A047461, A047524.

Programs

Magma

[n : n in [0..150] | n mod 8 in [1, 2, 4, 7]]; // Wesley Ivan Hurt, Jun 04 2016

Maple

A047541:=n->(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4: seq(A047541(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016

Mathematica

Table[(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
Select[Range[200],MemberQ[{1,2,4,7},Mod[#,8]]&] (* or  *) LinearRecurrence[ {2,-2,2,-1},{1,2,4,7},70] (* Harvey P. Dale, Jul 09 2020 *)

PARI

a(n)=n\4*8+[-1,1,2,4][n%4+1] \\ Charles R Greathouse IV, Nov 04 2011

Formula

From Wesley Ivan Hurt, Jun 04 2016: (Start)

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

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

a(n) = (1+i)*(n*(4-4*i)+3*i-3+i^(-n)-i^(1+n))/4 where i=sqrt(-1).

a(2k) = A047524(k), a(2k-1) = A047461(k). (End)

E.g.f.: (2 + sin(x) + cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, Jun 04 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)+1)*Pi/16 - log(2)/8. - Amiram Eldar, Dec 24 2021