A047291 Numbers that are congruent to {0, 1, 4, 6} mod 7.

0, 1, 4, 6, 7, 8, 11, 13, 14, 15, 18, 20, 21, 22, 25, 27, 28, 29, 32, 34, 35, 36, 39, 41, 42, 43, 46, 48, 49, 50, 53, 55, 56, 57, 60, 62, 63, 64, 67, 69, 70, 71, 74, 76, 77, 78, 81, 83, 84, 85, 88, 90, 91, 92, 95, 97, 98, 99, 102, 104, 105, 106, 109, 111

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A047336, A047345.

Programs

Magma

I:=[0, 1, 4, 6, 7]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Apr 26 2012

Maple

A047291:=n->(-13-(-1)^n+(3-I)*(-I)^n+(3+I)*I^n+14*n)/8: seq(A047291(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Mathematica

Select[Range[0,120], MemberQ[{0,1,4,6}, Mod[#,7]]&] (* Vincenzo Librandi, Apr 26 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,4,6,7},100] (* G. C. Greubel, Jun 01 2016 *)

PARI

x='x+O('x^100); concat(0, Vec(x^2*(1+3*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

Formula

From Colin Barker, Mar 13 2012: (Start)

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

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

a(n) = (-13 - (-1)^n + (3-i)*(-i)^n + (3+i)*i^n + 14*n)/8 where i=sqrt(-1). - Colin Barker, May 14 2012

a(2k) = A047336(k), a(2k-1) = A047345(k). - Wesley Ivan Hurt, Jun 01 2016

E.g.f.: (4 - sin(x) + 3*cos(x) + (7*x - 6)*sinh(x) + 7*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, Jun 01 2016