A047207 Numbers that are congruent to {0, 1, 3, 4} mod 5.

0, 1, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers not ending in 2 or 7. - Bruno Berselli, Oct 30 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..8000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A001622, A030308, A047209, A047218.

Programs

Magma

[n : n in [0..100] | n mod 5 in [0, 1, 3, 4]]; // Wesley Ivan Hurt, May 30 2016

Maple

seq(floor((5*n-3)/4), n=1..57); # Gary Detlefs, Mar 06 2010

Mathematica

Flatten[Table[5*n + {0, 1, 3, 4}, {n, 0, 20}]] (* T. D. Noe, Nov 12 2013 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,3,4,5},100] (* Harvey P. Dale, Jan 31 2022 *)

PARI

forstep(n=0,99,[1,2,1,1],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

Formula

a(n) = floor((5*n-3)/4). - Gary Detlefs, Mar 06 2010

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

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=3, b(k)=5*2^(k-2) for k>1. - Philippe Deléham, Oct 17 2011

From Wesley Ivan Hurt, May 30 2016: (Start)

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

a(n) = (10*n-9-i^(2*n)+(1-i)*i^(-n)+(1+i)*i^n)/8, where i=sqrt(-1).

a(2*k) = A047209(k), a(2*k-1) = A047218(k). (End)

E.g.f.: (4 - sin(x) + cos(x) + (5*x - 4)*sinh(x) + 5*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, May 30 2016

Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + 3*sqrt(5)*log(phi)/10 + sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021