A130877 Numbers that are congruent to {0, 5} mod 9.

0, 5, 9, 14, 18, 23, 27, 32, 36, 41, 45, 50, 54, 59, 63, 68, 72, 77, 81, 86, 90, 95, 99, 104, 108, 113, 117, 122, 126, 131, 135, 140, 144, 149, 153, 158, 162, 167, 171, 176, 180, 185, 189, 194, 198, 203, 207, 212, 216, 221, 225, 230, 234, 239, 243, 248, 252, 257

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Jul 25 2007

Keywords

• nonn
• easy

Comments

Numbers m such that m = digitsum(k*(m+k)) for some k>=0.

The first differences are 2-periodic: 5, 4, 5, 4, etc. The minimum numbers k associated to the first elements of the sequence are (m,k): (0,0), (5,2), (9,3), (14,5), (18,15), (23,44), (27,42), (32,119), etc.

Links

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

Crossrefs

Cf. A008591, A017221.

Programs

Maple

op(select(n->n mod 9=0 or n mod 9=5,[$0..257])); # Paolo P. Lava, Jul 12 2018
# second Maple program:
a:= n-> ceil(9*(n-1)/2):
seq(a(n), n=1..58);  # Alois P. Heinz, Apr 12 2025

Mathematica

Table[5n-5-Floor[(n-1)/2], {n,100}] (* Wesley Ivan Hurt, Oct 25 2013 *)
Select[Range[0,300],MemberQ[{0,5},Mod[#,9]]&] (* or *) LinearRecurrence[ {1,1,-1},{0,5,9},60] (* Harvey P. Dale, Aug 04 2019 *)

PARI

forstep(n=0,200,[5,4],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

Formula

a(n) = a(n-2) + 9 for n >= 3.

a(n) = 9/2*(n+1) - 4 + Sum{j=0..n} (-1)^j/2.

O.g.f.: x^2(5+4x)/((1+x)(1-x)^2). a(n) = 9(n-1)/2+(1+(-1)^n)/4. - R. J. Mathar, Jun 13 2008

a(n+1) = Sum_{k>=0} A030308(n,k)*A116453(k+1). - Philippe Deléham, Oct 17 2011

a(n) = 5n - 5 - floor((n-1)/2). - Wesley Ivan Hurt, Oct 25 2013

a(n) = ceiling(9*(n-1)/2). - Alois P. Heinz, Apr 12 2025