A087444 Numbers that are congruent to {1, 4} mod 9.

1, 4, 10, 13, 19, 22, 28, 31, 37, 40, 46, 49, 55, 58, 64, 67, 73, 76, 82, 85, 91, 94, 100, 103, 109, 112, 118, 121, 127, 130, 136, 139, 145, 148, 154, 157, 163, 166, 172, 175, 181, 184, 190, 193, 199, 202, 208, 211, 217, 220, 226, 229, 235, 238, 244, 247, 253

Offset: 1

Author

Paul Barry, Sep 04 2003

Keywords

• easy
• nonn

Comments

3*a(n) is conjectured to be the total number of sides (straight double bonds (long side) and single bond (short side)) of a certain equilateral triangle expansion shown in one of the links. The pattern is supposed to become the planar Archimedean net 3.3.3.3.6 when n -> infinity. 3*a(n) is also conjectured to be the total number of sided (bonds) in another version of an equilateral triangle expansion that is supposed to become the planar Archimedean net 3.6.3.6. See the illustrations in the links. - Kival Ngaokrajang, Nov 30 2014

Links

• David Lovler, Table of n, a(n) for n = 1..10000
• Kival Ngaokrajang, Illustration of initial terms (3.3.3.3.6), (3.6.3.6)
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001651, A047241, A087445, A087446.

Programs

Mathematica

Select[Range[300],MemberQ[{1,4},Mod[#,9]]&] (* or *) LinearRecurrence[ {1,1,-1},{1,4,10},60] (* Harvey P. Dale, Jan 22 2019 *)

PARI

a(n) = (18*n - 17 - 3*(-1)^n)/4 \\ David Lovler, Aug 20 2022

Formula

G.f.: x*(1+3*x+5*x^2)/((1+x)*(1-x)^2).

E.g.f.: 5 + ((9*x - 17/2)*exp(x) - (3/2)*exp(-x))/2.

a(n) = (18*n - 17 - 3*(-1)^n)/4.

a(n) = 9*n - a(n-1) - 13 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

Extensions

Kival Ngaokrajang's comment reworded by Wolfdieter Lang, Dec 05 2014

E.g.f. and formula adapted to offset by David Lovler, Aug 20 2022