A072065 Define a "piece" to consist of 3 mutually touching pennies welded together to form a triangle; sequence gives side lengths of triangles that can be made from such pieces.

0, 2, 9, 11, 12, 14, 21, 23, 24, 26, 33, 35, 36, 38, 45, 47, 48, 50, 57, 59, 60, 62, 69, 71, 72, 74, 81, 83, 84, 86, 93, 95, 96, 98, 105, 107, 108, 110, 117, 119, 120, 122, 129, 131, 132, 134, 141, 143, 144, 146, 153, 155, 156, 158, 165, 167, 168, 170, 177, 179, 180

Offset

1,2

Comments

The "piece" in question is also called a "tribone" [Ardila and Stanley]. - N. J. A. Sloane, Feb 27 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.

J. H. Conway and J. C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, Journal of Combinatorial Theory, Series A 53 (1990), 183-208. [From N. J. A. Sloane, Jul 04 2011]

Jim McCann, Triangle Sequence

N. C. Saldanha and C. Tomei, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.

Torsten Sillke, A Word Problem

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Formula

A number n is in the sequence iff n == 0, 2, 9 or 11 (mod 12). See Conway-Lagarias or the Sillke link. - Sascha Kurz, Mar 04 2003

a(1)=0, a(2)=2, a(3)=9, a(4)=11, a(5)=12, a(n) = a(n-1)+a(n-4)-a(n-5). - Harvey P. Dale, Jan 30 2015

From Colin Barker, Dec 12 2015: (Start)

a(n) = (3/4+(3*i)/4)*(i^n-i*(-i)^n)-(-1)^n/2+3*(n+1)-5 where i = sqrt(-1).

G.f.: x^2*(2+7*x+2*x^2+x^3) / ((1-x)^2*(1+x)*(1+x^2)). (End)

E.g.f.: (2 + 3*cos(x) + (6*x - 5)*cosh(x) - 3*sin(x) + (6*x - 3)*sinh(x))/2. - Stefano Spezia, May 05 2022

a(n) = (6*n-4-(-1)^n+3*(-1)^((2*n+1-(-1)^n)/4))/2. - Wesley Ivan Hurt, Nov 09 2023

Example

A possible side-9 arrangement:
          A
         A A
        B B C
       D B C C
      D D E E F
     G H H E F F
    G G H I I J J
   K L L M I N J O
  K K L M M N N O O

Maple

f:=r-> {seq(12*i+r, i=0..100)}; t1:= f(0) union f(2) union f(9) union f(11); t2:=sort(convert(t1, list)); # N. J. A. Sloane, Jul 04 2011

Mathematica

Select[Range[0, 200], MemberQ[{0, 2, 9, 11}, Mod[#, 12]]&] (* Harvey P. Dale, Dec 15 2011 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 2, 9, 11, 12}, 70] (* Harvey P. Dale, Jan 30 2015 *)

Prog

(Haskell)
a072065 n = a072065_list !! n
a072065_list = filter ((`elem` [0, 2, 9, 11]) . (`mod` 12)) [0..]
-- Reinhard Zumkeller, Jan 09 2013
(PARI) concat(0, Vec(x^2*(2+7*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100))) \\ Colin Barker, Dec 12 2015

Crossrefs

Union of A008594, A017545, A017629 and A017653.

Sequence in context: A371163 A207670 A176545 * A137000 A073634 A065554

Adjacent sequences: A072062 A072063 A072064 * A072066 A072067 A072068

Keyword

nonn,nice,easy

Author

Jim McCann (jmccann(AT)umich.edu), Aug 04 2002

Extensions

Offset corrected by Reinhard Zumkeller, Jan 09 2013

Status

approved