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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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)

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: A085069 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

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Last modified November 12 21:10 EST 2018. Contains 317116 sequences. (Running on oeis4.)