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%I #50 Nov 09 2023 13:54:52
%S 0,2,9,11,12,14,21,23,24,26,33,35,36,38,45,47,48,50,57,59,60,62,69,71,
%T 72,74,81,83,84,86,93,95,96,98,105,107,108,110,117,119,120,122,129,
%U 131,132,134,141,143,144,146,153,155,156,158,165,167,168,170,177,179,180
%N 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.
%C The "piece" in question is also called a "tribone" [Ardila and Stanley]. - _N. J. A. Sloane_, Feb 27 2014
%H Reinhard Zumkeller, <a href="/A072065/b072065.txt">Table of n, a(n) for n = 1..1000</a>
%H F. Ardila and R. P. Stanley, <a href="https://arxiv.org/abs/math/0501170">Tilings</a>, arXiv:math/0501170 [math.CO], 2005.
%H J. H. Conway and J. C. Lagarias, <a href="http://dx.doi.org/10.1016/0097-3165(90)90057-4">Tiling with Polyominoes and Combinatorial Group Theory</a>, Journal of Combinatorial Theory, Series A 53 (1990), 183-208. [From _N. J. A. Sloane_, Jul 04 2011]
%H Jim McCann, <a href="https://web.archive.org/web/20050425025837/http://www-personal.umich.edu/~jmccann/triseq.html">Triangle Sequence</a>
%H N. C. Saldanha and C. Tomei, <a href="https://arxiv.org/abs/math/9801111">An overview of domino and lozenge tilings</a>, arXiv:math/9801111 [math.CO], 1998.
%H Torsten Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/word_problem_L3">A Word Problem</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F 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
%F 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
%F From _Colin Barker_, Dec 12 2015: (Start)
%F a(n) = (3/4+(3*i)/4)*(i^n-i*(-i)^n)-(-1)^n/2+3*(n+1)-5 where i = sqrt(-1).
%F G.f.: x^2*(2+7*x+2*x^2+x^3) / ((1-x)^2*(1+x)*(1+x^2)). (End)
%F 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
%F a(n) = (6*n-4-(-1)^n+3*(-1)^((2*n+1-(-1)^n)/4))/2. - _Wesley Ivan Hurt_, Nov 09 2023
%e A possible side-9 arrangement:
%e A
%e A A
%e B B C
%e D B C C
%e D D E E F
%e G H H E F F
%e G G H I I J J
%e K L L M I N J O
%e K K L M M N N O O
%p 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
%t Select[Range[0,200],MemberQ[{0,2,9,11},Mod[#,12]]&] (* _Harvey P. Dale_, Dec 15 2011 *)
%t LinearRecurrence[{1,0,0,1,-1},{0,2,9,11,12},70] (* _Harvey P. Dale_, Jan 30 2015 *)
%o (Haskell)
%o a072065 n = a072065_list !! n
%o a072065_list = filter ((`elem` [0,2,9,11]) . (`mod` 12)) [0..]
%o -- _Reinhard Zumkeller_, Jan 09 2013
%o (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
%Y Union of A008594, A017545, A017629 and A017653.
%K nonn,nice,easy
%O 1,2
%A Jim McCann (jmccann(AT)umich.edu), Aug 04 2002
%E Offset corrected by _Reinhard Zumkeller_, Jan 09 2013
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