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%I #57 Jun 03 2022 15:40:22
%S 1,3,9,12,15,21,24,27,33,36,39,45,48,51,57,60,63,69,72,75,81,84,87,93,
%T 96,99,105,108,111,117,120,123,129,132,135,141,144,147,153,156,159,
%U 165,168,171,177,180,183,189,192,195,201,204,207,213,216,219,225,228,231,237
%N a(0)=1, a(1)=3, a(2)=9, a(3)=12, a(4)=15; thereafter a(n) = a(n-1) + a(n-3) - a(n-4).
%C This is the coordination sequence for Marjorie Rice's tiling of the plane shown in Fig. 15 of Schattschneider (1981), with respect to the central vertex. The Schattschneider illustration below shows that the first differences of the coordination sequence are 2, 6, 3, 3, 6, 3, 3, 6, 3, 3, ..., and so the sequence itself satisfies the recurrence in the definition. The tiling has symmetry group D_6 (the dihedral group of order 6).
%C Continuing from the arrangement of pennies described in A136289, we also wish to place dimes over the holes in the array, where the n-th generation of dimes can be placed only when all three of its supporting pennies are in place already; then a(n-1) is the number of dimes in generation n for >= 1. - _Colin Mallows_, Apr 13 2008
%D Doris Schattschneider, In Praise of Amateurs, pp. 140-166 in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981.
%H G. C. Greubel, <a href="/A136290/b136290.txt">Table of n, a(n) for n = 0..1000</a>
%H Colin Mallows, <a href="/A136290/a136290_1.pdf">Analysis of coordination sequence for Marjorie Rice tiling</a>
%H Doris Schattschneider, <a href="/A136290/a136290.pdf">Illustration of terms a(0) to a(12) of the coordination sequence for the Rice tiling</a>
%H N. J. A. Sloane, <a href="/A136290/a136290.png">Illustration of terms a(0) to a(9) of the coordination sequence for the Rice tiling</a> [Annotated copy of Fig. 15 of Schattschneider (1981)]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: (1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1-x)^2*(1+x+x^2)). - _Colin Barker_, Jul 12 2014
%F a(n) = [n=0] + 4*n - ChebyshevU(n-1, -1/2). - _G. C. Greubel_, Apr 13 2021
%F For n > 0, a(n) = 3*A042965(n+1). - _Jon E. Schoenfield_, Jun 03 2022
%p 1, seq(4*n -simplify(ChebyshevU(n-1, -1/2)), n = 1..20); # _G. C. Greubel_, Apr 13 2021
%t {1}~Join~LinearRecurrence[{1, 0, 1, -1}, {3, 9, 12, 15}, 59] (* _Jean-François Alcover_, Oct 23 2019 *)
%o (Magma) a:=[1,3,9,12,15]; [n le 5 select a[n] else Self(n-1)+Self(n-3)-Self(n-4):n in [1..60]]; // _Marius A. Burtea_, Oct 23 2019
%o (Sage) [1]+[4*n-chebyshev_U(n-1,-1/2) for n in (1..60)] # _G. C. Greubel_, Apr 13 2021
%Y Cf. A042965, A136289.
%K nonn
%O 0,2
%A _Colin Mallows_, Apr 13 2008
%E Entry revised by _N. J. A. Sloane_, Apr 06 2019, replacing the old definition with _Colin Barker_'s recurrence.
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