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%I #24 Mar 25 2019 16:50:37
%S 1,7,14,28,49,84,147,252,434,749,1288,2219,3822,6580,11333,19516,
%T 33607,57876,99666,171633,295568,508991,876526,1509452,2599401,
%U 4476388,7708715,13275052,22860754,39368133,67795224,116749059,201051662,346227812,596233309
%N Number of tiles in distance d from a given heptagon in the truncated order-3 tiling of the heptagonal plane (a.k.a. the "hyperbolic soccerball").
%H Colin Barker, <a href="/A290398/b290398.txt">Table of n, a(n) for n = 0..1000</a>
%H Eryk Kopczyński, Dorota Celińska and Marek Čtrnáct, <a href="http://archive.bridgesmathart.org/2017/bridges2017-9.html">HyperRogue: Playing with Hyperbolic Geometry</a>, Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture, Pages 9-16
%H This sequence is important in the game <a href="http://roguetemple.com/z/hyper/">HyperRogue</a> which uses this tiling.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Truncated_order-7_triangular_tiling">Truncated order-7 triangular tiling</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1).
%F a(n+4) = a(n+3) + a(n+2) + a(n+1) - a(n), for n >= 1. (proved)
%F G.f.: (1 + 6*x + 6*x^2 + 6*x^3 + x^4) / (1 - x - x^2 - x^3 + x^4). - _Colin Barker_, Jan 05 2018
%e There is only the original heptagon in distance 0, so a(0)=1. It is adjacent to 7 hexagons, so a(1)=7. These are adjacent to 7 new heptagons and 7 new hexagons, so a(2)=14.
%t LinearRecurrence[{1,1,1,-1},{1,7,14,28,49},40] (* _Harvey P. Dale_, Mar 25 2019 *)
%o (PARI) Vec((1 + 6*x + 6*x^2 + 6*x^3 + x^4) / (1 - x - x^2 - x^3 + x^4) + O(x^40)) \\ _Colin Barker_, Jan 05 2018
%K easy,nonn
%O 0,2
%A _Eryk Kopczynski_, Jul 29 2017
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