cp's OEIS Frontend

A204185 Number of quadrilaterals in a triangular matchstick arrangement of side n.

%I A204185 #34 Feb 17 2025 08:13:44
%S A204185 0,0,6,33,102,243,492,894,1500,2370,3570,5175,7266,9933,13272,17388,
%T A204185 22392,28404,35550,43965,53790,65175,78276,93258,110292,129558,151242,
%U A204185 175539,202650,232785,266160,303000,343536,388008,436662,489753,547542,610299,678300,751830,831180,916650,1008546,1107183,1212882,1325973,1446792,1575684,1713000,1859100,2014350
%N A204185 Number of quadrilaterals in a triangular matchstick arrangement of side n.
%C A204185 The total number of parallelograms and trapezoids that appear in a triangular matchstick array of side n.
%C A204185 Can always be split into three equal sets, parallelograms 'belonging' to the side of the triangle that none of its sides are parallel to, and trapezoids 'belonging' to the side of the triangle that two of its sides are parallel to.
%C A204185 Rhombuses belonging to each side are A173196(n).
%C A204185 Irregular parallelograms belonging to each side are 2*A001752(n-3).
%C A204185 'Upside down' trapezoids (those where the shorter of the two parallel sides is closest to the parallel side of the triangle) belonging to each side are A001752(n-3).
%C A204185 'Right side up' trapezoids belonging to each side are A000332(n+2).
%H A204185 Harvey P. Dale, <a href="/A204185/b204185.txt">Table of n, a(n) for n = 0..1000</a>
%H A204185 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A204185 a(n) = Sum_{k=1..n-1} b(k)*T(n-k), where b(m) = 3*floor(5*m/2) and T(m) is the m-th triangular number A000217.
%F A204185 a(n) = a(n-1) + floor((n+1)*(n-1)*(10*n-3)/8).
%F A204185 a(n) = 3*(A173196(n) + A000332(n+2) + 3*A001752(n-3)) (see comments above).
%F A204185 From _Colin Barker_, Mar 16 2015: (Start)
%F A204185 a(n) = (3-3*(-1)^n-16*n-16*n^2+16*n^3+10*n^4)/32.
%F A204185 a(n) = 4*a(n-1)-5*a(n-2)+5*a(n-4)-4*a(n-5)+a(n-6).
%F A204185 G.f.: -3*x^2*(3*x+2) / ((x-1)^5*(x+1)). (End)
%F A204185 E.g.f.: (x*(5*x^3 + 38*x^2 + 51*x - 3)*cosh(x) + (5*x^4 + 38*x^3 + 51*x^2 - 3*x + 3)*sinh(x))/16. - _Stefano Spezia_, Jul 19 2022
%e A204185 a(3) = 33 because the following figure contains 33 quadrilaterals (15 parallelograms and 18 trapezoids)
%e A204185 ....... /\
%e A204185 ...... /\/\
%e A204185 ..... /\/\/\
%e A204185 Size and quantity of each quadrilateral in above figure:
%e A204185 2 triangles: 9
%e A204185 3 triangles: 12
%e A204185 4 triangles: 6
%e A204185 5 triangles: 3
%e A204185 8 triangles: 3
%t A204185 nxt[{n_,a_}]:={n+1,a+Floor[n(n+2) (10(n+1)-3)/8]}; Transpose[ NestList[ nxt,{0,0},50]][[2]] (* _Harvey P. Dale_, Jan 11 2013 *)
%o A204185 (PARI) concat([0,0], Vec(-3*x^2*(3*x+2)/((x-1)^5*(x+1)) + O(x^100))) \\ _Colin Barker_, Mar 16 2015
%Y A204185 Cf. A173196 = number of rhombuses of a particular orientation; A001752, related to number of irregular parallelograms and number of 'upside down' trapezoids; A000332, related to number of 'right side up' trapezoids (see comments above); A002717 = number of triangles in a triangular matchstick arrangement; A000217 = triangular numbers.
%K A204185 nonn,nice,easy
%O A204185 0,3
%A A204185 _Elliott Line_ & Paul Bostock (enigma.mensa(AT)yahoo.co.uk), Jan 12 2012