cp's OEIS Frontend

A216172 Number of all possible tetrahedra of any size, having reverse orientation to the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts.

%I A216172 #32 Sep 08 2022 08:46:03
%S A216172 0,0,1,4,10,21,39,66,105,159,231,325,445,595,780,1005,1275,1596,1974,
%T A216172 2415,2926,3514,4186,4950,5814,6786,7875,9090,10440,11935,13585,15400,
%U A216172 17391,19569,21945,24531,27339,30381,33670,37219,41041,45150,49560,54285,59340
%N A216172 Number of all possible tetrahedra of any size, having reverse orientation to the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts.
%C A216172 The number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron is given by A000332(n+3).
%C A216172 Create a sequence wherein the sum of three consecutive numbers is a triangular number: 0,0,0,1,2,3,5,7...; then find the partial sums of this sequence: 0,0,0,1,3,6,11,18...; then take the partial sums of this sequence: 0,0,0,1,4,10,21,39,66... and after dropping the first two zeros, you get this sequence. - _J. M. Bergot_, Apr 14 2016
%H A216172 Vincenzo Librandi, <a href="/A216172/b216172.txt">Table of n, a(n) for n = 1..1000</a>
%H A216172 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-5,6,-4,1).
%F A216172 a(n) = (1/72)*(-6*n -5*n^2 +2*n^3 +n^4 +4 -4*(-1)^(n mod 3)).
%F A216172 G.f.: x^3/((1-x)^5*(1+x+x^2)). - _Bruno Berselli_, Sep 11 2012
%F A216172 a(3*n-1) = A000217(A115067(n)); a(3*n) = A000217(A095794(n)); a(3*n+1) = A000217(A143208(n+2)) + A000217(n). - _J. M. Bergot_, Apr 14 2016
%F A216172 E.g.f.: (1/216)*(8 - 24*x + 24*x^2 + 24*x^3 + 3*x^4)*exp(x) - (1/27)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))*exp(-x/2). - _Ilya Gutkovskiy_, Apr 14 2016
%e A216172 For n=9 the numbers of the reversely oriented tetrahedra, starting from the smallest size, are A000292(7)=84, A000292(4)=20, and A000292(1)=1, the sum being a(9)=105.
%t A216172 nnn = 100; Tev[n_] := (n - 2) (n - 1) n/6; Table[Sum[Tev[n - nn], {nn, 0, n - 1, 3}], {n, nnn}]
%t A216172 Table[(1/72) (-6 n - 5 n^2 + 2 n^3 + n^4 + 4 - 4 (-1)^Mod[n, 3]), {n, 50}]
%t A216172 CoefficientList[Series[x^2 / ((1 - x)^5*(1 + x + x^2)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 12 2012 *)
%t A216172 LinearRecurrence[{4,-6,5,-5,6,-4,1},{0,0,1,4,10,21,39},50] (* _Harvey P. Dale_, Feb 18 2018 *)
%o A216172 (Magma) I:=[0, 0, 1, 4, 10, 21, 39]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..50]]; // _Vincenzo Librandi_, Sep 12 2012
%o A216172 (PARI) a(n)=(n^4+2*n^3-5*n^2-6*n+4-4*(-1)^(n%3))/72 \\ _Charles R Greathouse IV_, Sep 12 2012
%Y A216172 Cf. A000292, A000332, A216173, A216175.
%K A216172 nonn,easy
%O A216172 1,4
%A A216172 _V.J. Pohjola_, Sep 03 2012