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 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. 4
 0, 0, 1, 4, 10, 21, 39, 66, 105, 159, 231, 325, 445, 595, 780, 1005, 1275, 1596, 1974, 2415, 2926, 3514, 4186, 4950, 5814, 6786, 7875, 9090, 10440, 11935, 13585, 15400, 17391, 19569, 21945, 24531, 27339, 30381, 33670, 37219, 41041, 45150, 49560, 54285, 59340 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron is given by A000332(n+3). 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1). FORMULA a(n) = (1/72)*(-6*n -5*n^2 +2*n^3 +n^4 +4 -4*(-1)^(n mod 3)). G.f.: x^3/((1-x)^5*(1+x+x^2)). - Bruno Berselli, Sep 11 2012 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 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 EXAMPLE 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. MATHEMATICA nnn = 100; Tev[n_] := (n - 2) (n - 1) n/6; Table[Sum[Tev[n - nn], {nn, 0, n - 1, 3}], {n, nnn}] Table[(1/72) (-6 n - 5 n^2 + 2 n^3 + n^4 + 4 - 4 (-1)^Mod[n, 3]), {n, 50}] CoefficientList[Series[x^2 / ((1 - x)^5*(1 + x + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 12 2012 *) LinearRecurrence[{4, -6, 5, -5, 6, -4, 1}, {0, 0, 1, 4, 10, 21, 39}, 50] (* Harvey P. Dale, Feb 18 2018 *) PROG (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 (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 CROSSREFS Cf. A000292, A000332, A216173, A216175. Sequence in context: A253687 A253688 A049480 * A055908 A023538 A085360 Adjacent sequences:  A216169 A216170 A216171 * A216173 A216174 A216175 KEYWORD nonn,easy AUTHOR V.J. Pohjola, Sep 03 2012 STATUS approved

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Last modified October 23 02:28 EDT 2018. Contains 316518 sequences. (Running on oeis4.)