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A113067 Expansion of -x/((x^2+x+1)*(x^2+3*x+1)); invert transform gives signed version of tetrahedral numbers A000292.

%I A113067 #33 May 05 2025 07:10:11
%S A113067 0,-1,4,-11,28,-72,188,-493,1292,-3383,8856,-23184,60696,-158905,
%T A113067 416020,-1089155,2851444,-7465176,19544084,-51167077,133957148,
%U A113067 -350704367,918155952,-2403763488,6293134512,-16475640049,43133785636,-112925716859,295643364940,-774004377960
%N A113067 Expansion of -x/((x^2+x+1)*(x^2+3*x+1)); invert transform gives signed version of tetrahedral numbers A000292.
%C A113067 Invert(a(n)) gives (0, -1, 4, -10, 20, -35, ...) = A000292 (with alternating signs).
%C A113067 Binomial(a(n)) gives (0, -1, 2, -2, 4, -7, 10, ...) = A094686 (with alternating signs, from 2nd term).
%C A113067 Floretion Algebra Multiplication Program, FAMP Code: 2basei[C*F]; C = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; F = + .5'i + .5'ii' + .5'ij' + .5'ik'
%D A113067 Creighton Dement, Floretion Integer Sequences (work in progress).
%H A113067 Colin Barker, <a href="/A113067/b113067.txt">Table of n, a(n) for n = 0..1000</a>
%H A113067 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-5,-4,-1).
%F A113067 a(n) + a(n+1) + a(n+2) = (-1)^n *A001906(n+2) = (-1)^n*F(2n+4).
%F A113067 a(n) + 3*a(n+1) + 3*a(n+2) + a(n+3) = ((-1)^(n+1))*A109961(n+2).
%F A113067 (|a(n)|) = A290890(n) for n >= 0, this being the p-INVERT of (1,2,3,4,...), where p(S) = 1 - S^2.  - _Clark Kimberling_, Aug 21 2017
%F A113067 a(n) = -4*a(n-1) - 5*a(n-2) - 4*a(n-3) - a(n-4) for n > 3. - _Colin Barker_, May 11 2019
%F A113067 2*a(n) = (-1)^n*A001906(n+1) - A049347(n). - _R. J. Mathar_, Sep 20 2020
%t A113067 -x/((x^2+x+1)*(x^2+3*x+1)) + O[x]^30 // CoefficientList[#, x]& (* _Jean-François Alcover_, Jun 15 2017 *)
%o A113067 (SageMath) [((lucas_number1(n,3,1)-lucas_number1(n,1,1)))/(-2) for n in range(1,32)] # _Zerinvary Lajos_, Jul 06 2008
%o A113067 (PARI) concat(0, Vec(-x / ((1 + x + x^2)*(1 + 3*x + x^2)) + O(x^30))) \\ _Colin Barker_, May 11 2019
%Y A113067 Cf. A000292, A113067, A113068, A094686, A001906, A109961, A290890.
%K A113067 easy,sign
%O A113067 0,3
%A A113067 _Creighton Dement_, Oct 13 2005