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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. 4
 0, -1, 4, -11, 28, -72, 188, -493, 1292, -3383, 8856, -23184, 60696, -158905, 416020, -1089155, 2851444, -7465176, 19544084, -51167077, 133957148, -350704367, 918155952, -2403763488, 6293134512, -16475640049, 43133785636, -112925716859, 295643364940, -774004377960 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Invert((a(n)) gives (0, -1, 4, -10, 20, -35, ) = A000292 (with alternating signs). Binomial(a(n)) gives (0, -1, 2, -2, 4, -7, 10) = A094686 (with alternating signs, from 2nd term). REFERENCES C. Dement, Floretion Integer Sequences (work in progress). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (-4,-5,-4,-1). FORMULA a(n) + a(n+1) + a(n+2) = (-1)^n *A001906(n+2) = (-1)^n*F(2n+4). a(n) + 3*a(n+1) + 3*a(n+2) + a(n+3) = ((-1)^(n+1))*A109961(n+2). (|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 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 2*a(n) = (-1)^n*A001906(n+1) - A049347(n). - R. J. Mathar, Sep 20 2020 MATHEMATICA -x/((x^2+x+1)*(x^2+3*x+1)) + O[x]^30 // CoefficientList[#, x]& (* Jean-François Alcover, Jun 15 2017 *) PROG 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' (Sage) [((lucas_number1(n, 3, 1)-lucas_number1(n, 1, 1)))/(-2) for n in range(1, 32)] # Zerinvary Lajos, Jul 06 2008 (PARI) concat(0, Vec(-x / ((1 + x + x^2)*(1 + 3*x + x^2)) + O(x^30))) \\ Colin Barker, May 11 2019 CROSSREFS Cf. A000292, A113067, A113068, A094686, A001906, A109961, A290890. Sequence in context: A005409 A245124 A020964 * A290890 A152689 A217918 Adjacent sequences:  A113064 A113065 A113066 * A113068 A113069 A113070 KEYWORD easy,sign AUTHOR Creighton Dement, Oct 13 2005 STATUS approved

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Last modified September 17 16:50 EDT 2021. Contains 347487 sequences. (Running on oeis4.)