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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). 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

REFERENCES

C. Dement, Floretion Integer Sequences (work in progress).

LINKS

Table of n, a(n) for n=0..29.

MATHEMATICA

-x/((x^2+x+1)*(x^2+3*x+1)) + O[x]^30 // CoefficientList[#, x]& (* Jean-Fran├žois 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 xrange(1, 32)] # Zerinvary Lajos, Jul 06 2008

CROSSREFS

Cf. A000292, A113067, A113068, A094686, A001906, A109961, A290890.

Sequence in context: A245124 A020964 * A290890 A152689 A217918 A000604

Adjacent sequences:  A113064 A113065 A113066 * A113068 A113069 A113070

KEYWORD

easy,sign,changed

AUTHOR

Creighton Dement, Oct 13 2005

STATUS

approved

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Last modified August 22 22:36 EDT 2017. Contains 290952 sequences.