Invert transform gives signed version of Tetrahedral numbers

[Creighton Dement]

Dear Seqfans, 

These two sequences appear interesting to me. Perhaps someone can dig
deeper than I and find out more of what they are about...

Many thanks,
Creighton  


%S A113067 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
%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).  
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)
%D A113067 C. Dement, Floretion Integer Sequences (work in progress).
%o 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'
%Y A113067 Cf. A000292, A113067, A113068, A094686, A001906, A109961
%O A113067 0,3
%K A113067 ,easy,sign,
%A A113067 Creighton Dement (crowdog at crowdog.de), Oct 13 2005



A113066 1, -2, 4, -10, 27, -72, 189, -494, 1292, -3382, 8855, -23184,
60697, -158906, 416020, -1089154, 2851443, -7465176, 19544085,
-51167078, 133957148, -350704366, 918155951, -2403763488, 6293134513,
-16475640050, 43133785636, -112925716858, 295643364939, -774004377960
%N A113066 Expansion of (1+x)^2/((x^2+x+1)*(x^2+3*x+1))
%C A113066 Binomial transform gives signed version of A093040. a(n) +
a(n+1) = ((-1)^(n+1))A109961(n+1).  a(n) + a(n+1) + a(n+2) =
((-1)^n)A001906(n+2) = ((-1)^n)F(2n+4)
%D A113066 C. Dement, Floretion Integer Sequences (work in progress).
%o A113066 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'
%Y A113066 Cf. A113067, A113068, A093040, A109961, A001906.
%O A113066 0,2
%K A113066 ,easy,sign,
%A A113066 Creighton Dement (crowdog at crowdog.de), Oct 13 2005