OFFSET
0,3
COMMENTS
See A105968 for a similar sequence. Observe the four periodic sequences (1,1,1,1,); (-1,-1,-1,-1); (1,-1,1,-1,); (-1,1,-1,1,); (a(n)) is the (Type 1A) jbasejfor-transform of the periodic sequence (1,1,1,1) with respect to the floretion given in the program code. A109438 is the (Type 1A) jbasejfor-transform of the periodic sequence (-1,-1,-1,-1) with respect to the floretion given in the program code. A001834 is the (Type 1A) jbasejfor-transform of the periodic sequence (1,-1,1,-1) with respect to the floretion given in the program code. A102871 is the (Type 1A) jbasejfor-transform of the periodic sequence (-1,1,-1,1) with respect to the floretion given in the program code.
Floretion Algebra Multiplication Program, FAMP Code: (-1)^(n+1)jbasejfor[ + .5'ii' + .5'kk' + .5'ij' + .5'ji' + .5'jk' + .5'kj'] 1vesfor = (1,1,1,1,)
REFERENCES
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.
LINKS
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).
FORMULA
G.f.: x/((x+1)(x^2-4x+1)).
a(n) = ((1 + sqrt(3))*(2 + sqrt(3))^n + (1 - sqrt(3))*(2 - sqrt(3))^n - 2*(-1)^n)/12. - Stefano Spezia, Sep 19 2023
EXAMPLE
x + 3*x^2 + 12*x^3 + 44*x^4 + 165*x^5 + 615*x^6 + 2296*x^7 + 8568*x^8 + ...
MAPLE
with(numtheory):a := cfrac (tan(Pi/3), 60): > b := cfrac (tan(Pi/6), 60): > seq(nthnumer (b, i)*nthdenom (a, i), i=0..24 ); # Zerinvary Lajos, Feb 08 2007
MATHEMATICA
LinearRecurrence[{3, 3, -1}, {0, 1, 3}, 40] (* Harvey P. Dale, Apr 21 2018 *)
PROG
(PARI) {a(n) = local(s=1); if( n<0, n = -1 - n; s=-1); s * polcoeff( x / ((x + 1) * (x^2 -4*x + 1)) + x * O(x^n), n)} /* Michael Somos, Jul 27 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Jun 28 2005
EXTENSIONS
a(25)-a(27) from Stefano Spezia, Sep 19 2023
STATUS
approved