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%I #36 Mar 13 2024 21:51:57
%S 0,1,3,12,44,165,615,2296,8568,31977,119339,445380,1662180,6203341,
%T 23151183,86401392,322454384,1203416145,4491210195,16761424636,
%U 62554488348,233456528757,871271626679,3251629977960,12135248285160,45289363162681,169022204365563,630799454299572
%N a(-1) = a(0) = 0, a(1) = 1; a(n) = 5a(n-1) - 5a(n-2) + a(n-3) + 2*(-1)^(n+1), alternatively a(n) = 3a(n-1) + 3a(n-2) - a(n-3).
%C 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.
%C 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,)
%D R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-1).
%F G.f.: x/((x+1)(x^2-4x+1)).
%F a(n) = A002530(n)*A002530(n+1). - _Zerinvary Lajos_, Feb 08 2007
%F a(-1 - n) = -a(n). a(2*n) = A011916(n). a(2*n + 1) = -A011916(-1 -n). - _Michael Somos_, Jul 27 2012
%F 6*a(n) = A001353(n)+A001353(n+1)-(-1)^n. - _R. J. Mathar_, Sep 07 2016
%F 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
%e x + 3*x^2 + 12*x^3 + 44*x^4 + 165*x^5 + 615*x^6 + 2296*x^7 + 8568*x^8 + ...
%p 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
%t LinearRecurrence[{3,3,-1},{0,1,3},40] (* _Harvey P. Dale_, Apr 21 2018 *)
%o (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 */
%Y Cf. A001353, A001834, A002530, A011916, A102871, A105968, A109438.
%K nonn,easy
%O 0,3
%A _Creighton Dement_, Jun 28 2005
%E a(25)-a(27) from _Stefano Spezia_, Sep 19 2023
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