A112533 - OEIS

%I #19 Oct 04 2024 15:37:53

%S 4,49,144,9,324,42849,46656,1347921,3175524,1896129,23619600,

%T 532917225,359254116,30866624721,59997563136,185622243921,

%U 917583904836,4659420127761,750046066704,604376350260489,964709560931076

%N Expansion of (4+49*x+108*x^2-432*x^3+54675*x^5)/((1-27*x^2)*(1-6*x+27*x^2)*(1+6*x+27*x^2)).

%C A floretion-generated sequence of squares.

%H Colin Barker, <a href="/A112533/b112533.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,9,0,-243,0,19683).

%F a(n) = 9*a(n-2) - 243*a(n-4) + 19683*a(n-6) for n>5. - _Colin Barker_, May 06 2019

%F a(n) = (p^n/12)*( 9*((2+p) + (-1)^n*(2-p)) + (49 - 37*(-1)^n)*ChebyshevU(n, 3/p) - (1/p)*(153 - 261*(-1)^n)*ChebyshevU(n-1, 3/p) ), where p = sqrt(27). - _G. C. Greubel_, Jan 12 2022

%t a[n_]:= With[{p=Sqrt[27]}, Simplify[(p^n/12)*(9*((2+p) + (-1)^n*(2-p)) + (49 - 37*(-1)^n)*ChebyshevU[n, 3/p] -(153-261*(-1)^n)/p*ChebyshevU[n-1, 3/p] )]];

%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jan 12 2022 *)

%o (PARI) Vec((4 + 49*x + 108*x^2 - 432*x^3 + 54675*x^5) / ((1 - 6*x + 27*x^2)*(1 - 27*x^2)*(1 + 6*x + 27*x^2)) + O(x^20)) \\ _Colin Barker_, May 06 2019

%o (Magma) I:=[4,49,144,9,324,42849]; [n le 6 select I[n] else 9*(Self(n-2) - 27*Self(n-4) +2187*Self(n-6)): n in [1..31]]; // _G. C. Greubel_, Jan 12 2022

%o (Sage)

%o U=chebyshev_U

%o p=sqrt(27)

%o def A112533(n): return (p^n/12)*( 9*((2+p) + (-1)^n*(2-p)) + (49 - 37*(-1)^n)*U(n, 3/p) - (1/p)*(153 - 261*(-1)^n)*U(n-1, 3/p) )

%o [A112533(n) for n in (0..30)] # _G. C. Greubel_, Jan 12 2022

%Y Cf. A112534, A112535, A112536, A112537, A112538.

%K nonn,easy

%O 0,1

%A _Creighton Dement_, Sep 11 2005