A112533 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)).
4, 49, 144, 9, 324, 42849, 46656, 1347921, 3175524, 1896129, 23619600, 532917225, 359254116, 30866624721, 59997563136, 185622243921, 917583904836, 4659420127761, 750046066704, 604376350260489, 964709560931076
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0,9,0,-243,0,19683).
Programs
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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
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Mathematica
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] )]]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 12 2022 *) -
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
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Sage
U=chebyshev_U p=sqrt(27) 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) ) [A112533(n) for n in (0..30)] # G. C. Greubel, Jan 12 2022
Formula
a(n) = 9*a(n-2) - 243*a(n-4) + 19683*a(n-6) for n>5. - Colin Barker, May 06 2019
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
Comments