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 A054485 Expansion of (1+3*x)/(1-4*x+x^2). 4
 1, 7, 27, 101, 377, 1407, 5251, 19597, 73137, 272951, 1018667, 3801717, 14188201, 52951087, 197616147, 737513501, 2752437857, 10272237927, 38336513851, 143073817477, 533958756057, 1992761206751, 7437086070947, 27755583077037 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193. E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (4,-1). FORMULA a(n) = (7*((2+sqrt(3))^n - (2-sqrt(3))^n) - ((2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1)))/2*sqrt(3). a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(0)=7. a(n) = ChebyshevU(n,2) + 3*Chebyshev(n-1,2) = ChebyshevT(n,2) + 5*ChebyshevU(n-1,2). - G. C. Greubel, Jan 19 2020 MAPLE seq( simplify(ChebyshevU(n, 2) +3*ChebyshevU(n-1, 2)), n=0..30); # G. C. Greubel, Jan 19 2020 MATHEMATICA LinearRecurrence[{4, -1}, {1, 7}, 40] (* Vincenzo Librandi, Jun 23 2012 *) Table[ChebyshevU[n, 2] +3*ChebyshevU[n-1, 2], {n, 0, 30}] (* G. C. Greubel, Jan 19 2020 *) PROG (MAGMA) I:=[1, 7]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in[1..30]]; // Vincenzo Librandi, Jun 23 2012 (PARI) Vec((1+3*x)/(1-4*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015 (PARI) vector(31, n, polchebyshev(n-1, 1, 2) +5*polchebyshev(n-2, 2, 2) ) \\ G. C. Greubel, Jan 19 2020 (Sage) [chebyshev_U(n, 2) + 3*chebyshev_U(n-1, 2) for n in (0..30)] # G. C. Greubel, Jan 19 2020 (GAP) a:=[1, 7];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020 CROSSREFS Cf. A054491. Sequence in context: A282642 A185080 A006350 * A090856 A055917 A056120 Adjacent sequences:  A054482 A054483 A054484 * A054486 A054487 A054488 KEYWORD easy,nonn AUTHOR Barry E. Williams, May 06 2000 STATUS approved

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Last modified April 23 05:06 EDT 2021. Contains 343199 sequences. (Running on oeis4.)