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A054491 a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(1)=6. 11
1, 6, 23, 86, 321, 1198, 4471, 16686, 62273, 232406, 867351, 3236998, 12080641, 45085566, 168261623, 627960926, 2343582081, 8746367398, 32641887511, 121821182646, 454642843073, 1696750189646, 6332357915511, 23632681472398 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Bisection (even part) of Chebyshev sequence with Diophantine property.
The odd part is A077234 with Diophantine companion A077235.
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.
LINKS
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
FORMULA
-3*a(n)^2 + A077236(n)^2 = 13.
a(n) = ( 6*((2+sqrt(3))^n-(2-sqrt(3))^n) - ((2+sqrt(3))^(n-1)-(2-sqrt(3))^(n-1)) )/(2*sqrt(3)).
a(n) = 6*S(n-1, 4) - S(n-2, 4) = S(n, 4) + 2*S(n-1, 4), with S(n, x) := U(n, x/2) Chebyshev's polynomials of 2nd kind, A049310. S(-1, x) := 0, S(-2, x) := -1, S(n, 4)= A001353(n+1).
G.f.: (1+2*x)/(1-4*x+x^2).
a(n+1) = A001353(n+2) + 2*A001353(n+1) - Creighton Dement, Nov 28 2004. Comment from Vim Wenders, Mar 26 2008: This is easily verified using a(n) = (6*( (2+sqrt(3))^n - (2-sqrt(3))^n ) - ( (2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1) ))/(2*sqrt(3)) and A001353(n) = ( (2+sqrt(3))^n - (2-sqrt(3))^n )/(2*sqrt(3)).
a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-7)^k. - Philippe Deléham, Mar 05 2014
E.g.f.: (1/3)*exp(2*x)*(3*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Jan 27 2020
MAPLE
seq( simplify(ChebyshevU(n, 2) +2*ChebyshevU(n-1, 2)), n=0..30); # G. C. Greubel, Jan 15 2020
MATHEMATICA
Table[ChebyshevU[n, 2] +2*ChebyshevU[n-1, 2], {n, 0, 30}] (* G. C. Greubel, Jan 15 2020 *)
LinearRecurrence[{4, -1}, {1, 6}, 30] (* Harvey P. Dale, Sep 04 2021 *)
PROG
(PARI) a(n) = if (n==0, 1, if (n==1, 6, 4*a(n-1)-a(n-2))) \\ Michel Marcus, Jun 23 2013
(PARI) a(n) = polchebyshev(n, 2, 2) + 2*polchebyshev(n-1, 2, 2); \\ Michel Marcus, Oct 13 2021
(Magma) I:=[1, 6]; [n le 2 select I[n] else 4*Self(n-1) -Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2020
(Sage) [chebyshev_U(n, 2) +2*chebyshev_U(n-1, 2) for n in (0..30)]; # G. C. Greubel, Jan 15 2020
(GAP) a:=[1, 6];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 15 2020
CROSSREFS
Sequence in context: A006815 A264690 A241394 * A282710 A295132 A013261
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, May 04 2000
EXTENSIONS
Chebyshev comments from Wolfdieter Lang, Nov 08 2002
STATUS
approved

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Last modified June 14 21:35 EDT 2024. Contains 373401 sequences. (Running on oeis4.)