|
|
A054488
|
|
Expansion of (1+2*x)/(1-6*x+x^2).
|
|
10
|
|
|
1, 8, 47, 274, 1597, 9308, 54251, 316198, 1842937, 10741424, 62605607, 364892218, 2126747701, 12395593988, 72246816227, 421085303374, 2454265004017, 14304504720728, 83372763320351, 485932075201378, 2832219687887917
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Bisection (even part) of Chebyshev sequence with Diophantine property.
b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).
|
|
REFERENCES
|
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=8.
a(n) = ((3 + 2*sqrt(2))^(n+1) - (3 - 2*sqrt(2))^(n+1) + 2*((3 + 2*sqrt(2))^n - (3 - 2*sqrt(2))^n))/(4*sqrt(2)).
a(n) = S(n, 6) + 2*S(n-1, 6), with S(n, x) Chebyshev's polynomials of the second kind, A049310. S(n, 6) = A001109(n+1).
a(n) = (Pell(2*n+2) + 2*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + Pell(2*n))/2. - G. C. Greubel, Jan 19 2020
E.g.f.: (1/4)*exp(3*x)*(4*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x)). - Stefano Spezia, Jan 27 2020
|
|
EXAMPLE
|
8 = a(1) = sqrt((A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) = 8.
|
|
MAPLE
|
a[0]:=1: a[1]:=8: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006
|
|
MATHEMATICA
|
LinearRecurrence[{6, -1}, {1, 8}, 30] (* Harvey P. Dale, Oct 09 2017 *)
Table[(LucasL[2*n+1, 2] + Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jan 19 2020 *)
|
|
PROG
|
(PARI) my(x='x+O('x^30)); Vec((1+2*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jan 19 2020
(PARI) apply( {A054488(n)=[1, 8]*([0, -1; 1, 6]^n)[, 1]}, [0..30]) \\ M. F. Hasler, Feb 27 2020
(Magma) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 21); Coefficients(R!( (1+2*x)/(1-6*x+x^2))); // Marius A. Burtea, Jan 20 2020
(Sage) [(lucas_number2(2*n+1, 2, -1) + lucas_number1(2*n, 2, -1))/2 for n in (0..30)] # G. C. Greubel, Jan 19 2020
(GAP) a:=[1, 8];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|