login
A077240
Bisection (even part) of Chebyshev sequence with Diophantine property.
5
5, 23, 133, 775, 4517, 26327, 153445, 894343, 5212613, 30381335, 177075397, 1032071047, 6015350885, 35060034263, 204344854693, 1191009093895, 6941709708677, 40459249158167, 235813785240325, 1374423462283783, 8010726988462373, 46689938468490455
OFFSET
0,1
COMMENTS
a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A054488(n).
The odd part is A077239(n) with Diophantine companion A077413(n).
FORMULA
a(n) = 6*a(n-1) - a(n-2), a(-1) = 7, a(0) = 5.
a(n) = T(n+1, 3)+2*T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).
G.f.: (5-7*x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-4+5*sqrt(2))+(3+2*sqrt(2))^n*(4+5*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015
EXAMPLE
23 = a(1) = sqrt(8*A054488(1)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.
MATHEMATICA
Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 19}] (* Jean-François Alcover, Dec 19 2013 *)
LinearRecurrence[{6, -1}, {5, 23}, 30] (* Harvey P. Dale, Mar 29 2017 *)
PROG
(PARI) Vec((5-7*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015
CROSSREFS
Cf. A077242 (even and odd parts).
Sequence in context: A009321 A078509 A239820 * A281231 A356010 A244786
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 08 2002
STATUS
approved