

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
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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).


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,1).


FORMULA

a(n) = 6*a(n1)  a(n2), 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.: (57*x)/(16*x+x^2).
a(n) = (((32*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}] (* JeanFrançois Alcover, Dec 19 2013 *)
LinearRecurrence[{6, 1}, {5, 23}, 30] (* Harvey P. Dale, Mar 29 2017 *)


PROG

(PARI) Vec((57*x)/(16*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 A244786 A129098
Adjacent sequences: A077237 A077238 A077239 * A077241 A077242 A077243


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Nov 08 2002


STATUS

approved



