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A077409 Bisection (even part) of Chebyshev sequence with Diophantine property. 6
7, 59, 583, 5771, 57127, 565499, 5597863, 55413131, 548533447, 5429921339, 53750679943, 532076878091, 5267018100967, 52138104131579, 516114023214823, 5109002128016651, 50573907256951687, 500630070441500219, 4955726797158050503, 49056637901139004811 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077251(n).

The odd part is A077250(n) with Diophantine companion A077249(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 (10,-1).

FORMULA

a(n) = 10*a(n-1)- a(n-2), a(-1)=11, a(0)=7.

a(n) = T(n+1, 5)+2*T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5) = A001079(n).

a(n) = sqrt(24*A077251(n)^2 + 25).

G.f.: (7-11*x)/(1-10*x+x^2).

EXAMPLE

59 = a(1) = sqrt(24*A077251(1)^2 + 25) = sqrt(24*12^2 + 25) = sqrt(3481) = 59.

MATHEMATICA

CoefficientList[Series[(7 - 11 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

PROG

(PARI) a(n)=if(n<0, 0, subst(poltchebi(n+1)+2*poltchebi(n), x, 5))

(PARI) Vec((7-11*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015

(PARI) a(n)=polchebyshev(n+1, , 5)+2*polchebyshev(n, , 5) \\ Charles R Greathouse IV, Jun 15 2015

(PARI) a(n)=([0, 1; -1, 10]^n*[7; 59])[1, 1] \\ Charles R Greathouse IV, Jun 15 2015

CROSSREFS

Sequence in context: A210397 A099659 A135150 * A192458 A203237 A099347

Adjacent sequences:  A077406 A077407 A077408 * A077410 A077411 A077412

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 08, 2002

STATUS

approved

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Last modified April 29 09:07 EDT 2017. Contains 285604 sequences.