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A097727 Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n)=A097726(n), n >= 0. 8

%I #43 Sep 08 2022 08:45:14

%S 1,101,10301,1050601,107151001,10928351501,1114584702101,

%T 113676711262801,11593909964103601,1182465139627304501,

%U 120599850332020955501,12300002268726510156601,1254479631559772015017801,127944622416828019021659101,13049097006884898168194210501

%N Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n)=A097726(n), n >= 0.

%C Hypotenuses of primitive Pythagorean triples in A195622 and A195623. - _Clark Kimberling_, Sep 22 2011

%H Vincenzo Librandi, <a href="/A097727/b097727.txt">Table of n, a(n) for n = 0..200</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (102,-1).

%F a(n) = S(n, 2*51) - S(n-1, 2*51) = T(2*n+1, sqrt(26))/sqrt(26), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

%F a(n) = ((-1)^n)*S(2*n, 10*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

%F G.f.: (1-x)/(1-102*x+x^2).

%F a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=101. - _Philippe Deléham_, Nov 18 2008

%e (x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 =-1.

%t LinearRecurrence[{102,-1},{1,101},20] (* _Harvey P. Dale_, Apr 12 2014 *)

%t CoefficientList[Series[(1-x)/(1-102x+x^2), {x,0,20}], x] (* _Vincenzo Librandi_, Apr 13 2014 *)

%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-102*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019

%o (Magma) I:=[1,101]; [n le 2 select I[n] else 102*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019

%o (Sage) ((1-x)/(1-102*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019

%o (GAP) a:=[1,101];; for n in [3..20] do a[n]:=102*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019

%Y Cf. A097725 for S(n, 102).

%Y Row 5 of array A188647.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 31 2004

%E More terms from _Harvey P. Dale_, Apr 12 2014

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)