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A057080 Even-indexed Chebyshev U-polynomials evaluated at sqrt(10)/2.

%I A057080 #76 May 16 2025 03:25:23
%S A057080 1,9,71,559,4401,34649,272791,2147679,16908641,133121449,1048062951,
%T A057080 8251382159,64962994321,511452572409,4026657584951,31701808107199,
%U A057080 249587807272641,1965000650073929,15470417393318791,121798338496476399
%N A057080 Even-indexed Chebyshev U-polynomials evaluated at sqrt(10)/2.
%C A057080 a(n) = L(n,-8)*(-1)^n, where L is defined as in A108299; see also A070997 for L(n,+8). - _Reinhard Zumkeller_, Jun 01 2005
%C A057080 General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878. a(1)=5 gives A001834. a(1)=6 gives A030221. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - _Ctibor O. Zizka_, Sep 02 2008
%C A057080 The primes in this sequence are 71, 34649, 16908641, 8251382159, 31701808107199,... - _Ctibor O. Zizka_, Sep 02 2008
%C A057080 The aerated sequence (b(n))n>=1 = [1, 0, 9, 0, 71, 0, 559, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -6, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - _Peter Bala_, Mar 22 2015
%C A057080 From _Klaus Purath_, May 06 2025: (Start)
%C A057080 Nonnegative solutions to the Diophantine equation 3*a(n)^2 - 5*b(n)^2 = -2. The corresponding b(n) are A070997(n). Note that (a(n)*a(n+2) - a(n+1)^2)/2 = -5 and (b(n)*b(n+2) - b(n+1)^2)/2 = 3.
%C A057080 (a(n) + b(n))/2 = (a(n+1) - b(n+1))/2 = A001090(n+1) = Lucas U(8,1). Also a(n)*b(n+1) - a(n+1)*b(n) = -2.
%C A057080 a(n) = (t(i+2n+1) - t(i))/(t(i+n+1) - t(i+n)) as long as t(i+n+1) - t(i+n) != 0 for integer i and n >= 0 where (t) is a sequence satisfying t(i+3) = 9*t(i+2) - 9*t(i+1) + t(i) or t(i+2) = 8*t(i+1) - t(i), regardless of the initial values and including this sequence itself. (End)
%H A057080 G. C. Greubel, <a href="/A057080/b057080.txt">Table of n, a(n) for n = 0..1000</a>
%H A057080 Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H A057080 K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H A057080 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H A057080 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A057080 Wolfdieter Lang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eq.(44), rhs, m=10.
%H A057080 Donatella Merlini and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2016.08.017">Arithmetic into geometric progressions through Riordan arrays</a>, Discrete Mathematics 340.2 (2017): 160-174.
%H A057080 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A057080 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume
%H A057080 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A057080 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-1).
%F A057080 For all elements x of the sequence, 15*x^2 + 10 is a square. Lim. n-> Inf. a(n)/a(n-1) = 4 + sqrt(15). - _Gregory V. Richardson_, Oct 13 2002
%F A057080 a(n) = 8*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.
%F A057080 a(n) = S(n, 8) + S(n-1, 8) = S(2*n, sqrt(10)) with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 8) = A001090(n).
%F A057080 G.f.: (1+x)/(1-8*x+x^2).
%F A057080 a(n) = ( ((4+sqrt(15))^(n+1) - (4-sqrt(15))^(n+1)) + ((4+sqrt(15))^n - (4-sqrt(15))^n) )/(2*sqrt(15)). - _Gregory V. Richardson_, Oct 13 2002
%F A057080 a(n) = sqrt((5*A070997(n)^2 - 2)/3) (cf. Richardson comment).
%F A057080 Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i) then a(n) = (-1)^n*q(n,-10). - _Benoit Cloitre_, Nov 10 2002
%F A057080 a(n) = Jacobi_P(n,1/2,-1/2,4)/Jacobi_P(n,-1/2,1/2,1); - _Paul Barry_, Feb 03 2006
%F A057080 a(n+1) = 4*a(n) + sqrt(5*(3*a(n)^2 + 2)). - _Richard Choulet_, Aug 30 2007
%F A057080 In addition to the first formula above: In general, the following applies to all recurrences (a(n)) of the form (8,-1) with a(0) = 1 and arbritrary a(1): 15*a(n)^2 + y = b^2 where y = x^2 + 8*x + 1 and x = a(1) - 8. Also y = a(k+1)^2 - a(k)*a(k+1) for any k >=0. - _Klaus Purath_, May 06 2025
%F A057080 From _Peter Bala_, May 09 2025: (Start)
%F A057080 a(n) = Dir(n, 4), where Dir(n, x) denotes the n-th row polynomial of the triangle A244419.
%F A057080 a(n)^2 - 8*a(n)*a(n+1) + a(n+1)^2 = 10.
%F A057080 More generally, for arbitrary x, a(n+x)^2 - 8*a(n+x)*a(n+x+1) + a(n+x+1)^2 = 10 with a(n) := ( ((4+sqrt(15))^(n+1) - (4-sqrt(15))^(n+1)) + ((4+sqrt(15))^n - (4-sqrt(15))^n) )/(2*sqrt(15)) as given above.
%F A057080 a(n+1/2) = sqrt(10) * A001090(n+1).
%F A057080 a(n+3/4) + a(n+1/4) = sqrt(10)*sqrt(sqrt(10) + 2) * A001090(n+1).
%F A057080 a(n+3/4) - a(n+1/4) = sqrt((sqrt(40) - 4)/3) * A001091(n+1).
%F A057080 Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/10 (telescoping series: for n >= 1, 10/(a(n) - 1/a(n)) = 1/A001090(n) + 1/A001090(n+1)).
%F A057080 Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(5/3) (telescoping product: Product_{n = 1..k} ((a(n) + 1)/(a(n) - 1))^2 = 5/3 * (1 - 2/(1 + A001091(k+1)))). (End)
%p A057080 A057080 := proc(n)
%p A057080     option remember;
%p A057080     if n <= 1 then
%p A057080         op(n+1,[1,9]);
%p A057080     else
%p A057080         8*procname(n-1)-procname(n-2) ;
%p A057080     end if;
%p A057080 end proc: # _R. J. Mathar_, Apr 30 2017
%t A057080 CoefficientList[Series[(1+x)/(1-8x+x^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Mar 22 2015 *)
%o A057080 (Sage) [(lucas_number2(n,8,1)-lucas_number2(n-1,8,1))/6 for n in range(1, 21)] # _Zerinvary Lajos_, Nov 10 2009
%o A057080 (PARI) Vec((1+x)/(1-8*x+x^2) + O(x^30)) \\ _Michel Marcus_, Mar 22 2015
%o A057080 (Magma) I:=[1,9]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 22 2015
%o A057080 (GAP) a:=[1,9];; for n in [3..30] do a[n]:=8*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 06 2019
%Y A057080 Cf. A033890, A100047.
%K A057080 nonn,easy
%O A057080 0,2
%A A057080 _Wolfdieter Lang_, Aug 04 2000