A054320 - OEIS

%I #113 May 17 2023 08:42:18

%S 1,11,109,1079,10681,105731,1046629,10360559,102558961,1015229051,

%T 10049731549,99482086439,984771132841,9748229241971,96497521286869,

%U 955226983626719,9455772314980321,93602496166176491,926569189346784589,9172089397301669399,90794324783669909401

%N Expansion of g.f.: (1 + x)/(1 - 10*x + x^2).

%C Chebyshev's even-indexed U-polynomials evaluated at sqrt(3).

%C a(n)^2 is a star number (A003154).

%C Any k in the sequence has the successor 5*k + 2*sqrt(3(2*k^2 + 1)). - _Lekraj Beedassy_, Jul 08 2002

%C {a(n)} give the values of x solving: 3*y^2 - 2*x^2 = 1. Corresponding values of y are given by A072256(n+1). x + y = A001078(n+1). - _Richard R. Forberg_, Nov 21 2013

%C The aerated sequence (b(n))n>=1 = [1, 0, 11, 0, 109, 0, 1079, 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 = -8, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - _Peter Bala_, Mar 22 2015

%H G. C. Greubel, <a href="/A054320/b054320.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).

%H Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_1.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).

%H Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_2.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).

%H Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_3.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).

%H 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>, Contrib. Discr. Math. 3 (2) (2008), pp. 76-114. See Section 13.

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarNumber.html">Star Number</a>

%H 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 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).

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

%F (a(n)-1)^2 + a(n)^2 + (a(n)+1)^2 = b(n)^2 + (b(n)+1)^2 = c(n), where b(n) is A031138 and c(n) is A007667.

%F a(n) = 10*a(n-1) - a(n-2).

%F a(n) = (sqrt(6) - 2)/4*(5 + 2*sqrt(6))^(n+1) - (sqrt(6) + 2)/4*(5 - 2*sqrt(6))^(n+1).

%F a(n) = U(2*(n-1), sqrt(3)) = S(n-1, 10) + S(n-2, 10) with Chebyshev's U(n, x) and S(n, x) := U(n, x/2) polynomials and S(-1, x) := 0. S(n, 10) = A004189(n+1), n >= 0.

%F 6*a(n)^2 + 3 is a square. Limit_{n->oo} a(n)/a(n-1) = 5 + 2*sqrt(6). - _Gregory V. Richardson_, Oct 13 2002

%F Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i), then (-1)^n*q(n, -12) = a(n). - _Benoit Cloitre_, Nov 10 2002

%F a(n) = L(n,-10)*(-1)^n, where L is defined as in A108299; see also A072256 for L(n,+10). - _Reinhard Zumkeller_, Jun 01 2005

%F From _Reinhard Zumkeller_, Mar 12 2008: (Start)

%F (sqrt(2) + sqrt(3))^(2*n+1) = a(n)*sqrt(2) + A138288(n)*sqrt(3);

%F a(n) = A138288(n) + A001078(n).

%F a(n) = A001079(n) + 3*A001078(n). (End)

%F a(n) = A142238(2n) = A041006(2n)/2 = A041038(2n)/4. - _M. F. Hasler_, Feb 14 2009

%F a(n) = sqrt(A006061(n)). - _Zak Seidov_, Oct 22 2012

%F a(n) = sqrt((3*A072256(n)^2 - 1)/2). - _T. D. Noe_, Oct 23 2012

%F (sqrt(3) + sqrt(2))^(2*n+1) - (sqrt(3) - sqrt(2))^(2*n+1) = a(n)*sqrt(8). - _Bruno Berselli_, Oct 29 2019

%F a(n) = A004189(n)+A004189(n+1). - _R. J. Mathar_, Oct 01 2021

%F E.g.f.: exp(5*x)*(2*cosh(2*sqrt(6)*x) + sqrt(6)*sinh(2*sqrt(6)*x))/2. - _Stefano Spezia_, May 16 2023

%e a(1)^2 = 121 is the 5th star number (A003154).

%t CoefficientList[Series[(1+x)/(1-10x+x^2), {x,0,30}], x] (* _Vincenzo Librandi_, Mar 22 2015 *)

%t a[c_, n_] := Module[{},

%t    p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];

%t    d := Numerator[Convergents[Sqrt[c], n p]];

%t    t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];

%t    Return[t];

%t ] (* Complement of A142238 *)

%t a[3/2, 20] (* _Gerry Martens_, Jun 07 2015 *)

%o (PARI) a(n)=subst(poltchebi(n+1)-poltchebi(n),x,5)/4;

%o (Magma) I:=[1,11]; [n le 2 select I[n] else 10*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 22 2015

%o (GAP) a:=[1,11];; for n in [3..30] do a[n]:=10*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jul 22 2019

%Y A member of the family A057078, A057077, A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, which are the expansions of (1+x) / (1-kx+x^2) with k = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - _Philippe Deléham_, May 04 2004

%Y Cf. A003154, A006061, A031138, A007667, A004189.

%Y Cf. A138281. Cf. A100047.

%Y Cf. A142238.

%K nonn,easy

%O 0,2

%A _Ignacio Larrosa Cañestro_, Feb 27 2000

%E Chebyshev comments from _Wolfdieter Lang_, Oct 31 2002