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A072256 a(n) = 10*a(n-1) - a(n-2) for n > 1, a(0) = a(1) = 1.

%I A072256 #141 May 16 2025 03:13:09
%S A072256 1,1,9,89,881,8721,86329,854569,8459361,83739041,828931049,8205571449,
%T A072256 81226783441,804062262961,7959395846169,78789896198729,
%U A072256 779939566141121,7720605765212481,76426118085983689,756540575094624409,7488979632860260401,74133255753507979601,733843577902219535609
%N A072256 a(n) = 10*a(n-1) - a(n-2) for n > 1, a(0) = a(1) = 1.
%C A072256 Any k in the sequence is followed by 5*k + 2*sqrt(2*(3*k^2 - 1)).
%C A072256 Gives solutions for x in 3*x^2 - 2*y^2 = 1. Corresponding y is given by A054320(n-1). [corrected by _Jon E. Schoenfield_, Jun 08 2018]
%C A072256 Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8,9} which do not end in 0. - _Tanya Khovanova_, Jan 10 2007
%C A072256 For n >= 2, a(n) equals the permanent of the (2n-2) X (2n-2) tridiagonal matrix with sqrt(8)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 08 2011
%C A072256 Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 8 = 0. - _Colin Barker_, Feb 09 2014
%C A072256 The aerated sequence [b(n)]n>=1 = [1, 0, 9, 0, 89, 0, 881, 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 = -12, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy. - _Peter Bala_, May 12 2025
%H A072256 Vincenzo Librandi, <a href="/A072256/b072256.txt">Table of n, a(n) for n = 0..200</a>
%H A072256 Christian Aebi and Grant Cairns, <a href="https://arxiv.org/abs/2401.08827">Equable Parallelograms on the Eisenstein Lattice</a>, arXiv:2401.08827 [math.CO], 2024. See p. 14.
%H A072256 Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, <a href="https://arxiv.org/abs/2401.13524">Combinatorics on words and generating Dirichlet series of automatic sequences</a>, arXiv:2401.13524 [math.CO], 2024.
%H A072256 S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 283).
%H A072256 Bruno Deschamps, <a href="http://dx.doi.org/10.1016/j.jnt.2010.06.006">Sur les bonnes valeurs initiales de la suite de Lucas-Lehmer</a>, Journal of Number Theory, Volume 130, Issue 12, December 2010, Pages 2658-2670.
%H A072256 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 A072256 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 A072256 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 A072256 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 A072256 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 A072256 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A072256 J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H A072256 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 A072256 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 A072256 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>
%H A072256 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).
%F A072256 a(n) = (3-sqrt(6))/6 * (5+2*sqrt(6))^n + (3+sqrt(6))/6 * (5-2*sqrt(6))^n.
%F A072256 a(n) = (2*A031138(n) + 1)/3 = sqrt((2*A054320(n-1)^2 + 1)/3), n >= 1.
%F A072256 a(n) = U(n-1, 5)-U(n-2, 5) = T(2*n-1, sqrt(3))/sqrt(3) with Chebyshev's U- and T- polynomials and U(-1, x) := 0, U(-2, x) := -1, T(-1, x) := x.
%F A072256 G.f.: (1-9*x)/(1-10*x+x^2).
%F A072256 6*a(n)^2 - 2 is a square. Limit_{n->oo} a(n)/a(n-1) = 5 + 2*sqrt(6). - _Gregory V. Richardson_, Oct 10 2002
%F A072256 Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then q(n, 8) = a(n+1). - _Benoit Cloitre_, Nov 10 2002
%F A072256 a(n)*a(n+3) = 80 + a(n+1)*a(n+2). - _Ralf Stephan_, May 29 2004
%F A072256 a(n) = L(n-1,10), where L is defined as in A108299; see also A054320 for L(n,-10). - _Reinhard Zumkeller_, Jun 01 2005
%F A072256 a(n) = A138288(n-1) for n > 0. - _Reinhard Zumkeller_, Mar 12 2008
%F A072256 a(n) = sqrt(A046172(n)). - _Paul Weisenhorn_, May 15 2009
%F A072256 a(n) = ceiling(((3-sqrt(6))*(5+2*sqrt(6))^n)/6). - _Paul Weisenhorn_, May 23 2020
%F A072256 E.g.f.: exp(5*x)*(3*cosh(2*sqrt(6)*x) - sqrt(6)*sinh(2*sqrt(6)*x))/3. - _Stefano Spezia_, Oct 25 2023
%F A072256 From _Peter Bala_, May 08 2025: (Start)
%F A072256 a(n) = (-1)^n * Dir(n-1, -5), where Dir(n, x) denotes the n-th row polynomial of A244419.
%F A072256 For arbitrary x, a(n+x)^2 - 10*a(n+x)*a(n+x+1) + a(n+x+1)^2 = -8 with a(n) := (3-sqrt(6))/6 * (5+2*sqrt(6))^n + (3+sqrt(6))/6 * (5-2*sqrt(6))^n as given above (the particular case x = 0 is noted in the Comments section).
%F A072256 a(n+1/2) = 1/sqrt(3) * A001079(n).
%F A072256 a(n+3/4) + a(n+1/4) = sqrt(2/3) * sqrt(1 + sqrt(3)) * A001079(n).
%F A072256 a(n+3/4) - a(n+1/4) = 4 * sqrt(sqrt(3) - 1) * A004189(n).
%F A072256 a(n) divides a(3*n-1); a(n) divides a(5*n-2); in general, for k >= 0, a(n) divides a((2*k+1)*n - k).
%F A072256 Sum_{n >= 2} 1/(a(n) - 1/a(n)) = 1/8 (telescoping series: for n >= 2, 1/(a(n) - 1/a(n)) = 1/A291181(n-2) - 1/A291181(n-1).)
%F A072256 Product_{n >= 2} ((a(n) + 1)/(a(n) - 1))^(-1)^n = sqrt(3/2) (telescoping product: Product_{n = 2..k} (((a(n) + 1)/(a(n) - 1))^(-1)^n)^2 = 3/2 * (1 - (-1)^(k+1)/(3*A098308(k))).) (End)
%p A072256 seq( simplify(ChebyshevU(n,5) -9*ChebyshevU(n-1,5)), n=0..20); # _G. C. Greubel_, Jan 14 2020
%t A072256 a[n_]:= a[n]= 10a[n-1] -a[n-2]; a[0]=a[1]=1; Table[ a[n], {n, 0, 20}]
%t A072256 CoefficientList[Series[(1-9x)/(1-10x+x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 10 2014 *)
%t A072256 Table[ChebyshevU[n, 5] -9*ChebyshevU[n-1, 5], {n,0,20}] (* _G. C. Greubel_, Jan 14 2020 *)
%t A072256 LinearRecurrence[{10,-1},{1,1},20] (* _Harvey P. Dale_, Jun 17 2022 *)
%o A072256 (Magma) [n le 2 select 1 else 10*Self(n-1)-Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Feb 10 2014
%o A072256 (PARI) a(n)=([0,1; -1,10]^n*[1;1])[1,1] \\ _Charles R Greathouse IV_, May 10 2016
%o A072256 (PARI) vector(21, n, polchebyshev(n-1,2,5) -9*polchebyshev(n-2,2,5) ) \\ _G. C. Greubel_, Jan 14 2020
%o A072256 (GAP) a:=[1,1];; for n in [3..20] do a[n]:=10*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 14 2020
%Y A072256 Cf. A031138, A046172, A054320, A108299.
%Y A072256 Row 10 of array A094954.
%Y A072256 First differences of A004189.
%Y A072256 Essentially the same as A138288.
%K A072256 nonn,easy
%O A072256 0,3
%A A072256 _Lekraj Beedassy_, Jul 08 2002
%E A072256 Edited by _Robert G. Wilson v_, Jul 17 2002