Search: id:A072256
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%I A072256
%S A072256 1,1,9,89,881,8721,86329,854569,8459361,83739041,828931049,8205571449,
%T A072256 81226783441,804062262961,7959395846169,78789896198729,779939566141121,
%U A072256 7720605765212481,76426118085983689,756540575094624409
%N A072256 a(n) = 10*a(n-1) - a(n-2); a(0) = a(1) = 1.
%C A072256 Any k in the sequence is followed by 5*k + 2sqrt{2(3*k^2 - 1)}. Gives 
               solutions for x in 3*x^2 - 2*y^2 = 1. Corresponding y is given by 
               A054320(n).
%C A072256 a(n) = L(n-1,10), where L is defined as in A108299; see also A054320 
               for L(n,-10). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 01 2005
%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 (tanyakh(AT)yahoo.com), 
               Jan 10 2007
%C A072256 a(n) = A138288(n-1) for n > 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 12 2008
%D A072256 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, 
               Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 
               283).
%H A072256 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A072256 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A072256 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%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)^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 For all members x of the sequence, 6*x^2 - 2 is a square. Lim. n -> Inf. 
               a(n)/a(n-1) = 5 + 2*sqrt(6). - Gregory V. Richardson (omomom(AT)hotmail.com), 
               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 (benoit7848c(AT)orange.fr), Nov 10 2002
%F A072256 a(n)a(n+3) = 80 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%t A072256 a[n_] := a[n] = 10a[n - 1] - a[n - 2]; a[0] = a[1] = 1; Table[ a[n], 
               {n, 0, 20}]
%o A072256 (Other) sage: [lucas_number1(n,10,1)-lucas_number1(n-1,10,1) for n in 
               xrange(0, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 10 2009]
%Y A072256 Cf. A054320, A031138.
%Y A072256 Row 10 of array A094954.
%Y A072256 First differences of A004189.
%Y A072256 A072256(n)=sqrt(A046172(n)) [From Weisenhorn Paul (paulweisenhorn(AT)online.de), 
               May 15 2009]
%Y A072256 Sequence in context: A064616 A133486 A015584 this_sequence A138288 A059482 
               A109002
%Y A072256 Adjacent sequences: A072253 A072254 A072255 this_sequence A072257 A072258 
               A072259
%K A072256 nonn
%O A072256 0,3
%A A072256 Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 08 2002
%E A072256 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 17 2002

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