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A200442 Expansion of 1/(1-31*x+x^2).

%I A200442 #31 Sep 08 2022 08:46:00
%S A200442 1,31,960,29729,920639,28510080,882891841,27341136991,846692354880,
%T A200442 26220121864289,811977085438079,25145069526716160,778685178242762881,
%U A200442 24114095455998933151,746758273957724164800,23125392397233450175649,716140406040279231280319
%N A200442 Expansion of 1/(1-31*x+x^2).
%C A200442 A Diophantine property of these numbers: (a(n+1)-a(n-1))^2 - 957*a(n)^2 = 4. (See also comment in A200441.)
%C A200442 For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,30}. - _Milan Janjic_, Jan 26 2015
%H A200442 Bruno Berselli, <a href="/A200442/b200442.txt">Table of n, a(n) for n = 0..500</a>
%H A200442 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A200442 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (31,-1).
%F A200442 G.f.: 1/(1-31*x+x^2).
%F A200442 a(n) = 31*a(n-1)-a(n-2) with a(0)=1, a(1)=31.
%F A200442 a(n) = -a(-n-2) = (t^(n+1)-1/t^(n+1))/(t-1/t) where t=(31+sqrt(957))/2.
%F A200442 a(n) = sum((-1)^k*binomial(n-k, k)*31^(n-2k), k=0..floor(n/2)).
%F A200442 a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*30^k. - _Philippe Deléham_, Feb 10 2012
%F A200442 Product {n >= 0} (1 + 1/a(n)) = 1/29*(29 + sqrt(957)). - _Peter Bala_, Dec 23 2012
%F A200442 Product {n >= 1} (1 - 1/a(n)) = 1/62*(29 + sqrt(957)). - _Peter Bala_, Dec 23 2012
%t A200442 LinearRecurrence[{31, -1}, {1, 31}, 17]
%o A200442 (PARI)  Vec(1/(1-31*x+x^2)+O(x^17))
%o A200442 (Magma) /* By the closed form: */ Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-957); S:=[(((31+r)/2)^n-1/((31+r)/2)^n)/r: n in [1..17]]; [Integers()!S[j]: j in [1..#S]];
%o A200442 (Maxima)  makelist(sum((-1)^k*binomial(n-k,k)*31^(n-2*k),k,0,floor(n/2)),n,0,16);
%Y A200442 Cf. A029548, A097313, A200441.
%K A200442 nonn,easy
%O A200442 0,2
%A A200442 _Bruno Berselli_, Nov 18 2011