A200441 - OEIS

%I #40 Apr 16 2023 20:37:00

%S 1,33,1088,35871,1182655,38991744,1285544897,42383989857,

%T 1397386120384,46071357982815,1518957427312511,50079523743330048,

%U 1651105326102579073,54436396237641779361,1794749970516076139840,59172312630792870835359,1950891566845648661427007

%N Expansion of g.f. 1/(1 - 33*x + x^2).

%C A Diophantine property of these numbers: (a(n+1)-a(n-1))^2 - 1085*a(n)^2 = 4.

%C More generally, for t(m)=(m+sqrt(m^2-4))/2 and u(n)=(t(m)^(n+1)-1/t(m)^(n+1))/(t(m)-1/t(m)), we can verify that (u(n+1)-u(n-1))^2-(m^2-4)*u(n)^2=4.

%C a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,32}. - _Milan Janjic_, Jan 26 2015

%H Bruno Berselli, <a href="/A200441/b200441.txt">Table of n, a(n) for n = 0..500</a>

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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (33,-1).

%F a(n) = 33*a(n-1)-a(n-2) with a(0)=1, a(1)=33.

%F a(n) = -a(-n-2) = (t^(n+1)-1/t^(n+1))/(t-1/t), where t=(33+sqrt(1085))/2.

%F a(n) = sum((-1)^k*binomial(n-k, k)*33^(n-2k), k=0..floor(n/2)).

%F a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*32^k. - _Philippe Deléham_, Feb 10 2012

%F From _Peter Bala_, Dec 23 2012: (Start)

%F Product {n >= 0} (1 + 1/a(n)) = 1/31*(31 + sqrt(1085)).

%F Product {n >= 1} (1 - 1/a(n)) = 1/66*(31 + sqrt(1085)). (End)

%F E.g.f.: exp(33*x/2)*cosh(sqrt(1085)*x/2) + 33*exp(33*x/2)*sinh(sqrt(1085)*x/2)/sqrt(1085). - _Stefano Spezia_, Apr 16 2023

%t LinearRecurrence[{33, -1}, {1, 33}, 17]

%o (PARI) Vec(1/(1-33*x+x^2)+O(x^17))

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-1085); S:=[(((33+r)/2)^n-1/((33+r)/2)^n)/r: n in [1..17]]; [Integers()!S[j]: j in [1..#S]];

%o (Maxima) makelist(sum((-1)^k*binomial(n-k,k)*33^(n-2*k),k,0,floor(n/2)),n,0,16);

%Y Cf. A004190, A056594, A077421; A029547, A029548; A200442.

%Y Cf. A101950.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Nov 18 2011