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A100237 Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n >= 0.

%I A100237 #25 Feb 06 2022 06:51:05
%S A100237 1,4,21,109,566,2939,15261,79244,411481,2136649,11094726,57610279,
%T A100237 299146121,1553340884,8065850541,41882593589,217478818486,
%U A100237 1129276686019,5863862248581,30448587928924,158106801893201,820982597394929,4263019788867846,22136081541734159
%N A100237 Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n >= 0.
%C A100237 G.f. equals the ratio of the g.f.s of any two adjacent diagonals of triangle A100235.
%C A100237 a(n) is the number of compositions of n when there are 4 types of 1 and 5 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%H A100237 Colin Barker, <a href="/A100237/b100237.txt">Table of n, a(n) for n = 0..1000</a>
%H A100237 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A100237 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,1).
%F A100237 a(n) = 5*a(n-1) + a(n-2) for n>1, with a(0)=1, a(1)=4.
%F A100237 G.f.: (1-x)/(1-5*x-x^2).
%F A100237 Numerators in continued fraction [1, 4, 5, 5, 5, ...]. Continued fraction [1, 4, 5, 5, 5, ...] = 0.807417596433..., the inradius of a right triangle with legs 2 and 5. n-th convergent (n>0) to [1, 4, 5, 5, 5, ...] = A100237(n)/A052918(n), the first few being 1/1, 4/5, 21/26, 109/135, ... - _Gary W. Adamson_, Dec 21 2007
%F A100237 If p[1]=4, p[i]=5, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=det A. - _Milan Janjic_, Apr 29 2010
%F A100237 a(n) = (2^(-1-n)*((5-sqrt(29))^n*(-3+sqrt(29)) + (3+sqrt(29))*(5+sqrt(29))^n))/sqrt(29). - _Colin Barker_, Oct 13 2015
%p A100237 a[0]:=1: a[1]:=4: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..21); # _Zerinvary Lajos_, Jul 26 2006
%o A100237 (PARI) a(n)=polcoeff((1-x)/(1-5*x-x^2)+x*O(x^n),n)
%o A100237 (PARI) Vec((1-x)/(1-5*x-x^2) + O(x^40)) \\ _Colin Barker_, Oct 13 2015
%Y A100237 Cf. A100234, A100235, A100236.
%Y A100237 First differences of A052918.
%Y A100237 Cf. A052918.
%K A100237 nonn,easy
%O A100237 0,2
%A A100237 _Paul D. Hanna_, Nov 30 2004