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

1, 4, 21, 109, 566, 2939, 15261, 79244, 411481, 2136649, 11094726, 57610279, 299146121, 1553340884, 8065850541, 41882593589, 217478818486, 1129276686019, 5863862248581, 30448587928924, 158106801893201, 820982597394929, 4263019788867846, 22136081541734159

Offset: 0

Author

Paul D. Hanna, Nov 30 2004

Keywords

• nonn
• easy

Comments

G.f. equals the ratio of the g.f.s of any two adjacent diagonals of triangle A100235.

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (5,1).

Crossrefs

Cf. A100234, A100235, A100236.

First differences of A052918.

Cf. A052918.

Programs

Maple

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

PARI

a(n)=polcoeff((1-x)/(1-5*x-x^2)+x*O(x^n),n)

PARI

Vec((1-x)/(1-5*x-x^2) + O(x^40)) \\ Colin Barker, Oct 13 2015

Formula

a(n) = 5*a(n-1) + a(n-2) for n>1, with a(0)=1, a(1)=4.

G.f.: (1-x)/(1-5*x-x^2).

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

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

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