A055991 a(n) is its own 4th difference.

1, 5, 19, 69, 250, 907, 3292, 11949, 43371, 157422, 571388, 2073943, 7527704, 27322992, 99173120, 359964521, 1306548149, 4742323107, 17213011605, 62477347458, 226771411939, 823102698260, 2987581397893, 10843899100203

Offset: 1

Author

Henry Bottomley, Jun 02 2000

Keywords

• nonn
• easy

Comments

a(n) is the number of distinct matrix products in (A+B+C+D+E)^n where A,B,C and D all commute with each other, but not with E. - Paul D. Hanna and Max Alekseyev, Feb 01 2006

Row sums of Riordan array (1,1/(1-x)^4). - Paul Barry, Feb 02 2006

Quadrisection of A003269: a(n)=A003269(4n-1). - Paul Barry, Feb 02 2006

From Gary W. Adamson, Apr 23 2009: (Start)

Equals the INVERT transform of the tetrahedral series.

a(4) = 69 = (1, 4, 10) dot (19, 5, 1) + 20; = (19 + 20 + 10) + 20. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 16.
• Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
• Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).

Crossrefs

Cf. A055988, A055989, A055990 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

Programs

Magma

I:=[1, 5, 19, 69]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 05 2012

Mathematica

LinearRecurrence[{5,-6,4,-1},{1,5,19,69},30] (* Harvey P. Dale, Feb 27 2013 *)

Formula

a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055990(n) = A055988(n+1)-A055988(n) = A055989(n+1)-2*A055989(n)+A055989(n-1).

Letting a(0)=1, we have a(n)=sum(u=0, n-1, sum(v=0, u, sum(w=0, v, sum(x=0, w, a(x))))) for n>0. - Benoit Cloitre, Jan 26 2003

a(n) = sum_{k=1..n} binomial(n+3*k-1, n-k). - Vladeta Jovovic, Mar 23 2003

a(n) = sum{k=0..n, binomial(4n-3k-1,k)}. - Paul Barry, Feb 02 2006

G.f.: x/(1-5x+6x^2-4x^3+x^4). - Paul Barry, Feb 02 2006