OFFSET
0,2
COMMENTS
An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).
REFERENCES
Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Ronald Cools, Encyclopaedia of Cubature Formulas
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
FORMULA
a(n) = 2^n + 2*n.
From Gary W. Adamson, Jul 20 2007: (Start)
Binomial transform of (1, 3, 1, 1, 1, ...).
For n > 0, a(n) = 2*A005126(n-1). (End)
From R. J. Mathar, Jun 13 2008: (Start)
G.f.: 1 + 2*x*(2 -4*x +x^2)/((1-x)^2*(1-2*x)).
a(n+1)-a(n) = A052548(n). (End)
From Colin Barker, Oct 16 2013: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (1 - 3*x^2)/((1-x)^2*(1-2*x)). (End)
E.g.f.: exp(2*x) + 2*x*exp(x). - Franck Maminirina Ramaharo, Dec 19 2018
MAPLE
a:= proc(n) 2^n + 2*n: end: seq(a(n), n=0..50); # Gary W. Adamson, Jul 20 2007
MATHEMATICA
LinearRecurrence[{4, -5, 2}, {1, 4, 8}, 34] (* Jean-François Alcover, Mar 19 2020 *)
PROG
(Magma) [2^n+2*n: n in [1..40]]; // Vincenzo Librandi, Oct 22 2011
(Maxima) makelist(2^n + 2*n, n, 0, 50); /* Franck Maminirina Ramaharo, Dec 19 2018 */
(GAP) List([0..40], n->2^n+2*n); # Muniru A Asiru, Dec 21 2018
(SageMath) [2^n +2*n for n in range(41)] # G. C. Greubel, Feb 01 2023
CROSSREFS
Row sums of A131830.
KEYWORD
nonn,easy
AUTHOR
Sergey Kitaev, Nov 13 2004
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Dec 21 2018
STATUS
approved