OFFSET
0,2
COMMENTS
Consider a figure made up of a row of n >= 1 adjacent congruent rectangles in which all possible diagonals of the rectangles have been drawn. The number of regions formed is A306302. If we distort all these diagonals very slightly so that no three lines meet at a point, the number of regions changes to a(n).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
Satisfies the identity a(n) = A306302(n) + Sum_{k=3..(n+1)} binomial(k-1,2)*A333275(n,2*k). E.g. for n=4 we have a(4) = 104 + 8*1 + 2*3 + 1*6 = 124.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)
MATHEMATICA
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 4, 17, 51, 124}, 50] (* or *)
PROG
(PARI) concat(0, Vec(x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5 + O(x^40))) \\ Colin Barker, May 27 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Scott R. Shannon and N. J. A. Sloane, May 19 2020
STATUS
approved