OFFSET
0,3
COMMENTS
For all n, a(n)=1*6^n-4*3^n+6*1^n-4*0^n+1*0^n [with 0^0=1] where powers are taken of triangular numbers and multiplied by binomial coefficients with alternating signs.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-27,18).
FORMULA
For n>0, a(n) = 6^n-4*3^n+6.
a(n) = 10*a(n-1)-27*a(n-2)+18*a(n-3) for n>3. G.f.: -6*x^2*(9*x+1) / ((x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Sep 14 2014
EXAMPLE
a(2)=6 since intervals a-a and b-b can be combined as a-a-b-b, a-b-a-b, a-b-b-a, b-a-b-a, b-a-a-b, or b-a-a-b.
MAPLE
PROG
(Magma) [1*6^n-4*3^n+6*1^n-4*0^n+1*0^n: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011
(PARI) concat([0, 0], Vec(-6*x^2*(9*x+1)/((x-1)*(3*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Sep 14 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jan 05 2001
STATUS
approved