OFFSET
0,3
COMMENTS
From Bruno Berselli, Jun 07 2013: (Start)
A050228(n) = a(n) -a(n-1), n>0.
A077868(n-1)= a(n) -2*a(n-1) +a(n-2), n>1.
A000217(n) = a(n) -a(n-1) -a(n-3), n>2.
A000930(n-1)= a(n) -3*a(n-1) +3*a(n-2) -a(n-3), n>2.
n = a(n) -2*a(n-1) +a(n-2) -a(n-3) +a(n-4), n>3.
1 = a(n) -3*a(n-1) +3*a(n-2) -2*a(n-3) +2*a(n-4) -a(n-5), n>4.
0 = a(n) -4*a(n-1) +6*a(n-2) -5*a(n-3) +4*a(n-4) -3*a(n-5) +a(n-6), n>5.
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-4,3,-1).
FORMULA
G.f.: x/((1-x-x^3)*(1-x)^3).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).
a(n) = A099567(n+2, 3). (End)
MAPLE
a:= n-> (Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -6, 5, -4, 3, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
MATHEMATICA
LinearRecurrence[{4, -6, 5, -4, 3, -1}, {0, 1, 4, 10, 21, 40}, 40] (* Bruno Berselli, Jun 07 2013 *)
CoefficientList[Series[x/((1-x-x^3)*(1-x)^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)
PROG
(PARI) my(x='x+O('x^50)); Vec(x/((1-x-x^3)*(1-x)^3)) \\ G. C. Greubel, Apr 28 2017
(Magma)
A226405:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+2, j+3): j in [0..Floor((n+2)/3)]]) >;
[A226405(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022
(SageMath)
def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
[A226405(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Jun 06 2013
STATUS
approved