OFFSET
0,3
COMMENTS
Number of binary pattern classes with 4 black beads in the (2,n)-rectangular grid; two patterns are in the same class if one of them can be obtained by reflection or rotation of the other one. - Yosu Yurramendi, Sep 12 2008
This sequence is the case k=n+3 of b(n,k) = n*(n+1)*((k-2)*n-(k-5))/6, which is the n-th k-gonal pyramidal number. Therefore, apart from 0, this sequence is the 3rd diagonal of the array in A080851. - Luciano Ancora, Apr 10 2015
REFERENCES
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n>4. - Harvey P. Dale, May 01 2013
a(n) = (binomial(2*n+2,4) + 3*binomial(n+1,2))/4 = (A053134(n-1) + 3*A000217(n))/4 . - Yosu Yurramendi and María Merino, Aug 21 2013
G.f.: x*(1+x+2*x^2) / (1-x)^5 . - R. J. Mathar, Aug 21 2013
E.g.f.: (1/6)*x*(6 + 12*x + 7*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 06 2024
MATHEMATICA
Table[(n(n+1)(n^2+2))/6, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 6, 22, 60}, 40] (* Harvey P. Dale, May 01 2013 *)
PROG
(Magma) [n*(n+1)*(n^2+2)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011
(R) a <- vector()
for(n in 1:40) a[n] <- (1/4)*(choose(2*n, 4) + 3*choose(n, 2))
a
# Yosu Yurramendi and María Merino, Aug 21 2013
(PARI) a(n)=n*(n+1)*(n^2+2)/6 \\ Charles R Greathouse IV, Oct 07 2015
(SageMath)
def A071239(n): return binomial(n+1, 2)*(n^2+2)//3
[A071239(n) for n in range(41)] # G. C. Greubel, Aug 06 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 12 2002
STATUS
approved