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A071239 a(n) = n*(n+1)*(n^2+2)/6. 6
0, 1, 6, 22, 60, 135, 266, 476, 792, 1245, 1870, 2706, 3796, 5187, 6930, 9080, 11696, 14841, 18582, 22990, 28140, 34111, 40986, 48852, 57800, 67925, 79326, 92106, 106372, 122235, 139810, 159216, 180576, 204017, 229670, 257670, 288156, 321271, 357162, 395980 (list; graph; refs; listen; history; text; internal format)
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, María Merino, Aug 21 2013

G.f. -x*(1+x+2*x^2) / (x-1)^5 . - R. J. Mathar, Aug 21 2013

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, María Merino, Aug 21 2013

(PARI) a(n)=n*(n+1)*(n^2+2)/6 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A080851, A226048.

Sequence in context: A127760 A320243 A066188 * A105450 A011888 A081282

Adjacent sequences:  A071236 A071237 A071238 * A071240 A071241 A071242

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jun 12 2002

STATUS

approved

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Last modified July 24 06:56 EDT 2021. Contains 346273 sequences. (Running on oeis4.)