A071239 - OEIS

%I #62 Aug 07 2024 02:26:14

%S 0,1,6,22,60,135,266,476,792,1245,1870,2706,3796,5187,6930,9080,11696,

%T 14841,18582,22990,28140,34111,40986,48852,57800,67925,79326,92106,

%U 106372,122235,139810,159216,180576,204017,229670,257670,288156,321271,357162,395980

%N a(n) = n*(n+1)*(n^2 + 2)/6.

%C 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

%C 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

%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

%H Vincenzo Librandi, <a href="/A071239/b071239.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F 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

%F 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

%F G.f.: x*(1+x+2*x^2) / (1-x)^5 . - _R. J. Mathar_, Aug 21 2013

%F E.g.f.: (1/6)*x*(6 + 12*x + 7*x^2 + x^3)*exp(x). - _G. C. Greubel_, Aug 06 2024

%t 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 *)

%o (Magma) [n*(n+1)*(n^2+2)/6: n in [0..40]]; // _Vincenzo Librandi_, Jun 14 2011

%o (R) a <- vector()

%o     for(n in 1:40) a[n] <- (1/4)*(choose(2*n,4) + 3*choose(n,2))

%o     a

%o # _Yosu Yurramendi_ and María Merino, Aug 21 2013

%o (PARI) a(n)=n*(n+1)*(n^2+2)/6 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (SageMath)

%o def A071239(n): return binomial(n+1,2)*(n^2+2)//3

%o [A071239(n) for n in range(41)] # _G. C. Greubel_, Aug 06 2024

%Y Cf. A080851, A226048.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Jun 12 2002