A070893 Let r, s, t be three permutations of the set {1,2,3,..,n}; a(n) = value of Sum_{i=1..n} r(i)*s(i)*t(i), with r={1,2,3,..,n}; s={n,n-1,..,1} and t={n,n-2,n-4,...,1,...,n-3,n-1}.

1, 6, 19, 46, 94, 172, 290, 460, 695, 1010, 1421, 1946, 2604, 3416, 4404, 5592, 7005, 8670, 10615, 12870, 15466, 18436, 21814, 25636, 29939, 34762, 40145, 46130, 52760, 60080, 68136, 76976, 86649, 97206, 108699, 121182, 134710, 149340

Offset: 1

Author

Wouter Meeussen, May 22 2002

Keywords

• nonn
• easy

Comments

See A070735 for the minimal values for these products. This sequence is an upper bound. The third permutation 't'= ceiling(abs(range(n-1/2,-n,-2))) is such that it associates its smallest factor with the largest factor of the product 'r'*'s'.

We observe that is the transform of A002717 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of v is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010

Examples

			{1,2,3,4,5,6,7}*{7,6,5,4,3,2,1}*{7,5,3,1,2,4,6} gives {49,60,45,16,30,48,42}, with sum 290, so a(7)=290.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Crossrefs

Cf. A070735, A082289. a(n)=A082290(2n-2).

Cf. A000034, A032766, A006578, A002717. - Richard Choulet, Jan 28 2010

Cf. A002717 (first differences). - Bruno Berselli, Aug 26 2011

Column k=3 of A166278. - Alois P. Heinz, Nov 02 2012

Programs

Magma

[(1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3): n in [1..40]]; // Vincenzo Librandi, Aug 26 2011

Mathematica

Table[Plus@@(Range[n]*Range[n, 1, -1]*Ceiling[Abs[Range[n-1/2, -n, -2]]]), {n, 49}];
(* or *)
CoefficientList[Series[ -(1+2x)/(-1+x)^5/(1+x), {x, 0, 48}], x]//Flatten

PARI

a(n)=sum(i=1,n,i*(n+1-i)*ceil(abs(n+3/2-2*i)))

PARI

a(n)=polcoeff(if(n<0,x^4*(2+x)/((1+x)*(1-x)^5),x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)),abs(n))

Formula

G.f.: x*(1+2*x)/((1+x)*(1-x)^5). - Michael Somos, Apr 07 2003

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 0 by this equation, then a(n)=0 for -3 <= n <= 0 and a(n)=A082289(-n) for n <= -4. - Michael Somos, Apr 07 2003

a(n) = (1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3). a(n) - a(n-2) = A002411(n). - Bruno Berselli, Aug 26 2011