A322844 a(n) = (1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2)).

0, 0, 5, 6, 68, 50, 333, 196, 1040, 540, 2525, 1210, 5220, 2366, 9653, 4200, 16448, 6936, 26325, 10830, 40100, 16170, 58685, 23276, 83088, 32500, 114413, 44226, 153860, 58870, 202725, 76880, 262400, 98736, 334373, 124950, 420228, 156066, 521645, 192660, 640400, 235340

Offset: 0

Author

Stefano Spezia, Dec 28 2018

Keywords

• nonn
• easy
• changed

Comments

Conjectures: (Start)

For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.

For k > 2, the trace of the k-th exterior power of the matrix M(n) is equal to zero.

(End)

Links

• Stefano Spezia, Table of n, a(n) for n = 0..10000
• Wikipedia, Characteristic polynomial
• Wikipedia, Exterior algebra
• Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Crossrefs

Cf. A317614 (trace of matrix M(n)).

Cf. A002415, A037270, A074147 (antidiagonals of M matrices), A241016 (row sums of M matrices), A317617 (column sums of M matrices), A322277 (permanent of matrix M(n)), A323723 (subdiagonal sum of M matrices), A323724 (superdiagonal sum of M matrices), A325516 (k-superdiagonal sum of M matrices), A325655 (k-subdiagonal sum of M matrices).

Programs

GAP

Flat(List([0..50], n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2))));

Magma

[IsEven(n) select (1/4)*n^2*(1 + n^2) else (1/12)*(- 1 + n)*n^2*(1 + n): n in [0..50]];

Maple

a:=n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*modp(n,2)): seq(a(n), n=0..50);

Mathematica

a[n_]:=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*Mod[n,2]); Array[a,50,0]
LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{0,0,5,6,68,50,333,196,1040,540},50] (* Harvey P. Dale, Aug 23 2025 *)

Maxima

a(n):=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*mod(n,2))$ makelist(a(n), n, 0, 50);

PARI

a(n) = (1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n % 2));

PARI

a(n) = abs(polcoeff(charpoly(matrix(n, n, i, j, if (i %2, j + n*(i-1), n*i - j + 1))), n-2)); \\ Michel Marcus, Feb 06 2019

Python

[int(n**2*(3*(1 + n**2) - 2*(2 + n**2)*pow(n, 1, 2))/12) for n in range(0,50)]

Formula

O.g.f.: -x^2*(5 + 6*x + 43*x^2 + 20*x^3 + 43*x^4 + 6*x^5 + 5*x^6)/((-1 + x)^5*(1 + x)^5).

E.g.f.: (1/(12*x^2))*exp(-x)*(24 - 60*exp(x) + 21*x + 9*x^2 + 2*x^3 + x^4 + exp(2*x)*(36 - 33*x + 15*x^2 - 4*x^3 + 2*x^4)).

a(n) = (1/4)*n^2*(1 + n^2) for n even.

a(n) = (1/2)*A037270(n) for n even.

a(n) = (1/12)*(-1 + n)*n^2*(1 + n) for n odd.

a(n) = A002415(n) for n odd.

a(2*n+1) = 5*a(2*n-1) - 10*a(2*n-3) + 10*a(2*n-5) - 5*a(2*n-7) + a(2*n-9), for n > 4.

a(2*n) = 5*a(2*n-2) - 10*a(2*n-4) + 10*a(2*n-6) - 5*a(2*n-8) + a(2*n-10), for n > 4.

O.g.f. for a(2*n+1): -x*(2*(3 + 10*x + 3*x^2))/(-1 + x)^5.

O.g.f. for a(2*n): x*(-5 - 43*x - 43*x^2 - 5*x^3)/(-1 + x)^5.

E.g.f. for a(2*n+1): (1/12)*(6*x*cosh(sqrt(x)) + sqrt(x)*(6 + x)*sinh(sqrt(x))).

E.g.f. for a(2*n): (1/4)*(x*(8 + x)*cosh(sqrt(x)) + 2*sqrt(x)*(1 + 3*x)*sinh(sqrt(x))).

Sum_{k>=1} 1/a(2*k) = (1/6)*(12 + Pi^2 - 6*Pi*coth(Pi/2)) = 0.21955691692893092525407699347398665248691900...

Sum_{k>=1} 1/a(2*k+1) = 3*(5 - Pi^2/2) = 0.1955933983659620717482635001857732970...

Sum_{k>=2} 1/a(k) = 17 - (4*Pi^2)/3 - Pi*coth(Pi/2) = 0.415150315294892997002340493659759949516369894...