A325655 -id:A325655 - cp's OEIS frontend

A323723 a(n) = (-2 - (-1)^n(-2 + n) + n + 2n^3)/4.

0, 0, 4, 14, 32, 64, 108, 174, 256, 368, 500, 670, 864, 1104, 1372, 1694, 2048, 2464, 2916, 3438, 4000, 4640, 5324, 6094, 6912, 7824, 8788, 9854, 10976, 12208, 13500, 14910, 16384, 17984, 19652, 21454, 23328, 25344, 27436, 29678, 32000, 34480, 37044, 39774

Offset: 0

Stefano Spezia, Jan 25 2019

For n > 1, a(n) is the subdiagonal sum of the matrix M(n) whose permanent is A322277(n).

All the terms of this sequence are even numbers (A005843).

Stefano Spezia, Table of n, a(n) for n = 0..10000
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000982, A004526, A005843, A033431, A033999, A317614, A322277, A323724, A325655.

Flat(List([0..50], n -> (-2-(-1)^n*(-2+n)+n+2*n^3)/4));

[(-2-(-1)^n*(-2+n)+n+2*n^3)/4: n in [0..50]];

a:=n->(-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4: seq(a(n), n=0..50);

a[n_]:=(6 + n + n^3 + 12 Floor[1/2 (-3 + n)] + 4 Floor[1/2 (-3 + n)]^2 - 2 (1 + n) Floor[1/2 (-1 + n)])/2; Array[a,50,0]

makelist((-2-(-1)^n*(-2+n)+n+2*n^3)/4, n, 0, 50);

PARI
```
a(n) = (-2-(-1)^n*(-2+n)+n+2*n^3)/4;
```

[(-2-(-1)**n*(-2+n)+n+2*n**3)/4 for n in range(50)]

O.g.f.: 2*x^2*(2 + 3*x + x^3)/((1 - x)^4*(1 + x)^2).
E.g.f.: (1/4)*exp(-x)*(2 + x)*(1 + exp(2*x)*(-1 + 2*x + 2* x^2)).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5.
a(n) = (6 + n + n^3 + 12*floor((n - 3)/2) + 4*floor((n - 3)/2)^2 - 2*(1 + n)*floor((n - 1)/2))/2.
a(n) = (-2 - A033999(n)*(-2 + n) + n + A033431(n))/4.
a(n) = n^3/2 for even n; a(n) = (n - 1)*(n^2 + n + 2)/2 otherwise. - Bruno Berselli, Feb 06 2019
a(n) = 2*A004526(n*A000982(n)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021

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

Original entry on oeis.org

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

Stefano Spezia, Dec 28 2018

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)

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).

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).

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

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

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

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 *)

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

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

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

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

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...

A325656 a(n) = (1/24)n((4n + 3)(2n^2 + 1) - 3(-1)^n).

Original entry on oeis.org

0, 1, 8, 36, 104, 245, 492, 896, 1504, 2385, 3600, 5236, 7368, 10101, 13524, 17760, 22912, 29121, 36504, 45220, 55400, 67221, 80828, 96416, 114144, 134225, 156832, 182196, 210504, 242005, 276900, 315456, 357888, 404481, 455464, 511140, 571752, 637621, 709004, 786240

Offset: 0

Stefano Spezia, May 13 2019

For n > 0, a(n) is the n-th row sum of the triangle A325655.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A173722, A317614, A322277, A325655.

Flat(List([0..50], n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3)));

[(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): n in [0..50]];

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..50);

a[n_]:=(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3); Array[a,50,0]

a(n) = (1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3);

O.g.f.: -x*(1 + 5*x + 13*x*2 + 9*x^3 + 4*x^4)/((-1 + x)^5*(1 + x)^2).
E.g.f.: (1/24)*exp(-x)*x*(3 + 21*exp(2*x) + 78*exp(2*x)*x + 54*exp(2*x)*x^2 + 8*exp(2*x)*x*3).
a(n) = a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7) for n > 6.
a(n) = (1/12)*n^2*(4*n^2 + 3*n + 2) if n is even.
a(n) = (1/12)*n*(n + 1)*(4*n^2 - n + 3) if n is odd.
a(n) = n*A173722(2*n). - Stefano Spezia, Dec 21 2021

A325657 a(n) = (1/2)(-1 + (-1)^n)(n-1) + n^2.

Original entry on oeis.org

0, 1, 4, 7, 16, 21, 36, 43, 64, 73, 100, 111, 144, 157, 196, 211, 256, 273, 324, 343, 400, 421, 484, 507, 576, 601, 676, 703, 784, 813, 900, 931, 1024, 1057, 1156, 1191, 1296, 1333, 1444, 1483, 1600, 1641, 1764, 1807, 1936, 1981, 2116, 2163, 2304, 2353, 2500, 2551

Offset: 0

Stefano Spezia, May 13 2019

For n > 0, a(n) is the n-th element of the diagonal of the triangle A325655. Equivalently, a(n) is the element M_{n,1} of the matrix M(n) whose permanent is A322277(n).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A054569, A317614, A322277, A325655.

Flat(List([0..55], n->(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2));

[(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2: n in [0..55]];

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..55);

Table[(1/2)*(- 1+(-1)^n)*(n-1)+n^2,{n,0,55}]

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

O.g.f.: (-1 - 3*x - x^2 - 3*x^3)/((-1 + x)^3*(1+x)^2).
E.g.f.: (1/2)*exp(-x)*(-1 - x + exp(2*x)*(1 + x + 2*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4
a(n) = n^2 if n is even.
a(n) = n^2 - n + 1 if n is odd.

cp's OEIS Frontend

A323723 a(n) = (-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4.

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

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Comments

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Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Formula

A325656 a(n) = (1/24)*n*((4*n + 3)*(2*n^2 + 1) - 3*(-1)^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A325657 a(n) = (1/2)*(-1 + (-1)^n)*(n-1) + n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A323723 a(n) = (-2 - (-1)^n(-2 + n) + n + 2n^3)/4.

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

A325656 a(n) = (1/24)n((4n + 3)(2n^2 + 1) - 3(-1)^n).

A325657 a(n) = (1/2)(-1 + (-1)^n)(n-1) + n^2.