A325516 -id:A325516 - cp's OEIS frontend

A323724 a(n) = n(2(n - 2)*n + (-1)^n + 3)/4.

0, 0, 2, 6, 20, 40, 78, 126, 200, 288, 410, 550, 732, 936, 1190, 1470, 1808, 2176, 2610, 3078, 3620, 4200, 4862, 5566, 6360, 7200, 8138, 9126, 10220, 11368, 12630, 13950, 15392, 16896, 18530, 20230, 22068, 23976, 26030, 28158, 30440, 32800, 35322, 37926, 40700

Offset: 0

Stefano Spezia, Jan 25 2019

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

All the terms of this sequence are even numbers (A005843), but do not end with 4.

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. A000290, A000982, A004526, A005843, A005997, A317614, A322277, A323723, A325516.

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

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

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

a[n_] := 1/2 (-1 + n)^2 n - (-1 + n) Floor[n/2] + 2 Floor[n/2]^2; Array[a, 50, 0];
Table[n (2 (n - 2) n + (-1)^n + 3)/4, {n, 0, 50}] (* Bruno Berselli, Feb 06 2019 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,0,2,6,20,40},50] (* Harvey P. Dale, Jan 13 2024 *)

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

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

T(i,j,n) = if (i %2, j + n*(i-1), n*i - j + 1);
a(n) = sum(k=1, n-1, T(k,k+1,n)); \\ Michel Marcus, Feb 06 2019

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

O.g.f.: 2*x^2*(1 + x + 3*x^2 + x^3)/((1 - x)^4*(1 + x)^2).
E.g.f.: (1/2)*x*(exp(x)*x*(1 + x) + sinh(x)).
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) = (1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*floor(n/2)^2.
a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*A004526(n) + 2*A000290(A004526(n)).
a(n) = (n/2)*((n - 1)^2 + 1) for even n; a(n) = (n/2)*(n - 1)^2 otherwise. - Bruno Berselli, Feb 06 2019
a(n) = 2*A004526(n*A000982(n-1)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021
a(n) = 2*A005997(n-1) for n >= 2. - Hugo Pfoertner, Dec 13 2021

Definition by Bruno Berselli, Feb 06 2019

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

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...

A325517 a(n) = n((2n + 1)(2n^2 + 2n + 3) - 3(-1)^n)/24.

Original entry on oeis.org

0, 1, 6, 24, 64, 145, 282, 504, 832, 1305, 1950, 2816, 3936, 5369, 7154, 9360, 12032, 15249, 19062, 23560, 28800, 34881, 41866, 49864, 58944, 69225, 80782, 93744, 108192, 124265, 142050, 161696, 183296, 207009, 232934, 261240, 292032, 325489, 361722, 400920, 443200

Offset: 0

Stefano Spezia, May 07 2019

For n > 0, a(n) is the n-th row sum of the triangle A325516.

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. A131941, A317614, A322277, A325516.

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

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

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

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

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

O.g.f.: x*(1 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/((1 - x)^5*(1 + x)^2).
E.g.f.: (1/24)*exp(-x)*x*(3 + 21*exp(2*x) + 54*exp(2*x)*x + 30*exp(2*x)*x^2 + 4*exp(2*x)*x^3).
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) = n^2*(2*n^2 + 3*n + 4)/12 if n is even.
a(n) = n*(n + 1)*(2*n^2 + n + 3)/12 if n is odd.
a(n) = n*A131941(n). - Stefano Spezia, Dec 21 2021

cp's OEIS Frontend

A323724 a(n) = n*(2*(n - 2)*n + (-1)^n + 3)/4.

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A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

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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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A325517 a(n) = n*((2*n + 1)*(2*n^2 + 2*n + 3) - 3*(-1)^n)/24.

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GAP

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Formula

A323724 a(n) = n(2(n - 2)*n + (-1)^n + 3)/4.

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

A325517 a(n) = n((2n + 1)(2n^2 + 2n + 3) - 3(-1)^n)/24.