A074147 -id:A074147 - cp's OEIS frontend

A340804 Triangle read by rows: T(n, k) = 1 + k(n - 1) + (2k - n - 1)*(k mod 2) with 0 < k <= n.

1, 1, 3, 1, 5, 9, 1, 7, 11, 13, 1, 9, 13, 17, 25, 1, 11, 15, 21, 29, 31, 1, 13, 17, 25, 33, 37, 49, 1, 15, 19, 29, 37, 43, 55, 57, 1, 17, 21, 33, 41, 49, 61, 65, 81, 1, 19, 23, 37, 45, 55, 67, 73, 89, 91, 1, 21, 25, 41, 49, 61, 73, 81, 97, 101, 121, 1, 23, 27, 45, 53, 67, 79, 89, 105, 111, 131, 133

Offset: 1

Stefano Spezia, Jan 22 2021

T(n, k) is the k-th diagonal element of an n X n square matrix M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern.

It includes exclusively all the odd numbers (A005408). Except the term 1, all the other odd numbers appear a finite number of times.

			1
1,  3
1,  5,  9,
1,  7, 11, 13
1,  9, 13, 17, 25
1, 11, 15, 21, 29, 31
1, 13, 17, 25, 33, 37, 49
...

Stefano Spezia, Table of n, a(n) for n = 1..11325 (first 150 rows of the triangle, flattened).

Cf. A005408, A317614 (row sums).

Cf. A000012 (1st column), A006010 (sum of the first n rows), A060747 (2nd column), A074147 (antidiagonals of M matrices), A241016 (row sums of M matrices), A317617 (column sums of M matrices), A322277 (permanent of M matrices), A323723 (subdiagonal sum of M matrices), A323724 (superdiagonal sum of M matrices).

Table[1+k(n-1)+(2k-n-1)Mod[k,2],{n,12},{k,n}]//Flatten

T(n, k) = 1 + k*(n - 1) + (2*k - n - 1)*(k % 2); \\ Michel Marcus, Jan 25 2021

O.g.f.: (1 + y - 3*y^2 + y^3 + x*(-1 - y + 5*y^2 + y^3))/((-1 + x)^2*(-1 + y)^2*(1+y)^2).

E.g.f.: exp(x - y)*(1 + x + 2*y + exp(2*y)*(1 + x*(-1 + 2*y)))/2.

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...

A109857 Next 2n - 1 odd numbers in decreasing order followed by next 2n even numbers in decreasing order.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 12, 10, 8, 6, 17, 15, 13, 11, 9, 24, 22, 20, 18, 16, 14, 31, 29, 27, 25, 23, 21, 19, 40, 38, 36, 34, 32, 30, 28, 26, 49, 47, 45, 43, 41, 39, 37, 35, 33, 60, 58, 56, 54, 52, 50, 48, 46, 44, 42, 71, 69, 67, 65, 63, 61, 59, 57, 55, 53, 51, 84, 82, 80, 78, 76, 74

Offset: 1

Amarnath Murthy, Jul 08 2005

This sequence is a permutation of the positive integers. - Werner Schulte, Jul 29 2023

			 1;
 4,  2;
 7,  5,  3;
12, 10,  8,  6;
17, 15, 13, 11,  9;
24, 22, 20, 18, 16, 14;
31, 29, 27, 25, 23, 21, 19;
40, 38, 36, 34, 32, 30, 28, 26;

Cf. A074147 (row reversed), A074149 (row sums), A074148 (column 1), A001844, A061925 (main diagonal).

T(n,k)=n*(n+1)/2+floor(n/2)-2*(k-1) \\ Werner Schulte, Jul 29 2023

From Werner Schulte, Jul 29 2023: (Start)
T(n, k) = n*(n+1)/2 + floor(n/2) - 2*(k-1)  for 1 <= k <= n.
T(n, n) = (n^2-3*n+4)/2 + floor(n/2)  for n > 0.
T(2*n-1, n) = n^2 + (n-1)^2 = A001844(n-1)  for n > 0. (End)

More terms from Joshua Zucker, May 05 2006

A376468 Triangle T read by rows: T(n, k) = (n^2 - 2n + 3 - (-1)^n + n^2 mod 8) / 2 + 4k.

Original entry on oeis.org

1, 2, 6, 3, 7, 11, 4, 8, 12, 16, 5, 9, 13, 17, 21, 10, 14, 18, 22, 26, 30, 15, 19, 23, 27, 31, 35, 39, 20, 24, 28, 32, 36, 40, 44, 48, 25, 29, 33, 37, 41, 45, 49, 53, 57, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96

Offset: 0

Werner Schulte, Sep 23 2024

This triangle seen as a sequence yields a permutation of the natural numbers. For similar triangles see A000027 (seen as a triangle), A074147, and A367844 (row reversed).

			Triangle T(n, k) for 0 <= k <= n starts:
n \k :   0   1   2   3   4   5   6   7   8   9  10  11
======================================================
   0 :   1
   1 :   2   6
   2 :   3   7  11
   3 :   4   8  12  16
   4 :   5   9  13  17  21
   5 :  10  14  18  22  26  30
   6 :  15  19  23  27  31  35  39
   7 :  20  24  28  32  36  40  44  48
   8 :  25  29  33  37  41  45  49  53  57
   9 :  34  38  42  46  50  54  58  62  66  70
  10 :  43  47  51  55  59  63  67  71  75  79  83
  11 :  52  56  60  64  68  72  76  80  84  88  92  96
  etc.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Cf. A001844

Table[Range[#, #+n*4, 4] & [(Mod[n^2, 8] + n*(n-2) - (-1)^n + 3)/2], {n, 0, 15}] (* Paolo Xausa, Nov 13 2024 *)

PARI
```
T(n,k)=(n^2-2*n+3-(-1)^n+n^2%8)/2+4*k
```

from math import comb, isqrt
def A376468(n): return ((a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(a-2)+3+(1 if a&1 else -1)+(a**2&7)>>1)+(n-comb(a+1,2)<<2) # Chai Wah Wu, Nov 12 2024

T(n, k) = T(n, k-1) + 4.
T(n+4, 0) = T(n, n) + 4 for n > 3.
T(2*n, n) = 2 * (n^2 + n + 1) - (-1)^n = A001844(n) + 1 - (-1)^n.

cp's OEIS Frontend

A340804 Triangle read by rows: T(n, k) = 1 + k*(n - 1) + (2*k - n - 1)*(k mod 2) 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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A109857 Next 2*n - 1 odd numbers in decreasing order followed by next 2*n even numbers in decreasing order.

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A376468 Triangle T read by rows: T(n, k) = (n^2 - 2*n + 3 - (-1)^n + n^2 mod 8) / 2 + 4*k.

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A340804 Triangle read by rows: T(n, k) = 1 + k(n - 1) + (2k - n - 1)*(k mod 2) with 0 < k <= n.

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

A109857 Next 2n - 1 odd numbers in decreasing order followed by next 2n even numbers in decreasing order.

A376468 Triangle T read by rows: T(n, k) = (n^2 - 2n + 3 - (-1)^n + n^2 mod 8) / 2 + 4k.