A067725 -id:A067725 - cp's OEIS frontend

A302576 Numbers k such that k/10 + 1 is a square.

-10, 0, 30, 80, 150, 240, 350, 480, 630, 800, 990, 1200, 1430, 1680, 1950, 2240, 2550, 2880, 3230, 3600, 3990, 4400, 4830, 5280, 5750, 6240, 6750, 7280, 7830, 8400, 8990, 9600, 10230, 10880, 11550, 12240, 12950, 13680, 14430, 15200, 15990, 16800, 17630, 18480, 19350, 20240

Offset: 1

Bruno Berselli, Apr 10 2018

Equivalently, numbers k such that (k + 10)*10 is a square.

The positive terms belong to the fourth column of the array in A185781.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008592, A033583, A185781.
After -10, subsequence of A174133 because a(n) = ((n-1)^2-1)*(3^2+1).
Similar lists of k for which k/j + 1 is a square: A067998 (j=1), A054000 (j=2), A067725 (j=3), A134582 (j=4), A067724 (j=5), A067726 (j=6), A067727 (j=7), second bisection of A067728 (j=8), A147651 (j=9), this sequence (j=10), A067705 (j=11), second bisection of A067707 (j=12).

GAP
```
List([1..50], n -> 10*n*(n-2));
```
Julia
```
[10*n*(n-2) for n in 1:50] |> println
```
Magma
```
[10*n*(n-2): n in [1..50]];
```
Mathematica
```
Table[10 n (n - 2), {n, 1, 50}]
```
Maxima
```
makelist(10*n*(n-2), n, 1, 50);
```
PARI
```
vector(50, n, nn; 10*n*(n-2))
```
Python
```
[10*n*(n-2) for n in range(1, 50)]
```
Sage
```
[10*n*(n-2) for n in (1..50)]
```

O.g.f.: -10*x*(1 - 3*x)/(1 - x)^3.
E.g.f.: -10*x*(1 - x)*exp(x).
a(n) = a(2-n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 10*n*(n - 2) = 10*A067998(n).
a(n) = A033583(n-1) - 10. - Altug Alkan, Apr 10 2018

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 5, 4, 0, 0, 9, 12, 6, 0, 0, 14, 24, 21, 8, 0, 0, 20, 40, 45, 32, 10, 0, 0, 27, 60, 78, 72, 45, 12, 0, 0, 35, 84, 120, 128, 105, 60, 14, 0, 0, 44, 112, 171, 200, 190, 144, 77, 16, 0, 0, 54, 144, 231, 288, 300, 264, 189, 96, 18, 0

Offset: 0

Peter Luschny, Mar 22 2023

A detailed combinatorial interpretation can be found in A361682.

			[0] 0,  0,  0,   0,   0,   0,   0,    0, ...  A000004
[1] 0,  2,  5,   9,  14,  20,  27,   35, ...  A000096
[2] 0,  4, 12,  24,  40,  60,  84,  112, ...  A046092
[3] 0,  6, 21,  45,  78, 120, 171,  231, ...  A081266
[4] 0,  8, 32,  72, 128, 200, 288,  392, ...  A139098
[5] 0, 10, 45, 105, 190, 300, 435,  595, ...
[6] 0, 12, 60, 144, 264, 420, 612,  840, ...  A153792
[7] 0, 14, 77, 189, 350, 560, 819, 1127, ...
       | A028347 |     A163761
     A005843  A067725
.
[0] 0;
[1] 0,  0;
[2] 0,  2,   0;
[3] 0,  5,   4,   0;
[4] 0,  9,  12,   6,   0;
[5] 0, 14,  24,  21,   8,   0;
[6] 0, 20,  40,  45,  32,  10,   0;
[7] 0, 27,  60,  78,  72,  45,  12,  0;
[8] 0, 35,  84, 120, 128, 105,  60, 14,  0;
[9] 0, 44, 112, 171, 200, 190, 144, 77, 16, 0;

Rows: A000004, A000096, A046092, A081266, A139098, A153792.
Columns: A000004, A005843, A028347, A067725, A163761, A068380.
Cf. A361682.

A := (n, k) -> n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;

A(n, k) = n*k*(4 + n*(k - 1))/2.
T(n, k) = k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = A361682(n, k) - 1.

A384616 A(m,n) is the maximum sum of absolute differences of the labels of adjacent vertices of the grid graph P_m X P_n where the mn labels are exactly 1, 2, ..., mn.

Original entry on oeis.org

0, 1, 8, 3, 23, 58, 7, 44, 115

Offset: 1

Sela Fried, Jun 07 2025

A(m, n) ~ Theta((m*n)^2) (see link).

			Array begins (values in parentheses are conjectural):
  [1]  0
  [2]  1    8
  [3]  3   23    58
  [4]  7   44   115   (216)
  [5] 11   71  (182)  (347)  (554)
  [6] 17  104  (271)  (508)  (815) (1192)
  [7] 23  143  (370)  (699) (1118) (1639) (2250)
  [8] 31 (188) (491)  (920) (1475) (2156) (2963) (3896)
  [9] 39 (239) (622) (1171) (1874) (2743) (3766) (4955) (6298)

Sela Fried, On the maximal sum of absolute differences of certain labelings of the grid graph, 2025.

Column 1 is A047838.

Cf. A067725.

import itertools
import numpy as np
def max_difference_sum(m, n):
    nums = list(range(1, m * n + 1))
    max_sum = 0
    best_matrix = None
    for perm in itertools.permutations(nums):
        matrix = np.array(perm).reshape((m, n))
        diff_sum = np.sum(np.abs(matrix[:,1:]-matrix[:,:-1])) + np.sum(np.abs(matrix[1:,:]-matrix[:-1,:]))
        if diff_sum > max_sum:
            max_sum = diff_sum
            best_matrix = matrix.copy()
    return max_sum, best_matrix
for m in range(1, 10):
    for n in range(1, m+1):
        max_sum, best = max_difference_sum(m, n)
        print(max_sum, end=', ')

Conjecture: A(m,2) = A067725(m-1) - 1.

cp's OEIS Frontend

A302576 Numbers k such that k/10 + 1 is a square.

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PARI

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Formula

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

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Maple

Formula

A384616 A(m,n) is the maximum sum of absolute differences of the labels of adjacent vertices of the grid graph P_m X P_n where the mn labels are exactly 1, 2, ..., mn.

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Python

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cp's OEIS Frontend

A302576 Numbers k such that k/10 + 1 is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Julia

Magma

Mathematica

Maxima

PARI

Python

Sage

Formula

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A384616 A(m,n) is the maximum sum of absolute differences of the labels of adjacent vertices of the grid graph P_m X P_n where the m*n labels are exactly 1, 2, ..., m*n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A384616 A(m,n) is the maximum sum of absolute differences of the labels of adjacent vertices of the grid graph P_m X P_n where the mn labels are exactly 1, 2, ..., mn.