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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A117560 a(n) = n*(n^2 - 1)/2 - 1.

Original entry on oeis.org

2, 11, 29, 59, 104, 167, 251, 359, 494, 659, 857, 1091, 1364, 1679, 2039, 2447, 2906, 3419, 3989, 4619, 5312, 6071, 6899, 7799, 8774, 9827, 10961, 12179, 13484, 14879, 16367, 17951, 19634, 21419, 23309, 25307, 27416, 29639, 31979, 34439, 37022, 39731, 42569
Offset: 2

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Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 29 2006

Keywords

Comments

a(n-1) is an approximation for the lower bound of the "antimagic constant" of an antimagic square of order n. The antimagic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. By analogy with the magic constant. This approximation follows from the observation that (2*Sum_{k=1..n^2} k) + (m) + (m+1) <= Sum_{k=0..2*n+1} (m + k) where m is the antimagic constant for an antimagic square of order n. a(n) = A027480(n+1) - 1. Stricter bounds seem likely to exist. See A117561 for the upper bounds. Note there exist no antimagic squares of order two or three, but the values are indexed here for completeness.

Examples

			a(3) = 29 because the antimagic constant of an antimagic square of order 4 must be at least 29 (see comments).

Links

Crossrefs

Cf. A027480, A049475, A050257, A117561.

Programs

  • Magma
    [n*(n^2-1)/2 - 1: n in [2..50]]; // Vincenzo Librandi, Jun 20 2011
  • Mathematica
    Table[n*(n^2-1)/2 - 1, {n, 2, 50}]

Formula

a(n) = n*(n^2 - 1)/2 - 1.
G.f.: x^2*(2 + 3*x - 3*x^2 + x^3)/(1-x)^4. - Colin Barker, Mar 29 2012
From Elmo R. Oliveira, Aug 19 2025: (Start)
E.g.f.: 1 + x + (1 + x)*(-2 + 2*x + x^2)*exp(x)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 6. (End)

A117561 a(n) = floor(n*(n^3-n-3)/(2*(n-1))).

Original entry on oeis.org

3, 15, 38, 73, 124, 194, 286, 403, 548, 724, 934, 1181, 1468, 1798, 2174, 2599, 3076, 3608, 4198, 4849, 5564, 6346, 7198, 8123, 9124, 10204, 11366, 12613, 13948, 15374, 16894, 18511, 20228, 22048, 23974, 26009, 28156, 30418, 32798, 35299, 37924
Offset: 2

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 29 2006

Keywords

Comments

a[n-1] is one approximation for the upper bound of the "antimagic constant" of an antimagic square of order n. The antimagic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. By analogy with the magic constant. This approximation follows from the observation that Sum[m + k, {k, 0, 2*n + 1}] <= (2*Sum[k, {k, 1, n^2}]) + (2*m) + (2*m + 1) where m is the antimagic constant for an antimagic square of order n. Stricter bounds seem likely to exist. See A117560 for the lower bounds. Note there exist no antimagic squares of order two or three, but the values are indexed here for completeness.

Examples

			a[3] = 38 because the antimagic constant of an antimagic square of order 4 cannot exceed 38 (see comments)

Links

Crossrefs

Cf. A117560, A050257, A049475.

Programs

  • Mathematica
    Table[Floor[n(n^3-n-3)/(2*(n-1))], {n, 2, 50}]

Formula

a(n) = floor(n*(n^3-n-3)/(2*(n-1))).
G.f.: x^2*(3+3*x-4*x^2-x^3+3*x^4-x^5)/(1-x)^4. - Colin Barker, Mar 29 2012

A364527 Triangle read by rows giving the number of square arrays composed of the numbers from 1 to n^2, counted up to rotation and reflection, with heterogeneity k, i.e., number of k different sums of rows, columns or diagonals with 1 <= k <= 2*n+2 for n > 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 0, 1, 22, 346, 2060, 7989, 17160, 14662, 3120, 880
Offset: 1

Views

Author

Martin Renner, Jul 27 2023

Keywords

Comments

T(n,1) gives the number of magic squares A006052(n).
For n > 1, T(n,2*n+2) gives the number of squares with maximum heterogeneity, i.e., all sums are different (but do not necessarily form a sequence of consecutive integers), sometimes called (super)heterogeneous squares or antimagic squares.
Subsets of T(n,2) or T(n,3) with one or both of the diagonal sums not equal to the magic constant are sometimes called semimagic squares.
Sum_{k=1..2*n+2} T(n,k) = A086829(n) = (n^2)!/8 for n > 1.

Examples

			T(n,k) starts with
  n = 1: 1;
  n = 2: 0, 0, 0, 0, 3, 0;
  n = 3: 1, 22, 346, 2060, 7989, 17160, 14662, 3120;
etc.
For n = 2 there are only three square arrays up to rotation and reflection, all of heterogeneity k = 5, i.e.,
  [1 2] [1 2] [1 3]
  [3 4] [4 3] [4 2]
since there are always the five different sums of rows, columns and diagonals 3, 4, 5, 6 and 7.
For n = 3 the lexicographically first square arrays of heterogeneity 1 <= k <= 8 are
  [2 7 6] [1 2 6] [1 2 5] [1 2 3] [1 2 3] [1 2 3] [1 2 3] [1 2 3]
  [9 5 1] [5 9 4] [3 9 6] [5 6 4] [4 5 6] [4 5 7] [4 5 6] [4 5 8]
  [4 3 8] [3 7 8] [4 7 8] [9 7 8] [7 8 9] [6 9 8] [7 9 8] [6 9 7]
For k = 1 we have the famous Lo Shu square with magic sum (n^3+n)/2 = 15. The other sums for the given examples are (9, 18), (8, 18, 19), (6, 15, 18, 24), (6, 12, 15, 18, 24), (6, 11, 14 16, 18, 23), (6, 12, 14, 15, 16, 17, 24) and (6, 11, 13, 14, 16, 17, 18, 22). Note that there are different sets of sums, namely a total of 6 with two values, 61 with three, 348 with four, 1295 with five, 2880 with six, 3845 with seven and 1538 with eight.

References

  • Pierre Berloquin, Garten der Sphinx. 150 mathematische Denkspiele, München 1984, p. 20, nr. 15 (Heterogene Quadrate), p. 20, nr. 16 (Antimagie), p. 86, nr. 148 (Höhere Antimagie), pp. 99-100, 178 (Solutions).

Links

Crossrefs

Cf. A006003, A006052, A050257, A086829, A088020.
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