A351831 -id:A351831 - cp's OEIS frontend

A351830 Distance from the n-th square pyramidal number (sum of the first n positive squares) to the nearest square.

0, 0, 1, 2, 5, 6, 9, 4, 8, 4, 15, 22, 25, 22, 9, 15, 25, 21, 7, 30, 46, 53, 49, 32, 0, 49, 40, 41, 30, 91, 46, 12, 9, 15, 4, 26, 77, 114, 25, 91, 61, 105, 15, 122, 129, 66, 22, 1, 1, 24, 76, 157, 170, 37, 131, 141, 91, 139, 165, 15, 174, 247, 150, 80, 39, 29

Offset: 0

Paolo Xausa, Feb 21 2022

As noted by Conway and Sloane (1999), the only zero terms appear at n = 0, n = 1 and n = 24, and the n = 24 case allows for the Lorentzian construction of the Leech lattice through the A351831 vector.

The zero terms are equivalently the subject of the "pile of cannonballs" problem posed by Lucas and solved by Watson. - Peter Munn, Aug 03 2023

			a(4) = 5 because the sum of the first 4 positive squares is 1 + 4 + 9 + 16 = 30, the nearest square is 25 and 30 - 25 = 5. - _Paolo Xausa_, Jul 05 2022

W. Ljunggren, New solution of a problem proposed by E. Lucas, Norsk Mat. Tidsskr. 34 (1952), pp 65-72.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, entry 24, p 101.

Paolo Xausa, Table of n, a(n) for n = 0..9999
Michael A. Bennett, Lucas' Square Pyramid Problem Revisited.
Richard E. Borcherds, How to construct the Leech lattice, YouTube video, 2022.
J. H. Conway and N. J. A. Sloane, Lorentzian forms for the Leech lattice, Bulletin (New Series) of the American Mathematical Society, Volume 6, Number 2, March 1982.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer, New York, NY, 1999, pp. 524-528.
Anji Dong, Katerina Saettone, Kendra Song, and Alexandru Zaharescu, An Equidistribution Result for Differences Associated to Square Pyramidal Numbers II, arXiv:2505.04166 [math.NT], 2025.
E. Lucas, Problem 1180, Nouvelles Ann. Math. (2) 14 (1875), p 336.
G. N. Watson, The problem of the square pyramid, Messenger of Mathematics 48 (1918), pp. 1-22.
Wikipedia, Leech lattice.

Cf. A000290, A000330, A053188, A351831, A353295, A354330, A363284.

nterms=66;Array[Abs[(s=#(#+1)(2#+1)/6)-Round[Sqrt[s]]^2]&,nterms,0]

from math import isqrt
def a(n):
    t = n*(n+1)*(2*n+1)//6
    r = isqrt(t)
    return min(t - r**2, (r+1)**2 - t)
print([a(n) for n in range(66)]) # Michael S. Branicky, Feb 21 2022

From Paolo Xausa, Jul 05 2022: (Start)
a(n) = A053188(A000330(n)).
a(n) = abs(A000330(n) - A353295(n)). (End)

Name edited by Peter Munn, Aug 04 2023

A353294 A generator matrix for the Leech lattice, multiplied by sqrt(8), read by rows.

Original entry on oeis.org

8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Paolo Xausa, Apr 12 2022

There are infinitely many such matrices, this just happens to be a concrete example that we gave in the Sphere-Packing book. It is not unique in any way. - N. J. A. Sloane, Jun 04 2022

			As depicted by Conway and Sloane (1999), p. 133, the full 24 X 24 matrix is given below, in standard MOG (Miracle Octad Generator) coordinates.
.
   8 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 4 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 4 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 4 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 4 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 0 4 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   2 2 2 2 | 2 2 2 2 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 0 0 0 0 | 4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 0 0 0 | 0 4 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 0 0 0 | 0 0 4 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   2 2 2 2 | 0 0 0 0 | 2 2 2 2 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 4 0 0 0 | 0 0 0 0 | 0 0 0 0
   2 2 0 0 | 2 2 0 0 | 2 2 0 0 | 2 2 0 0 | 0 0 0 0 | 0 0 0 0
   2 0 2 0 | 2 0 2 0 | 2 0 2 0 | 2 0 2 0 | 0 0 0 0 | 0 0 0 0
   2 0 0 2 | 2 0 0 2 | 2 0 0 2 | 2 0 0 2 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 4 0 0 0 | 0 0 0 0
   2 0 2 0 | 2 0 0 2 | 2 2 0 0 | 0 0 0 0 | 2 2 0 0 | 0 0 0 0
   2 0 0 2 | 2 2 0 0 | 2 0 2 0 | 0 0 0 0 | 2 0 2 0 | 0 0 0 0
   2 2 0 0 | 2 0 2 0 | 2 0 0 2 | 0 0 0 0 | 2 0 0 2 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   0 2 2 2 | 2 0 0 0 | 2 0 0 0 | 2 0 0 0 | 2 0 0 0 | 2 0 0 0
   0 0 0 0 | 0 0 0 0 | 2 2 0 0 | 2 2 0 0 | 2 2 0 0 | 2 2 0 0
   0 0 0 0 | 0 0 0 0 | 2 0 2 0 | 2 0 2 0 | 2 0 2 0 | 2 0 2 0
  -3 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1

Paolo Xausa, Table of n, a(n) for n = 1..576 (rows 1..24 of the matrix, flattened)
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer, New York, NY, 1999, pp. 131-133.
Wikipedia, Leech lattice.
Wikipedia, Miracle Octad Generator.

Cf. A008408, A260646, A351831.

det(A/sqrt(8)) = 1, where A is the present matrix.

cp's OEIS Frontend

A351830 Distance from the n-th square pyramidal number (sum of the first n positive squares) to the nearest square.

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