A371835 Triangle read by rows: T(n,k) is the number of points (x,y,z) satisfying |x|+|y|+|z|<=n and max(|x|,|y|,|z|)<=k; 0<=k<=n.

1, 1, 7, 1, 19, 25, 1, 27, 57, 63, 1, 27, 93, 123, 129, 1, 27, 117, 195, 225, 231, 1, 27, 125, 263, 341, 371, 377, 1, 27, 125, 311, 461, 539, 569, 575, 1, 27, 125, 335, 569, 719, 797, 827, 833, 1, 27, 125, 343, 649, 895, 1045, 1123, 1153, 1159

Offset

0,3

Comments

For all pairs of positive integers (a,b), T(a*m,b*m) satisfies a cubic polynomial in m.

Links

Table of n, a(n) for n=0..54.

Sela Fried, On a problem related to the integer lattice and its layers, 2024.

Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 5.

Formula

T(n,k) = 8*n^3 + 12*n^2 + 6*n + 1 = A016755(k) if k <= n/3.

T(m,m) = (4*n^3 + 6*n^2 + 8*n + 3)/3 = A001845(m).

T(2m,m) = (20*n^3 + 24*n^2 + 10*n + 3)/3 = A371532(m).

T(3m,2m) = 32*n^3 + 18*n^2 + 6*n + 1 = A371515(m).

T(4m,3m) = (244*n^3 + 96*n^2 + 26*n + 3)/3.

T(5m,2m) = (188*m^3 + 132*m^2 + 28*m + 3)/3.

T(5m,3m) = (404*m^3 + 150*m^2 + 28*m + 3)/3.

T(5m,4m) = (488*m^3 + 150*m^2 + 34*m + 3)/3.

Conjectures:

T(n,k) = (-84*k^3 + 108*k^2*n - 72*k^2 - 36*k*n^2 + 72*k*n - 6*k + 4*n^3 - 12*n^2 + 8*n + 3)/3 for (n-2)/3 <= k <= n/2.

T(n,k) = (12*k^3 - 36*k^2*n + 36*k*n^2 + 6*k - 8*n^3 + 6*n^2 + 2*n + 3)/3 for (n-1)/2 <= k <= n.

The two conjectures are true. See links. - Sela Fried, Jul 05 2024

Example

Table begins:
  n\k| 0  1   2   3   4    5    6    7    8    9   10
  ---+-----------------------------------------------
   0 | 1
   1 | 1  7
   2 | 1 19  25
   3 | 1 27  57  63
   4 | 1 27  93 123 129
   5 | 1 27 117 195 225  231
   6 | 1 27 125 263 341  371  377
   7 | 1 27 125 311 461  539  569  575
   8 | 1 27 125 335 569  719  797  827  833
   9 | 1 27 125 343 649  895 1045 1123 1153 1159
  10 | 1 27 125 343 697 1051 1297 1447 1525 1555 1561

Crossrefs

Sequence in context: A028325 A245484 A215503 * A195220 A119546 A173204

Adjacent sequences: A371832 A371833 A371834 * A371836 A371837 A371838

Keyword

nonn,tabl

Author

Peter Kagey, Apr 07 2024

Status

approved