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.
%I A371835 #35 Oct 14 2024 11:21:25 %S A371835 1,1,7,1,19,25,1,27,57,63,1,27,93,123,129,1,27,117,195,225,231,1,27, %T A371835 125,263,341,371,377,1,27,125,311,461,539,569,575,1,27,125,335,569, %U A371835 719,797,827,833,1,27,125,343,649,895,1045,1123,1153,1159 %N 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. %C A371835 For all pairs of positive integers (a,b), T(a*m,b*m) satisfies a cubic polynomial in m. %H A371835 Sela Fried, <a href="/A371835/a371835_2.pdf">On a problem related to the integer lattice and its layers</a>, 2024. %H A371835 Sela Fried, <a href="https://arxiv.org/abs/2410.07237">Proofs of some Conjectures from the OEIS</a>, arXiv:2410.07237 [math.NT], 2024. See p. 5. %F A371835 T(n,k) = 8*n^3 + 12*n^2 + 6*n + 1 = A016755(k) if k <= n/3. %F A371835 T(m,m) = (4*n^3 + 6*n^2 + 8*n + 3)/3 = A001845(m). %F A371835 T(2m,m) = (20*n^3 + 24*n^2 + 10*n + 3)/3 = A371532(m). %F A371835 T(3m,2m) = 32*n^3 + 18*n^2 + 6*n + 1 = A371515(m). %F A371835 T(4m,3m) = (244*n^3 + 96*n^2 + 26*n + 3)/3. %F A371835 T(5m,2m) = (188*m^3 + 132*m^2 + 28*m + 3)/3. %F A371835 T(5m,3m) = (404*m^3 + 150*m^2 + 28*m + 3)/3. %F A371835 T(5m,4m) = (488*m^3 + 150*m^2 + 34*m + 3)/3. %F A371835 Conjectures: %F A371835 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. %F A371835 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. %F A371835 The two conjectures are true. See links. - _Sela Fried_, Jul 05 2024 %e A371835 Table begins: %e A371835 n\k| 0 1 2 3 4 5 6 7 8 9 10 %e A371835 ---+----------------------------------------------- %e A371835 0 | 1 %e A371835 1 | 1 7 %e A371835 2 | 1 19 25 %e A371835 3 | 1 27 57 63 %e A371835 4 | 1 27 93 123 129 %e A371835 5 | 1 27 117 195 225 231 %e A371835 6 | 1 27 125 263 341 371 377 %e A371835 7 | 1 27 125 311 461 539 569 575 %e A371835 8 | 1 27 125 335 569 719 797 827 833 %e A371835 9 | 1 27 125 343 649 895 1045 1123 1153 1159 %e A371835 10 | 1 27 125 343 697 1051 1297 1447 1525 1555 1561 %K A371835 nonn,tabl %O A371835 0,3 %A A371835 _Peter Kagey_, Apr 07 2024