A306568 Regular triangle read by rows where T(n,k) is obtained by partitioning the set of the first n squares into k groups, then summing up each group and taking the minimum width, where the width is the difference between its greatest and its smallest term.

3, 4, 8, 2, 11, 15, 3, 11, 20, 24, 1, 10, 22, 31, 35, 0, 4, 23, 35, 44, 48, 0, 4, 19, 38, 50, 59, 63, 1, 3, 15, 36, 55, 67, 76, 80, 1, 2, 6, 34, 55, 74, 86, 95, 99, 0, 2, 4, 27, 55, 76, 95, 107, 116, 120, 0, 1, 3, 19, 50, 78, 99, 118, 130, 139, 143, 1, 0, 3, 8, 44, 75, 103, 124, 143, 155, 164, 168

Offset

2,1

Comments

Collatz proves that for a sufficiently large n (depending on k), the k-th column is periodic with period 2*k^2, but remarks that column 2 and 3 appear to have periods 4 and 9, so he asks if the period is actually k^2.

Links

Table of n, a(n) for n=2..79.

L. Collatz, Equipartition with square numbers, Discrete Mathematics, Volume 42, Issues 2-3, 1982, pp. 313-316.

Formula

T(n,n) = n^2 - 1.

Example

For n=3 and k=2, the partitions are (1,4; 9) or (1; 4,9) or (1,9; 4);
                 the sums are (5,9) or (1,13) or (4,10);
                 the widths are 4, 12, 6; so a(3,2) = 4, the minimum.
The triangle begins:
  3;
  4,  8;
  2, 11, 15;
  3, 11, 20, 24;
  1, 10, 22, 31, 35;
  0,  4, 23, 35, 44, 48;
  0,  4, 19, 38, 50, 59, 63;
  1,  3, 15, 36, 55, 67, 76, 80;
  ...

Prog

(PARI) vj(j, b, n, p) = {vd = select(x->(x==j), b, 1); sum(k=1, #vd, vd[k]^2); }
a(n, p) = {my(res = sum(k=1, n, k^2), b, vone); for (k=0, p^n-1, b = digits(k, p); if (#b < n, b = concat(vector(n-#b), b)); if (#Set(b) == p, vone = vector(p, j, vj(j-1, b, n, p)); res = min(res, vecmax(vone) - vecmin(vone)); ); ); res; }
tabl(nn) = for (n=2, nn, for (p=2, n, print1(a(n, p), ", ")); print); \\ Michel Marcus, Feb 24 2019

Crossrefs

Cf. A000110, A000290 (the squares), A005563, A133872.

Sequence in context: A199618 A088745 A213922 * A154743 A020812 A021291

Adjacent sequences: A306565 A306566 A306567 * A306569 A306570 A306571

Keyword

nonn,tabl

Author

Michel Marcus, Feb 24 2019

Extensions

42 terms more (rows 10-13) from Alois P. Heinz, Mar 16 2019

Status

approved