A132193 Triangle whose n-th row is the list in increasing order of the integers which are the sum of squares of positive integers with sum n. The n-th row begins with n and ends with n^2.

0, 1, 2, 4, 3, 5, 9, 4, 6, 8, 10, 16, 5, 7, 9, 11, 13, 17, 25, 6, 8, 10, 12, 14, 18, 20, 26, 36, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 37, 49, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 38, 40, 50, 64, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 45, 51, 53, 65, 81

Offset

0,3

Comments

The n-th row is the list of possible dimensions of the commutant space of an n X n matrix A, i.e. the set of matrices M such that AM=MA. The number of elements in the n-th row is given by the sequence A069999. - Corrected by Ricardo C. Santamaria, Nov 08 2012

Links

Alois P. Heinz, Rows n = 0..50, flattened (first 1000 terms from Jean-François Alcover)

Example

T(4,1)=4 because 4=1+1+1+1 and 1^2+1^2+1^2+1^2=4 ; T(4,2)=6 because 4=2+1+1 and 2^2+1^2+1^2=6.
Triangle T(n,k) begins:
  0;
  1;
  2, 4;
  3, 5,  9;
  4, 6,  8, 10, 16;
  5, 7,  9, 11, 13, 17, 25;
  6, 8, 10, 12, 14, 18, 20, 26, 36;
  7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 37, 49;
  ...

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, {n},
     {b(n, i-1)[], map(x-> x+i^2, b(n-i, min(n-i, i)))[]})
    end:
T:= n-> sort([b(n$2)[]])[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 06 2022

Mathematica

selQ[n_][p_] := MemberQ[#.# & /@ IntegerPartitions[n], p]; row[n_] := Select[Range[n, n^2], selQ[n] ]; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

Crossrefs

Cf. A069999.

Sequence in context: A215898 A246162 A243573 * A185910 A306779 A349947

Adjacent sequences: A132190 A132191 A132192 * A132194 A132195 A132196

Keyword

nonn,look,tabf

Author

Roger Cuculière, Nov 05 2007

Extensions

More terms from Ricardo C. Santamaria, Nov 08 2012

Row n=0 prepended by Alois P. Heinz, Jun 06 2022

Status

approved