cp's OEIS Frontend

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.

%I A132193 #30 May 19 2025 08:16:56
%S A132193 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,
%T A132193 7,9,11,13,15,17,19,21,25,27,29,37,49,8,10,12,14,16,18,20,22,24,26,28,
%U A132193 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
%N 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.
%C A132193 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 A*M=M*A. The number of elements in the n-th row is given by the sequence A069999. - Corrected by _Ricardo C. Santamaria_, Nov 08 2012
%H A132193 Alois P. Heinz, <a href="/A132193/b132193.txt">Rows n = 0..50, flattened</a> (first 1000 terms from Jean-François Alcover)
%e A132193 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.
%e A132193 Triangle T(n,k) begins:
%e A132193   0;
%e A132193   1;
%e A132193   2, 4;
%e A132193   3, 5,  9;
%e A132193   4, 6,  8, 10, 16;
%e A132193   5, 7,  9, 11, 13, 17, 25;
%e A132193   6, 8, 10, 12, 14, 18, 20, 26, 36;
%e A132193   7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 37, 49;
%e A132193   ...
%p A132193 b:= proc(n, i) option remember; `if`(n=0 or i=1, {n},
%p A132193      {b(n, i-1)[], map(x-> x+i^2, b(n-i, min(n-i, i)))[]})
%p A132193     end:
%p A132193 T:= n-> sort([b(n$2)[]])[]:
%p A132193 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jun 06 2022
%t A132193 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 *)
%Y A132193 Cf. A069999.
%K A132193 nonn,look,tabf
%O A132193 0,3
%A A132193 _Roger Cuculière_, Nov 05 2007
%E A132193 More terms from _Ricardo C. Santamaria_, Nov 08 2012
%E A132193 Row n=0 prepended by _Alois P. Heinz_, Jun 06 2022