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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.
1

%I #25 Jun 21 2022 18:08:17

%S 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 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 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 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 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

%H Alois P. Heinz, <a href="/A132193/b132193.txt">Rows n = 0..50, flattened</a> (first 1000 terms from Jean-François Alcover)

%e 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 Triangle T(n,k) begins:

%e 0;

%e 1;

%e 2, 4;

%e 3, 5, 9;

%e 4, 6, 8, 10, 16;

%e 5, 7, 9, 11, 13, 17, 25;

%e 6, 8, 10, 12, 14, 18, 20, 26, 36;

%e 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 37, 49;

%e ...

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, {n},

%p {b(n, i-1)[], map(x-> x+i^2, b(n-i, min(n-i, i)))[]})

%p end:

%p T:= n-> sort([b(n$2)[]])[]:

%p seq(T(n), n=0..10); # _Alois P. Heinz_, Jun 06 2022

%t 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 Cf. A069999.

%K nonn,look,tabf

%O 0,3

%A _Roger Cuculière_, Nov 05 2007

%E More terms from Ricardo C. Santamaria, Nov 08 2012

%E Row n=0 prepended by _Alois P. Heinz_, Jun 06 2022