%I A135419
%S A135419 1,2,3,4,7,10,12,13,16,1,2,3,4,8,9,11,14,16,1,2,3,5,6,10,11,14,16,1,2,
               3,5,7,8,
%T A135419 12,14,16,1,2,4,5,6,8,11,15,16,1,2,7,8,11,12,13,14,1,3,6,8,10,12,13,15,
               1,3,6,9,
%U A135419 10,11,12,16,1,4,5,8,10,11,14,15,1,4,6,7,9,12,14,15,1,4,6,7,10,11,13,16,
               1,5,6,7,8,11,14,16
%N A135419 Array read by rows, showing the ways of splitting the numbers from 1 
               to 16 into two groups so that the numbers in each group have the 
               same sum (68) and the same sum of squares (748).
%H A135419 Colm Mulcahy, <a href="http://www.maa.org/columns/colm/cardcolm.html">
               Dec 2007 Card Colm</a>
%e A135419 There are exactly 657 ways of splitting the numbers 1..16 into two groups 
               so that the numbers in each group have the same sum s1=68.
%e A135419 There are exactly 57 ways of splitting the numbers 1..16 into two groups 
               so that the numbers in each group have the same sum of squares s2=748.
%e A135419 And there is the only one way of splitting the numbers 1..16 into two 
               groups so that the numbers in each group have the same sum of cubes 
               s3=9248: {{1, 4, 6, 7, 10, 11, 13, 16} (=A133483) and {2, 3, 5, 8, 
               9, 12, 14, 15} (=A133484)}.
%e A135419 Amazingly, this last way also appears in previous two cases! This is 
               given in A133483 and A133484.
%e A135419 (Also, there is the only one way of splitting the numbers 1..16 into 
               two groups so that the numbers in each group have the same sum of 
               fourth powers s4=121924: {{1,2,3,4,8,9,10,11,12,16},{5,6,7,13,14,
               15}}. But this splitting does not give equal sum of powers 1..3.)
%e A135419 Intersection of first two cases gives 12 ways of splitting the numbers 
               1..16 into two groups so that the numbers in each group have the 
               same sum s1=68 and the same sum of squares s2=748;
%e A135419 Here we list only groups containing 1 (the corresponding 2nd groups are 
               their complements):
%e A135419 {1,2,3,4,8,9,11,14,16},
%e A135419 {1,2,3,5,6,10,11,14,16},
%e A135419 {1,2,3,5,7,8,12,14,16},
%e A135419 {1,2,4,5,6,8,11,15,16},
%e A135419 {1,2,7,8,11,12,13,14},
%e A135419 {1,3,6,8,10,12,13,15},
%e A135419 {1,3,6,9,10,11,12,16},
%e A135419 {1,4,5,8,10,11,14,15},
%e A135419 {1,4,6,7,9,12,14,15},
%e A135419 {1,4,6,7,10,11,13,16},
%e A135419 {1,5,6,7,8,11,14,16}.
%e A135419 This table read by rows gives the present sequence.
%Y A135419 Cf. A133483, A133484, A135418.
%Y A135419 Sequence in context: A047546 A139759 A030292 this_sequence A051914 A104519 
               A117220
%Y A135419 Adjacent sequences: A135416 A135417 A135418 this_sequence A135420 A135421 
               A135422
%K A135419 fini,full,nonn,tabf
%O A135419 1,2
%A A135419 Zak Seidov (zakseidov(AT)yahoo.com), Dec 01 2007
%E A135419 Error in example lines corrected by Colm Mulcahy (colm(AT)spelman.edu), 
               Dec 30 2007