Generalising Unique Sums beyond A025582 and A051912

[Wouter Meeussen]

Generalising Unique Sums beyond A025582 and A051912

Origin:
in a  net (lattice?) with j neighbourhoods, the 'energy' of a cell 
is the sum of the interaction-energies with its neighbours.
Say that there are m types of neighbours (20 amino-acids, ..). 
Choose a basis such that there is no accidental energy equality for 
any combination of j neighbours.



seq_j = {0 , 1 , j+1 , j^2+j+1 , ...  ==  { seq[j,0],seq[j,1],seq[j,2],... }
seq[j,m] is least integer such that the sums of all combinations of j terms taken from
seq[j,0] upto seq[j,m-1] incl. are unique, giving Pochhammer[m , j]/j! different terms.
                               


seq_2={0,1, 3,  7, 12,20,30,44,65,80,96,122,147,181,203,251,289,360,400,474,564,592,661,774,
        821,915,969,1015,1158,1311,1394,1522} = A025582

seq_3={0,1, 4, 13, 32,71,124,218,375,572,744,1208,1556,2441,3097,4047,5297,6703,7838,
        10986,12331,15464,19143,24545,28973,34405,37768,45863,50876} = A051912

seq_4={0,1, 5, 21, 55,153,368,856,1424,2603,4967,8194}
seq_5={0,1, 6, 31, 108,366,926,2286,5733 }

looking vertically (columnwise):

Table[seqn[j,i],{j,1,10},{i,5}]

 {0, 1, 2, 3, 4, ...
 {0, 1, 3, 7, 12}, 
 {0, 1, 4, 13, 32},
 {0, 1, 5, 21, 55},
 {0, 1, 6, 31,108},
 {0, 1, 7, 43, 154},
 {0, 1, 8, 57, 256},
 {0, 1, 9, 73, 333},
 {0, 1,10, 91, 500},
 {0, 1,11, 111, 616,
 ...

{3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273} is known:

ID Number: A002061 (Formerly M2638 and N1049)
Sequence:  1,1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,
           307,343,381,421,463,507,553,601,651,703,757,813,871,931,993,
           1057,1123,1191,1261
Name:      Central polygonal numbers: n^2 - n + 1.
References Archimedeans Problems Drive, Eureka, 22 (1959), 15.
           L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1,
           Chanticleer Press, NY, 1950, p. 22.
           R. Honsberger, Ingenuity in Math., Random House, 1970, p. 87.
Keywords:  nonn,easy,nice
Offset:    0
Author(s): njas


the next column isn't:
{4,12,32,55,108,154,256,333,500,616,864,1027,1372,1590,2048,2329,...}

I am sorry, but the terms
12,32,55,108,154,256,333,500,616 
do not match anything in the table

What? No GF? can sums be hard?