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A248605 Partitions into parts of the form k(3k plus or minus 1)/2 (in other words: 1,2,5,7,12,15,...) with a set of frequencies which has no binary carry.

%I A248605 #39 May 31 2017 14:50:10
%S A248605 1,1,2,1,3,3,3,3,4,6,6,6,7,7,7,10,9,11,11,14,15,14,16,19,17,22,20,22,
%T A248605 20,23,28,28,29,29,32,35,35,37,39,43,46,45,50,49,53,58,60,60,63,61,70,
%U A248605 73,77,77,75,84,83,84,88,92,99,101,110,99,112,118,118,121
%N A248605 Partitions into parts of the form k(3k plus or minus 1)/2 (in other words: 1,2,5,7,12,15,...) with a set of frequencies which has no binary carry.
%C A248605 The expression "a set of frequencies which has no binary carry," means the following: For a given partition take the set of frequencies of the summands expressed as binary numbers and add them together. If there is a carry in the addition, then this is not an allowed set of frequencies. See the example for more explanation.
%C A248605 Elements of this sequence have the same parity (A040051) as the corresponding elements of the sequence of unrestricted partitions (A000041). See lemma 2.2.ii of the paper by Cooper, Eichorn and O'Bryant.
%C A248605 From _David S. Newman_, May 30 2017: (Start)
%C A248605 Also the number of partitions of n into parts which are powers of 2 used with a frequency which is k(3k plus or minus 1)/2.
%C A248605 Every set of partitions defined with the "no binary carry" condition has a dual of this sort. (End)
%H A248605 Alois P. Heinz, <a href="/A248605/b248605.txt">Table of n, a(n) for n = 0..1000</a>
%H A248605 J. N. Cooper, D. Eichhorn and K. O'Bryant, <a href="http://arXiv.org/abs/math.NT/0506496">Reciprocals of binary power series</a>, arXiv:math/0506496 [math.NT], 2005.
%e A248605 For n=5, there are 4 partitions which have summands coming from {1,2,5,7,...} namely: 5; 2+2+1; 2+1+1+1; and 1+1+1+1. The third of these has frequencies 1 and 3. These frequencies when written in binary are 1 and 11. If we add these two binary numbers there will be a carry from the units column; therefore this set of frequencies is not allowed and the partition 2+1+1+1 is not counted.
%t A248605 <<"Combinatorica`";
%t A248605 nend=20;
%t A248605 For[n=1,n<=nend,n++,
%t A248605 summands={1,2,5,7,12,15,22,26,35,40};
%t A248605 p=Partitions[n];preduced=p;
%t A248605 For[i=Length[p],i>=1,i--,
%t A248605 For[j=1,j<=Length[p[[i]]],j++,
%t A248605 If[MemberQ[summands,p[[i]][[j]]]= =False,preduced=Delete[preduced,i];
%t A248605 Break[]]]];
%t A248605 For[i=Length[preduced],i>=1,i--,
%t A248605 t=Tally[preduced[[i]]];
%t A248605 For[j=1,j<=nend,j++,sum[j]=0];
%t A248605 For[j=1,j<=Length[t],j++,
%t A248605 IntDig=IntegerDigits[t[[j,2]],2,7];
%t A248605 For[k=1,k<=7,k++,sum[k]=sum[k]+IntDig[[k]]]];
%t A248605 table=Table[sum[k],{k,1,7}];
%t A248605 If[Max[table]>1,preduced=Delete[preduced,i]]];
%t A248605 a[n]=Length[preduced]];
%t A248605 Print[Table[a[i],{i,1,nend}]]
%Y A248605 Cf. A000041, A001318, A040051.
%K A248605 nonn
%O A248605 0,3
%A A248605 _David S. Newman_, Oct 09 2014
%E A248605 More terms from _Alois P. Heinz_, Oct 13 2014