A278575 Number of decompositions of n as a sum of nonnegative multiples of 5, 7, even numbers greater than 2, and the two partitions obtained by expressing the numbers 6 and 8 (each of them exactly once) as a sum using positive integers less than 4.

0, 0, 0, 1, 1, 2, 1, 3, 1, 4, 3, 8, 5, 10, 7, 16, 11, 23, 17, 34, 25, 46, 36, 68, 52, 91, 73, 128, 103, 173, 142, 236, 194, 313, 265, 424, 357, 555, 476, 737, 634, 961, 837, 1256, 1098, 1621, 1433, 2102, 1860, 2687, 2401, 3445, 3089, 4379, 3952, 5563, 5034, 7015, 6391, 8852, 8082, 11087, 10177

Offset

1,6

Comments

According to the text of the paper (see Corollary 13), this should be the number of decompositions of n as a sum of nonnegative multiples of 3 and 4. But then a(3) would be 1, and a(5) would be zero. So that is not the definition. But they give 100 terms, so it should be possible to find the correct definition.

Old name was: Number of decompositions of n as a "sum of nonnegative multiples of 3 and 4". - Arkadiusz Wesolowski, Feb 03 2017

Links

Best, Darcy, Hadi Kharaghani, and Hugh Ramp, Table of n, a(n) for n = 1..100

Darcy Best, Hadi Kharaghani, and Hugh Ramp, On unit weighing matrices with small weight, Discrete Mathematics 313.7 (2013): 855-864. Also arXiv:1209.4581 [math.CO], 2012. See Table 2.

Example

From Arkadiusz Wesolowski, Feb 03 2017: (Start)
Let 6 be partitioned as 3 + 3, and let 8 be partitioned as 1 + 2 + 2 + 3. Then:
4 = 4, so a(4) = 1.
5 = 5, so a(5) = 1.
6 = 6 =
  = 3 + 3, so a(6) = 2.
7 = 7, so a(7) = 1.
8 = 8 =
  = 2*4 =
  = 1 + 2 + 2 + 3, so a(8) = 3.
9 = 5 + 4, so a(9) = 1.
10 = 10 =
   = 6 + 4 =
   = 2*5 =
   = 4 + (3 + 3), so a(10) = 4.
11 = 7 + 4 =
   = 6 + 5 =
   = 5 + (3 + 3), so a(11) = 3.
12 = 12 =
   = 8 + 4 =
   = 5 + 7 =
   = 2*6 =
   = 6 + (3 + 3) =
   = 3*4 =
   = 4 + (1 + 2 + 2 + 3) =
   = 2*(3 + 3), so a(12) = 8. (End)

Prog

(Magma) /* computes the first 27 elements of the sequence */ lst:=[]; m:=27; if m lt 28 then b:=SequenceToSet([p: p in [4..m by 2]]); o:={5, 7} join b; for n in [1..m] do p:=RestrictedPartitions(n, o); a:=#p; b:=0; for y in [1..#p] do lst1:=p[y]; if #lst1 eq 1 then if lst1[1] eq 6 or lst1[1] eq 8 then b+:=#p[y]; end if; else t:=0; for r in [1..#lst1] do if lst1[r] eq 6 or lst1[r] eq 8 then t+:=1; end if; end for; if t eq 1 then b+:=1; end if; if t ge 2 then if 6 in lst1 and not 8 in lst1 then b+:=t; end if; if 8 in lst1 and not 6 in lst1 then b+:=t; end if; if 6 in lst1 and 8 in lst1 then b+:=2*t-1; end if; end if; end if; end for; Append(~lst, a+b); end for; end if; lst; // Arkadiusz Wesolowski, Feb 06 2017

Crossrefs

Sequence in context: A225045 A361736 A379314 * A333879 A175126 A142884

Adjacent sequences: A278572 A278573 A278574 * A278576 A278577 A278578

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2016

Extensions

Name changed by Arkadiusz Wesolowski, Feb 03 2017

Status

approved