OFFSET
1,22
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 361 terms from Antti Karttunen)
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 6)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020
MAPLE
b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
b(n, i-1, t, s)+(h-> `if`(h in {0, 3}, add(b(n-i*j, i-1,
`if`(h=0, 1, t), `if`(h=3, 1, s)), j=1..n/i), 0))(irem(i, 4))))
end:
a:= n-> `if`(n::odd, 0, b(n/2$2, 0$2)):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 17 2020
MATHEMATICA
nmax = 87; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 87; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 13 2020 *)
PROG
(PARI)
parts8katleast(up_to, n) = select(x -> (x>=n), vector(((up_to+0)>>3), k, ((k<<3)-0)));
parts8kplus6(up_to) = vector(((up_to+2)>>3), k, ((k<<3)-2));
partitions_for_A035677(n, parts, from=1, has8k6parts=0) = if(!n, (has8k6parts>0), my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_for_A035677(n-parts[i], parts, i, (has8k6parts+(6==(parts[i]%8)))))); (s));
A035677(n) = if(n%2, 0, sum(i=1, n>>3, my(k = i*8); partitions_for_A035677(n-k, vecsort(setunion(parts8katleast(n-k, k), parts8kplus6(n-k)), , 4)))); \\ Antti Karttunen, Feb 06 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved