A035677 Number of partitions of n into parts 8k and 8k + 6 with at least one part of each type.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 3, 0, 6, 0, 1, 0, 3, 0, 7, 0, 12, 0, 3, 0, 7, 0, 15, 0, 21, 0, 7, 0, 16, 0, 28, 0, 36, 0, 16, 0, 31, 0, 50, 0, 60, 0, 32, 0, 57, 0, 85, 0, 98, 0, 60, 0, 100, 0, 141, 0, 157, 0, 107, 0, 169, 0, 226, 0, 248, 0
Offset: 1
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 361 terms from Antti Karttunen)
Crossrefs
Programs
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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 *) -
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
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