A282249 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k < j_k, j_k > j_{k+1} and all factors distinct.
1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 4, 4, 6, 5, 6, 8, 8, 9, 11, 10, 14, 15, 14, 14, 21, 18, 21, 25, 25, 30, 34, 33, 42, 45, 41, 55, 62, 58, 66, 79, 76, 94, 95, 97, 115, 131, 120, 148, 153, 159, 175, 203, 189, 226, 232, 243, 268, 299, 271, 340, 349, 363, 389
Offset: 0
Keywords
Examples
a(0) = 1: the empty sum. a(6) = 2: 1*6 = 2*3. a(8) = 2: 1*8 = 2*4. a(10) = 3: 1*10 = 2*5 = 1*4+2*3. a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3. a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4. a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4. a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2. a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5. a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Maple
h:= proc(n) option remember; (((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12 end: g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0, b(n, i, select(x-> x<=i, s)))): b:= proc(n, i, s) option remember; g(n, i-1, s)+ `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j, min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i))) end: a:= n-> g(n$2, {}): seq(a(n), n=0..100); -
Mathematica
h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12; g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]]; b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j, Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i - 1, n/i]}]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)
Comments