bogus OGF in A002113 (and very many others)

[Peter Pein]

N. J. A. Sloane schrieb:
> Mitch,  I found the letter that Elizabeth Morgan
> wrote to me on April 6 1978, and i will add
> a more precise definition to A005045.
> 
> Neil
> 

Neil, I am afraid you made three typos. The formula reads:
" Let n = 3k, 3k-1 or 3k-2 according as n == 0, 2 or 1 mod 3, for n>=3.

Then a(n) = Sum_{ i=0,...,n-k } Sum_{ m=max(0,2i-n),...,floor(i/2) } Sum_{
r=0,...,floor(i/2)-m } c(i,m,r), where c(i,m,r) = n-2i+m+1 when m+r != i/2, or
= floor((m-2i+m+2)/2) when m+r = i2. "

But I get the listed values using:
" Let n = 3k, 3k-1 or 3k-2 according as n == 0, 2 or 1 mod 3, for n>=3.

Then a(n) = Sum_{ i=1,...,n-k } Sum_{ m=max(0,2i-n),...,floor(i/2) } Sum_{
r=0,...,floor(i/2)-m } c(i,m,r), where c(i,m,r) = n-2i+m+1 when m+r != i/2, or
= floor((n-2i+m+2)/2) when m+r = i/2. "

(i goes from *1* to n-k; c(i,m,r) has the value floor(*n*-2i+m+2)/2) if m+r ==
i */* 2)

Could some of the readers please check these changes?


Cheers,
 Peter

P.S.: Mma-code

In[1]:=
an = Block[{k = Floor[(n + 2)/3]},
    Sum[Sum[Sum[If[m + r == i/2, Floor[(n - 2*i + m + 2)/2], n - 2*i + m + 1],
       {r, 0, Floor[i/2 - m]}], {m, Max[2*i - n, 0], Floor[i/2]}],
     {i, 1, n - k}]];
In[2]:=
Table[an, {n, 2, 100}]
Out[2]=
{1, 3, 6, 10, 17, 25, 37, 51, 70, 92, 121, 153, 194, 240, 296, 358, 433, 515,
  612, 718, 841, 975, 1129, 1295, 1484, 1688, 1917, 2163, 2438, 2732, 3058, 3406,
  3789, 4197, 4644, 5118, 5635, 6183, 6777, 7405, 8084, 8800, 9571, 10383, 11254,
  12170, 13150, 14178, 15275, 16425, 17648, 18928, 20287, 21707, 23211, 24781,
  26440, 28170, 29995, 31895, 33896, 35978, 38166, 40440, 42827, 45305, 47902,
  50596, 53415, 56337, 59391, 62553, 65854, 69270, 72831, 76513, 80348, 84310,
  88432, 92688, 97111, 101675, 106414, 111300, 116369, 121593, 127007, 132583,
  138358, 144302, 150453, 156781, 163324, 170052, 177004, 184148, 191525, 199103,
  206922}