OFFSET
1,3
COMMENTS
Also the sum of the floors of half the parts of all partitions of n, because the sum of one kind for a partition equals the sum of the other kind for the conjugate partition. Furthermore, this generalizes to taking m-th indices and dividing by m. - George Beck, Apr 15 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
EXAMPLE
For n = 5, write the partitions of 5 and below write the sums of their even-indexed parts:
. 5
. 3+2
. 4+1
. 2+2+1
. 3+1+1
. 2+1+1+1
. 1+1+1+1+1
------------
. 8 + 2 = 10
The sum of the even-indexed parts is 10, so a(5) = 10.
From George Beck, Apr 15 2017: (Start)
Alternatively, sum the floors of the parts divided by 2:
. 2
. 1+1
. 2+0
. 1+1+0
. 1+0+0
. 1+0+0+0
. 0+0+0+0+0
The sum is 10, so a(5) = 10. (End)
MAPLE
b:= proc(n, i) option remember; local g, h;
if n=0 then [1, 0$2]
elif i<1 then [0$3]
else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
[g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
fi
end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50); # Alois P. Heinz, Mar 12 2012
MATHEMATICA
b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0, 0}, i<1, {0, 0, 0}, True, g = b[n, i-1]; h = If[i>n, {0, 0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
a[n_]:= Total@Flatten@Quotient[IntegerPartitions[n], 2];
Table [a[n], {n, 1, 50}] (* George Beck, Apr 15 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 17 2012
EXTENSIONS
More terms from Alois P. Heinz, Mar 12 2012
STATUS
approved