[seqfan] Cumulative sum is odd accordingly : D and Dcomplementary

[Eric Angelini]

Hello SeqFans,

  D = 1, 2, 5, 6, 7, 8, 10, 12, 13, 15, 17, 19, 20, 21, 23, 25, 27, 29, 31, ...
Dcomp = 3, 4, 9, 11, 14, 16, 18, 22, 24, 26, 28, 30, ...

All Natural numbers are split between D and Dcomp

Q is the cumulative sum of D (which is reproduced here, above Q):

  D = 1, 2, 5, 6, 7, 8, 10, 12, 13, 15, 17, 19, 20, 21, 23, 25, 27, 29, 31, ...
  Q = 1, 3, 8,14,21,29, 39, 51, 64, 79, 96,115,135,156,179,204,231,260,291, ...

Even terms in Q are marked "E" and odd ones are marked "o"; "n" is the rank
of the even terms in Q:

  Q = 1, 3, 8,14,21,29, 39, 51, 64, 79, 96,115,135,156,179,204,231,260,291, ...
      o  o  E  E  o  o   o   o   E   o   E  o   o   E   o   E   o   E   o
  n = .  .  3  4  .  .   .   .   9   .  11  .   .  14   .  16   .  18   . 

We see that Dcomp is equal to the above "n" line (without the dots).

Pick thus any term of Dcomp (e.g. "11"); this term says that the sum of
the first 11 terms of D is even (here 96).

Pick any term of D (e.g. "12"), this term says that the sum of the
first 12 terms of D is odd (here 115).

If this is of interest, could someone check, compute and submit (in
september)? Thanks!
Best,
É.

(strange behavior of D so far; is a pattern to come?)