A sequence S describing its partial sums

[Max A.]

On 8/13/06, Eric Angelini <Eric.Angelini at kntv.be> wrote:

> based on an equivalent idea and building rule as A121053,
> here is a sequence I would appreciate someone to check,
> define properly in good mathematical english, and submit
> to the OEIS -- if of interest (I'll be back at my PC in
> two weeks time -- and will do it myself if no one moves ;-)
>
> a(n) says : "At position a(n) in S you will find the sum
> of all terms from a(1) to a(n)"

Should it be "...from a(1) to a(n-1)"? Otherwise, it may be impossible
to determine a(n).

> S should be (computed by hand) :
>
> 1,3,4,8,6,22,9,16,53,11,133,13,279,15,573,69,18,1233,20,
> 2486,23,44,4995,25,10059,27,20145,29,40319,31,80669,33,
> 161371,35,322777,37,645591,39,12911221,41,25822483,43,
> 51645009,5039,103295057,47,...
>
> S reads (from the beginning) :
> - at position 1 there is the sum of all previously written terms [indeed,
> nil + 1=1]
> - at position 3 there is the sum of all previously written terms [indeed, 1+
> 3=4]

There is a discrepancy. From your definition: a(1)=a(1) but not
a(3)=a(1)+a(2)+a(3).

Let's adopt the rule:
for n is S, a(n)=a(1)+a(2)+...+a(n-1), and for n not in S, a(n)=n+1.
Then a(1)=1 cannot happen. Therefore, the sequence S starts with
a(1)=2 as follows:

2, 2, 4, 8, 6, 22, 8, 52, 10, 114, 12, 240, 14, 494, 16, 1004, 18,
2026, 20, 4072, 22, 8166, 24, 16356, 26, 32738, 28, 65504, 30, 131038,
32, 262108, 34, 524250, 36, 1048536, 38, 2097110, 40, 4194260, 42,
8388562, 44, 16777168, 46, 33554382, 48, 67108812, 50, 134217674, 52,
268435400, 54, 536870854, 56, 1073741764, 58, 2147483586, 60,
4294967232, 62, 8589934526, 64, 17179869116, 66, 34359738298, 68,
68719476664, 70, 137438953398, 72, 274877906868, 74, 549755813810, 76,
1099511627696, 78, 2199023255470, 80, 4398046511020, 82,
8796093022122, 84, 17592186044328, 86, 35184372088742, 88,
70368744177572, 90, 140737488355234, 92, 281474976710560, 94,
562949953421214, 96, 1125899906842524, 98, 2251799813685146, 100,
4503599627370392

Note that a(n)=a(1)+a(2)+...+a(n-1) can hold even if n is not in S.
The smallest example is n=3.

Max