A sequence S describing its partial sums

[Max A.]

On 8/14/06, Max A. <maxale at gmail.com> wrote:

> 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.

An alternative rule imposing all terms distinct is:
(i) a(1)=3
(ii) for n is S, a(n)=a(1)+a(2)+...+a(n-1)
(iii) for n not in S, a(n)=the smallest number different from a(1),
..., a(n-1) not breaking condition (ii).

Then the sequence S is

3, 4, 7, 14, 6, 34, 68, 9, 145, 11, 301, 13, 615, 1230, 16, 2476, 18,
4970, 20, 9960, 22, 19942, 24, 39908, 26, 79842, 28, 159712, 30,
319454, 32, 638940, 35, 1277915, 2555830, 37, 5111697, 39, 10223433,
41, 20446907, 43, 40893857, 45, 81787759, 47, 163575565, 49,
327151179, 51, 654302409, 53, 1308604871, 55, 2617209797, 57,
5234419651, 59, 10468839361, 61, 20937678783, 63, 41875357629, 65,
83750715323, 67, 167501430713, 335002861426, 70, 670005722922, 72,
1340011445916, 74, 2680022891906, 76, 5360045783888, 78,
10720091567854, 80, 21440183135788, 82, 42880366271658, 84,
85760732543400, 86, 171521465086886, 88, 343042930173860, 90,
686085860347810, 92, 1372171720695712, 94, 2744343441391518, 96,
5488686882783132, 98, 10977373765566362, 100, 21954747531132824

We can also take a(1)=2 making all terms distinct except for a(1)=a(2)=2:

2, 2, 4, 8, 3, 7, 26, 52, 10, 114, 12, 240, 14, 494, 16, 1004, 18,
2026, 20, 4072, 22, 8166, 24, 16356, 27, 32739, 65478, 29, 130985, 31,
262001, 33, 524035, 35, 1048105, 37, 2096247, 39, 4192533, 41,
8385107, 43, 16770257, 45, 33540559, 47, 67081165, 49, 134162379, 51,
268324809, 536649618, 54, 1073299290, 56, 2146598636, 58, 4293197330,
60, 8586394720, 62, 17172789502, 64, 34345579068, 66, 68691158202, 68,
137382316472, 70, 274764633014, 72, 549529266100, 74, 1099058532274,
76, 2198117064624, 78, 4396234129326, 80, 8792468258732, 82,
17584936517546, 84, 35169873035176, 86, 70339746070438, 88,
140679492140964, 90, 281358984282018, 92, 562717968564128, 94,
1125435937128350, 96, 2250871874256796, 98, 4501743748513690, 100,
9003487497027480

Max