Puzzle 55.- Primes by Generation (Patrick De Geest)

Define the sequence of primes 2, 3, 5, 7, 11, 13, 17, 19, ... as generation 1. Starting from this generation 1 add the previous and next term of
each number thus creating generation 2. Apply the same procedure over and over again to make the next generations N. The following table summarizes everything for the first nine generations:

PRIMES Gen1	Gen2	Gen3	Gen4	Gen5	Gen6	Gen7	Gen8	Gen9	PLOT Nrs
2	3	7	13	30	56	127	237	530	prime cells
3	7	13	30	56	127	237	530	994	Odd composite
5	10	23	43	97	181	403	757	1662
7	16	30	67	125	276	520	1132	2156
11	20	44	82	179	339	729	1399	2970
13	28	52	112	214	453	879	1838	3598
17	32	68	132	274	540	1109	2199	4491
19	40	80	162	326	656	1320	2653	5335
23	48	94	194	382	780	1544	3136	6231
29	54	114	220	454	888	1816	3578	7278
31	66	126	260	506	1036	2034	4142	8178
37	72	146	286	582	1146	2326	4600	9308
41	80	160	322	640	1290	2566	5166	10290
43	88	176	354	708	1420	2840	5690	11382
	2	3	3	3	4	6	5	3	Odd's

With each generation the 'last odd term' moves down one place (see darkgreen background cells)! They form a very beautiful new sequence starting with primes but soon these become very rare:

7, 23, 67, 179, 453, 1109, 2653, 6231, 14409, 32877, 74137, 165429, 365691, 801747,...

Highlighting the primes versus the composites give the following Last Odd Term sequence (primes in blue, composites in orange):

7, 23, 67, 179,453,1109, 2653, 6231, 14409, 32877, 74137, 165429, 365691, 801747, 1745331, 3776605, 8130401, 17427659, 37217597, 79224121, 168170537, 356107787, 752453861, 1586875049, 3340696135, 7021048691, 14731810645,...

I was able to track down these entire Prime Last Odd Terms sequence- PLOT's - up to generation 50. They occur in the following generations: 2, 3, 4, 5, 7, 19, 25, 27 (next one >50).

This yield the next appealing series of eight primes for now :
7, 23, 67, 179, 1109, 17427659, 1586875049, 7021048691
(See sequences A047844, A048448 up to A048466 at Neil's Sloane site)

b) Three generations (2, 3 and 4) have only 'prime' odd terms in their ranks. Exist there a fourth or even a fifth generation where this fact occurs?

Yves Gallot wrote (5/6/99):
"I extended the search of Puzzle No. 55.

Each term of the sequence 7, 23, 67, 179, 453, 1109, ... [This is the 'Last Odd Term' sequence] are of the form

S(n) = Sum(j = 0 to n - 1, C(n - 1, j) * P(2*j + 1))
where C(n, p) = n! / (p! * (n-p)!)
and P(1) = 2,... P(i) is the ith prime number.

The first probable-primes of the sequence occur for n equal :
1, 2, 3, 4, 5, 7, 19, 25, 27, 53, 59, 68, 148, 176, 241, 347,
441, 444, 509, 844, 990 [found previously by Patrick, in red].
The search was extended up to 1000"