Graham-Pollak Sequences

I  was reminded of my previous post titled Pisot Sequences when I came across the topic of this current post relating to Graham-Pollak sequences. An example of this type of sequence is the following:$$a(n) = \text{ floor} \sqrt{2 \times a(n-1) \times (a(n-1)+1)}\\ \text{ where } a(0)=1$$The initial members of this sequence (OEIS A001521) are (permalink):

1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 54, 77, 109, 154, 218, 309, 437, 618, 874, 1236, 1748, 2472, 3496, 4944, 6992, 9888, 13984, 19777, 27969, 39554, 55938, 79108, 111876, 158217, 223753, 316435, 447507, 632871, 895015, 1265743, 1790031, 2531486, 3580062, 5062972

This sequence has apparently some amazing properties that can be read about here in Wolfram Mathworld but I have to confess to not understanding their significance. Note that from 1 to 9, the numbers 5 and 8 are missing and so two others sequences, listed in the OEIS, can be generated using these numbers as starting points instead of 1. Starting with 5 produces OEIS A091522 (permalink):

A091522

Graham-Pollak sequence with initial term 5.

The initial members are:

5, 7, 10, 14, 20, 28, 40, 57, 81, 115, 163, 231, 327, 463, 655, 927, 1311, 1854, 2622, 3708, 5244, 7416, 10488, 14832, 20976, 29665, 41953, 59331, 83907, 118663, 167815, 237326, 335630, 474653, 671261, 949307, 1342523, 1898614, 2685046 

Starting with 8 produces OEIS A091523 (permalink):

A091523

Graham-Pollak sequence with initial term 8.

The initial members are:

8, 12, 17, 24, 34, 48, 68, 96, 136, 193, 273, 386, 546, 772, 1092, 1545, 2185, 3090, 4370, 6180, 8740, 12360, 17480, 24721, 34961, 49443, 69923, 98886, 139846, 197772, 279692, 395544, 559384, 791089, 1118769, 1582179, 2237539, 3164358

I came across the Graham-Pollak sequences through a reference in the OEIS to the number associated my diurnal age of 27207. The number arises from the following formula:$$a(n) = \text{ floor} \sqrt{3 \times a(n-1) \times (a(n-1)+1)}\\ \text{ where } a(0)=1$$Notice that the multiplier is not 2 but 3. The numbers generated by this formula produce OEIS A100671 (permalink):

A100671

A Graham-Pollak-like sequence with multiplier 3 instead of 2.

1, 2, 4, 7, 12, 21, 37, 64, 111, 193, 335, 581, 1007, 1745, 3023, 5236, 9069, 15708, 27207, 47124, 81622, 141374, 244867, 424122, 734601, 1272367, 2203805, 3817103, 6611417, 11451311, 19834253, 34353934, 59502759, 103061802, 178508278, 309185407, 535524834 

Apparently it's not known whether this sequence has properties similar to the original.