A323796 Numbers (both the reverse and the add numbers) occurring in the Reverse and Add! graph with seed 196.

97, 196, 295, 394, 493, 592, 691, 790, 689, 788, 887, 986, 496, 586, 676, 766, 856, 946, 1495, 1585, 1675, 1765, 1855, 1945, 2494, 2584, 2674, 2764, 2854, 2944, 3493, 3583, 3673, 3763, 3853, 3943, 4492, 4582, 4672, 4762, 4852, 4942, 5491, 5581, 5671, 5761, 5851, 5941, 4079, 4169, 4259, 4349, 4439, 4529, 4619

Offset: 1

Author

A.H.M. Smeets, Jan 28 2019

Keywords

• base
• nonn

Comments

The graph has a tree structure with reversed edges. The tree structure is in fact a branched horsetail (a botanical term). The vertices are pairs (Added number and its Reverse). Both the Add as well as the Reverse terms are included in the sequence.

Each term in the sequence represents the number (either Added or Reversed) of the tail of the edge of the directed graph, lexicographical ordered by first the head of its edge and second the tail of the edge. The heads in the graph are the numbers in A006960.

In general, a(n) for n = A323797(m)..A323797(m+1)-1 point to A006960(m) for m > 0; for example: a(n) for n = 1..8 point to A006960(1) and a(n) for n = 9..12 point to A006960(2).

The structure seems is somewhat surprising:

a(1)..a(8) is given by 97 + 99*n0 for n0 = 0..7;

a(9)..a(12) is given by 689 + 99*n0 for n0 = 0..3

a(13)..a(54) is given by 496 + 90*n0 + 999*n1 for n0 = 0..5 and n1 = 0..6;

a(55)..a(102) is given by 4079 + 90*n0 + 999*n1 for n0 = 0..7 and n1 = 0..5;

a(103)..a(142) is given by 2794 + 90*n0 + 999*n1 for n0 = 0..7 and n1 = 0..4;

a(143)..a(182) is given by 539 + 990*n0 + 9999*n1 for n0 = 0..3 and n1 = 0..9;

a(183)..a(190) is given by 97009 + 990*n0 for n0 = 0..7;

a(191)..a(430) is given by 70799 + 900*n0 + 9990*n1 + 99999*n2 for n0 = 0..7, n1 = 0..2 and n2 = 0..8;

a(431)..a(744) is given by 1057969 + 9900*n0 + 99990*n1 + 999990*n2 for n0 = 0..7, n1 = 0..8 and n2 = 0..8, where (n0,n1,n2) = (2,3,4) must be excluded due to the fact that it results in a palindrome, 5377735.

Examples

			.   196--+--887--+  1495--+
.   691  |  788  |  5941  |
.        |       |        |
.   295--+  689--+  1585--+
.   592  |  986  |  5851  |
.        |       |        |
.   394--+       +--1675--+--7436
.   493  |          5761  |  6347
.        |                |
.   790--+          1765--+
.    97             5671  |
.                         |
.                   1855--+
.                   5581  |
.                         |
.                   1945--+
.                   5491  |
.                         |
.                   2494--+
.                   4942  |
.                         |
.                   ....  :
.                         |
.                   6940--+
.                    496
		

Links

• Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80 No. 3 2012, 375-384.

Crossrefs

Cf. A006960, A323797.