Sequence A094534, Centered Hexamorphic, or Automorphic Hexagonal, Numbers  

This sequence, Sloane's A094534, is discussed by Cliff Pickover on this page. It is a subset of A003215, the centered hexagonal numbers, given by A003215[n] = 3n(n-1)+1:

Sequence A003215: A[1] = 1 o   o o A[2] = 7 o o o o o   o o o o o o o A[3] = 19 o o o o o o o o o o o o   o o o o o o o o o o o o o o o A[4] = 37 o o o o o o o o o o o o o o o o o o o o o o

This sequence, the Centered Hexamorphic Numbers, consists of those values of n for which the hexagonal number 3n(n-1)+1 ends in digits equal to n. For example, 417 is in the sequence because if n=417, 3n(n-1)+1=520417, which ends in 417. This is similar to the concept of sequence A003226.

The sequence runs: 1, 7, 17, 51, 67, 167, 251, 417, 501, 667, 751, 917, 1251, 1667, 5001, 5417, 6251, 6667, 10417, 16667, 50001, 56251, 60417, 66667, 166667, 260417, 406251, 500001, 666667, 760417, 906251, 1406251, 1666667, 5000001, 5260417, 6406251, 6666667, 16666667, 25260417, 41406251, 50000001, 66666667, 75260417, 91406251, 166666667, 191406251, 475260417, 500000001, 666666667, 691406251, 975260417, 1191406251, 1666666667, 5000000001, 5475260417, 6191406251, 6666666667, 16666666667, 21191406251, 45475260417, 50000000001, 66666666667, 71191406251, 95475260417, 166666666667, 221191406251, 445475260417, 500000000001, 666666666667, 721191406251, 945475260417, 1666666666667, 1721191406251, 4945475260417, 5000000000001, 6666666666667, 6721191406251, 9945475260417, 16666666666667, 26721191406251, 39945475260417, 50000000000001, 66666666666667, 76721191406251, 89945475260417, 166666666666667, 326721191406251, 339945475260417, 500000000000001, 666666666666667, 826721191406251, 839945475260417, 1666666666666667, 5000000000000001, 5826721191406251, 5839945475260417, 6666666666666667, 16666666666666667, 20839945475260417, 45826721191406251, 50000000000000001, 66666666666666667, 70839945475260417, 95826721191406251, 166666666666666667, 170839945475260417, 495826721191406251, 500000000000000001, 666666666666666667, 670839945475260417, 995826721191406251, ...

Pickover suggests looking for patterns that occur in the sequence, in particular "families" of numbers that all have the same digit pattern. He gives the example of 51, 501, 5001, 50001, etc. As you can see, there are two other families of this type, and there appear to be lots of others without a pattern. Notice that for any number in the sequence, you can remove one or more digits from the beginning and get another number in the sequence. This property is shared by A003226, but this sequence has more terms (6 or 7 for each number of digits, versus 2 or 3 for A003226). It makes it very easy to find new terms, because all you have to do is try prepending new digits to existing terms in the sequence (possibly including extra zeros). For example, to 6251 you can prepend 5 to get a 5-digit term, or 40 or 90 to get a 6-digit term.

Some other sequences are discussed here.



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