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 henryzz 2018-08-20 19:45

Concatenated Triangular Numbers

Recently someone mentioned to me at a meal that their favourite number was 136 because it was the first 3 triangular numbers concatenated together and was also triangular itself. I wondered whether there were any more numbers with this property.

I ran a search today and found no more sets of 3 consecutive triangular numbers that are triangular.

Is there any particular reason why this should be the case other than I just haven't searched far enough?

I suppose there is pretty good probabilistic argument for there to be no more. The probability of a random number x being triangular is [$]O(1/{\sqrt{2x}})[/$]. The size of the number that needs to be triangular is O(n^6) for the nth set of 3 consecutive triangular numbers concatenated together. I believe that the sum of the probabilities probably converges to a little above 1.

Is anyone able to beat my probabilistic argument for there to be no more?

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