Limit_{n->infinity} 2*n*a(n-1)/a(n) = Pi, for Boustrophedon transform sequences that don't grow too much?

[Davide Rotondo]

A000667 Boustrophedon transform of all-1's sequence.

1, 2, 4, 9, 24, 77, 294, 1309, 6664, 38177, 243034, 1701909, 13001604, 107601977, 959021574, 9157981309, 93282431344, 1009552482977, 11568619292914, 139931423833509, 1781662223749884, 23819069385695177, 333601191667149054, 4884673638115922509

Dear seq fans I found really interesting the formula Limit_{n->infinity} 2*n*a(n-1)/a(n) = Pi by Gerald McGarvey and I wonder if this formula can be proper of all sequences where Boustrophedon transform is applied that don't grow too much. This because if I'm not wrong this formula is valid for A000747 too and Fibonacci and others.

What do you think

Regards

Davide

[Md. Rad Sarar Anando]

Dear,
Pi arises from the location of the dominant singularity, not from the Boustrophedon transform itself. Though your idea is plausible but it's only upto some specific observation. Some alternative sequences like Transforms of fast-growing sequences, Transforms producing multiple competing singularities, Sequences with oscillatory or stretched-exponential growth don't follow your idea fully rather act as a coincidence. But nice observation!

Regards,
Rad

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[Davide Rotondo]

Thanks a lot Rad!

Davide

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