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A086639 Write decimal expansion of Pi in triangular form; sequence gives left edge.

%I A086639 #27 Mar 06 2023 11:46:37
%S A086639 3,1,1,2,5,3,2,2,4,9,9,7,8,3,8,7,2,1,8,9,5,3,6,6,3,5,7,6,2,2,9,9,4,0,
%T A086639 4,2,3,0,4,1,6,7,8,9,9,1,2,3,0,1,7,2,2,4,7,8,3,1,8,3,0,2,7,9,1,6,2,2,
%U A086639 6,7,6,8,1,5,7,3,7,7,2,4,9,3,2,1,9,8,9,1,2,7,7,9,4,0,9,2,9,8,4,9,9,2,0,7,0
%N A086639 Write decimal expansion of Pi in triangular form; sequence gives left edge.
%C A086639 In the second formula, "if" can most probably be strengthened to "if and only if": Indeed, a(n) = 0 can be equal to A000030(A090897(n)) only if A090897(n) = 0, i.e., there would be a string of n consecutive zeros in the decimals of Pi from position T(n-1)+1 to position T(n). The probability that this happens appears to be zero. (Notice how A096764(n), first occurrence of n consecutive zeros, grows incredibly much faster than T(n).) Maybe this could be proved considering, e.g., a continued fraction expansion of Pi whose coefficients follow some pattern of moderate growth (as e.g. in A046126), while a very long string of zeros in the decimal expansion would mean that it is exceptionally close to the rational number given by the truncation. - _M. F. Hasler_, Jan 06 2023
%F A086639 a(n) = A000796(1-n(n-1)/2). - _M. F. Hasler_, Oct 20 2011
%F A086639 a(n) = A000030(A090897(n)) if (and probably only if) a(n) is nonzero. - _Michel Marcus_ and _M. F. Hasler_, Jan 06 2023
%e A086639 Triangle is
%e A086639   3
%e A086639   14
%e A086639   159
%e A086639   2653
%e A086639   58979
%e A086639   323846
%e A086639   2643383
%e A086639   27950288
%e A086639   419716939
%e A086639   9375105820
%e A086639 a(34) = 0 because in the decimals of Pi there is a 0 at position 562, following the triangular number A000217(33) = 561, i.e., in the first column of the 34th row in the above triangle. - _Michel Marcus_ and _M. F. Hasler_, Jan 06 2023
%t A086639 pi = RealDigits[Pi, 10, 5461][[1]]; Table[ pi[[n(n + 1)/2 + 1]], {n, 0, 104}]
%t A086639 Module[{nn=110,pid},pid=RealDigits[Pi,10,(nn(nn+1))/2][[1]];TakeList[ pid,Range[ nn]]][[;;,1]] (* _Harvey P. Dale_, Mar 06 2023 *)
%Y A086639 Cf. A000796, A083437.
%Y A086639 Cf. A000030, A090897.
%K A086639 easy,nonn,base
%O A086639 1,1
%A A086639 _Cino Hilliard_, Jul 24 2003
%E A086639 Edited by _Robert G. Wilson v_, Jul 26 2003