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

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, 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, 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

Offset: 1

Author

Cino Hilliard, Jul 24 2003

Keywords

• easy
• nonn
• base

Comments

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

Examples

			Triangle is
  3
  14
  159
  2653
  58979
  323846
  2643383
  27950288
  419716939
  9375105820
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
		

Crossrefs

Cf. A000796, A083437.

Cf. A000030, A090897.

Programs

Mathematica

pi = RealDigits[Pi, 10, 5461][[1]]; Table[ pi[[n(n + 1)/2 + 1]], {n, 0, 104}]
Module[{nn=110,pid},pid=RealDigits[Pi,10,(nn(nn+1))/2][[1]];TakeList[ pid,Range[ nn]]][[;;,1]] (* Harvey P. Dale, Mar 06 2023 *)

Formula

a(n) = A000796(1-n(n-1)/2). - M. F. Hasler, Oct 20 2011

a(n) = A000030(A090897(n)) if (and probably only if) a(n) is nonzero. - Michel Marcus and M. F. Hasler, Jan 06 2023

Extensions

Edited by Robert G. Wilson v, Jul 26 2003