A061233 Pierce expansion for 4 - Pi.

1, 7, 112, 115, 157, 372, 432, 1340, 7034, 8396, 9200, 18846, 29558, 34050, 89754, 101768, 1361737, 48461857, 81164005, 145676139, 163820009, 182446527, 5021656281, 8401618827, 22255558907, 28334352230, 127113921970, 310272097461, 782301280193, 5560255100022, 9925600136870, 85169484256928, 2542699818508737, 3145584963639199, 397021758001902006, 467746771316089905

Offset: 0

Author

Frank Ellermann, May 15 2001

Keywords

• nonn
• easy
• nice

Comments

Also, alternating Engel expansion for Pi.

Pi = 4 - 1/1 + 1/(1*7) - 1/(1*7*112) + 1/(1*7*112*115) - ...

Pierce expansions are always strictly increasing.

Links

• G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 0..1000 (terms 0 to 400 computed by T. D. Noe; terms 401 to 1000 computed by G. C. Greubel, Dec 31 2016)
• Eric Weisstein's World of Mathematics, Pierce Expansion
• Index entries for sequences related to Engel expansions

Crossrefs

A014014 and A015884 are inferior versions of this sequence.

Cf. A154956 (analog for 2/Pi).

Programs

Maple

Digits := 1000: x0 := 4-Pi-4^(-1000): x1 := 4-Pi+4^(-1000): ss := []: # when expansions of x0 and x1 differ, halt
k0 := floor(1/x0): k1 := floor(1/x1): while k0=k1 do ss := [op(ss),k0]: x0 := 1-k0*x0: x1 := 1-k1*x1: k0 := floor(1/x0): k1 := floor(1/x1): od:

Mathematica

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[4 - Pi, 7!], 25] (* G. C. Greubel, Dec 31 2016 *)

PARI

A061233(N=199)={localprec(N); my(c=4-Pi, d=c+c/10^N, a=[1\c]); while(a[#a]==1\d&&c=1-c*a[#a], d=1-d*a[#a]; a=concat(a, 1\c)); a[^-1]} \\ optional arg fixes precision, roughly equal to total number of digits in the result. - M. F. Hasler, Nov 24 2020

Extensions

More terms from Eric Rains (rains(AT)caltech.edu), May 31 2001