A229939 -id:A229939 - cp's OEIS frontend

A385994 Lexicographically greatest increasing expansion Pi = Sum_{n>=0} a(n)/10^n, where a(n+1) >= a(n).

2, 10, 12, 19, 23, 26, 29, 32, 40, 48, 50, 53, 53, 61, 62, 65, 74, 75, 79, 85, 86, 92, 95, 102, 111, 111, 115, 119, 128, 133, 134, 139, 144, 146, 151, 160, 165, 172, 179, 186, 190, 195, 197, 201, 206, 215, 219, 222, 229, 234, 243, 248, 250, 253, 261, 269, 276, 283, 287

Offset: 0

Pierre-Alain Sallard, Jul 14 2025

Each successive term is maximal consistent with the sum approaching Pi from below.

Each difference d = a(n) - a(n-1) (and reckoning an a(-1)=0) effectively repeats in all subsequent terms and so contributes (10/9)*d/10^n into the sum, and for that reason those differences are the decimal digits of (9/10)*Pi and the terms are partial sums of those digits.

Pierre-Alain Sallard, Proof

Cf. A000796.

Partial sums of A229939.

a[n_]:=Sum[Part[RealDigits[9*Pi, n+1][[1]],i],{i,1,n+1}]; Array[a,59,0] (* Stefano Spezia, Jul 14 2025 *)

a(n) = Sum_{i=0..n} A229939(i+1).

A265729 Decimal expansion of 32*Pi.

Original entry on oeis.org

1, 0, 0, 5, 3, 0, 9, 6, 4, 9, 1, 4, 8, 7, 3, 3, 8, 3, 6, 3, 0, 8, 0, 4, 5, 8, 8, 2, 6, 4, 9, 4, 4, 0, 9, 2, 2, 9, 4, 3, 0, 9, 4, 2, 0, 7, 8, 0, 0, 0, 3, 3, 8, 6, 2, 7, 1, 1, 9, 8, 2, 2, 6, 9, 5, 3, 8, 5, 0, 1, 2, 5, 0, 0, 1, 1, 5, 8, 6, 8, 7, 9, 5, 6, 0, 9, 7, 1, 1, 4, 4, 1, 0, 9, 4, 7, 7, 4, 6, 1, 7, 5, 4, 2, 8

Offset: 3

Eric W. Weisstein and Robert G. Wilson v, Dec 14 2015

"The integral corresponds to integration over a spherical cone with opening angle Pi/2 and radius 4. However, it is not clear what the integrand physically represents (it resembles computation of a moment of inertia, but that would give a factor (rho*sin(phi))^2 rather than the given rho*cos(phi))."

			100.53096491487338363080458826494409229430942078000338627119822695385012500...

The Jun 02 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007)

Eric Weisstein's World of Mathematics, Definite Integral
Index entries for transcendental numbers

Cf. A000796, A019692, A122952, A019694, A019669, A228719, A228721, A228824, A229939, A061146.

RealDigits[32 Pi, 10, 111][[1]] (* or *)
Integrate[\[Rho] Cos[\[Phi]] \[Rho]^2 Sin[\[Phi]], {\[Rho], 0, 4}, {\[Phi], 0, Pi/4}, {\[Theta], 0, 2 Pi}]

PARI
```
32*Pi \\ Altug Alkan, Dec 14 2015
```

Equals Integral_{theta=0..2*Pi} Integral_{phi=0..Pi/4} Integral_{rho=0..4} (rho*cos(phi))*rho^2*sin(phi) d(rho) d(phi) d(theta).

cp's OEIS Frontend

A385994 Lexicographically greatest increasing expansion Pi = Sum_{n>=0} a(n)/10^n, where a(n+1) >= a(n).

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