A110621 - OEIS

%I #16 Jul 29 2023 14:36:37

%S 1,4,18,33,42,44,50,55,90,98,195,288,311,395,457,521,859,891,898,1848,

%T 1876,2717,3688,3757,3796,4733,5243,5301,5321,6295,6389,6434,6526,

%U 6556,6634,6650,6690,7318,7938,8027,9013,9293,9327,9409,9462,9883,10053

%N Shadow of Pi.

%C First differences are Pi's shadow. Never twice the same integer in sequence or first differences.

%C a(20) is 1848, not 993, since the latter would leave out the next 0 in Pi. - _Michael S. Branicky_, Jun 15 2021

%H Michael S. Branicky, <a href="/A110621/b110621.txt">Table of n, a(n) for n = 1..10000</a>

%e The first line hereunder is the sequence, the second line are the first differences:

%e 1.4..18..33.42.44.50.55..90.98..195..288..311...

%e .3.14..15..9..2..6..5..35..8..97...93...23 <-- Pi shadow

%e (Pi=3.141592653589793238462643383279502884197169399...)

%t a[1] = 1; a[n_] := a[n] = Block[{c = RealDigits[Pi, 10, 300][[1]], k = 1, t = Table[a[i], {i, n - 1}]}, d = Drop[t, 1] - Drop[t, -1]; b = Drop[c, Length[ Flatten[ IntegerDigits /@ d]]]; e = Union[ Join[t, d]]; While[f = FromDigits[ Take[b, k]]; Position[e, f] != {} || b[[k + 1]] == 0, k++ ]; f + a[n - 1]]; Table[ a[n], {n, 47}] (* _Robert G. Wilson v_ *)

%o (Python)

%o # download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then

%o # with open('pi-billion.txt', 'r') as f: pi = f.readline()

%o from sympy import S

%o pi = str(S.Pi.n(3*10**5)).replace('.', '') # alternatively

%o def aupton(terms):

%o     global pi

%o     alst, idx, seen = [1], 0, {0, 1}

%o     while len(alst) < terms:

%o         k = 1

%o         while int(pi[idx:idx+k]) in seen or pi[idx+k] == '0': k += 1

%o         diffn = int(digits_of_pi[idx:idx+k])

%o         alst.append(alst[-1] + diffn)

%o         seen |= {diffn, alst[-1]}

%o         idx += k

%o     return alst

%o print(aupton(47)) # _Michael S. Branicky_, Jun 15 2021

%Y Cf. A000796.

%K easy,nonn,base

%O 1,2

%A _Eric Angelini_ and _Alexandre Wajnberg_, Sep 14 2005

%E More terms from _Robert G. Wilson v_, Oct 10 2005