1,2

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

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

Michael S. Branicky, Table of n, a(n) for n = 1..10000

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

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

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

(Pi=3.141592653589793238462643383279502884197169399...)

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 *)

(Python)

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

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

from sympy import S

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

def aupton(terms):

global pi

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

while len(alst) < terms:

k = 1

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

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

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

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

idx += k

return alst

print(aupton(47)) # Michael S. Branicky, Jun 15 2021

Cf. A000796.

Sequence in context: A095823 A092116 A083969 * A124978 A031081 A009956

Adjacent sequences: A110618 A110619 A110620 * A110622 A110623 A110624

easy,nonn,base

Eric Angelini and Alexandre Wajnberg, Sep 14 2005

More terms from Robert G. Wilson v, Oct 10 2005

approved