A110621 Shadow of Pi.

1, 4, 18, 33, 42, 44, 50, 55, 90, 98, 195, 288, 311, 395, 457, 521, 859, 891, 898, 1848, 1876, 2717, 3688, 3757, 3796, 4733, 5243, 5301, 5321, 6295, 6389, 6434, 6526, 6556, 6634, 6650, 6690, 7318, 7938, 8027, 9013, 9293, 9327, 9409, 9462, 9883, 10053

Offset

1,2

Comments

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

Links

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

Example

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...)

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000796.

Sequence in context: A095823 A092116 A083969 * A124978 A031081 A009956

Adjacent sequences: A110618 A110619 A110620 * A110622 A110623 A110624

Keyword

easy,nonn,base

Author

Eric Angelini and Alexandre Wajnberg, Sep 14 2005

Extensions

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

Status

approved