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A330128
a(n) is the number of terms in the analog of A121805 but starting with n, or -1 if that sequence is infinite.
18
2137453, 194697747222394, 2, 199900, 19706, 209534289952018960, 15, 198104936410, 19511030, 20573, 20572, 2137452, 20534, 19238, 2, 2089707, 20670629294, 1, 21482, 19278442756937613, 2074, 19278442756937612, 20571, 194697747222393, 193, 197062, 1, 197, 2061823
OFFSET
1,1
COMMENTS
The final terms of the corresponding sequences are given in A330129.
LINKS
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube
MATHEMATICA
nxt[x_] := Block[{p=1, n=x}, While[n >= 10, n = Floor[n/10]; p *= 10]; p (n + 1)]; a[n_] := Block[{nT=1, nX=n, w1, w2, w3, x, it, stp, oX}, stp = 100; w1 = w2 = w3 = 0; While[True, oX = nX; nT++; x = 10*Mod[oX, 10]; nX = SelectFirst[Range[9], IntegerDigits[oX + x + #][[1]] == # &, 0]; If[nX == 0, Break[], nX = nX + oX + x]; If[nT == stp, stp += 100; w1=w2; w2=w3; w3=nX; If[w3 + w1 == 2 w2 && Mod[w3 - w2, 100] == 0, it = Floor[(nxt[nX] - nX - 1)/(w3 - w2)]; nT += it*100; nX += (w3 - w2)*it; w3=nX; stp += it*100]]]; nT - 1]; Array[a, 30]
PROG
(Python)
def nxt(x):
p, n = 1, x
while n >= 10:
n //= 10
p *= 10
return p * (n + 1)
def a(n):
nT, nX, w1, w2, w3, stp = 1, n, 0, 0, 0, 100
while True:
oX = nX
nT += 1
x = 10*(oX%10)
nX = next((y for y in range(1, 10) if str(oX+x+y)[0] == str(y)), 0)
if nX == 0: break
else: nX += oX + x
if nT == stp:
stp += 100
w1, w2, w3 = w2, w3, nX
if w3 + w1 == 2*w2 and (w3 - w2)%100 == 0:
it = (nxt(nX) - nX - 1)//(w3 - w2)
nT += it*100
nX += (w3 - w2)*it
w3 = nX
stp += it*100
return nT - 1
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Nov 18 2023 after Giovanni Resta
CROSSREFS
Cf. A330129 (corresponding last term), A121805, A139284, A366492.
For record high points see A367364 and A367365.
Sequence in context: A231198 A237224 A244072 * A367601 A367364 A367602
KEYWORD
nonn,base
AUTHOR
Giovanni Resta, Dec 02 2019
EXTENSIONS
Escape clause added to definition by N. J. A. Sloane, Nov 14 2023
STATUS
approved