A330128 a(n) is the number of terms in the analog of A121805 but starting with n, or -1 if that sequence is infinite.

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

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

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

Adjacent sequences: A330125 A330126 A330127 * A330129 A330130 A330131

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