A330129 a(n) is the last term of the analogous sequence to A121805, but with initial term n, or -1 if that sequence is infinite.

99999945, 9999999999999918, 36, 9999945, 999945, 9999999999999999936, 936, 9999999999972, 999999936, 999936, 999936, 99999945, 999954, 999918, 72, 99999918, 999999999927, 18, 999981, 999999999999999963, 99981, 999999999999999963, 999936, 9999999999999918, 9963

Offset: 1

Author

Giovanni Resta, Dec 02 2019

Keywords

• nonn
• base

Comments

The numbers of terms of the corresponding sequences are in A330128.

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

Crossrefs

Cf. A330128, A121805, A139284, A366492.

Programs

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]]]; oX]; Array[a, 30]

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 oX
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Nov 18 2023 after Giovanni Resta

Extensions

Escape clause added to definition by N. J. A. Sloane, Nov 14 2023