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
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, Fibonacci Quarterly 62:3 (2024), 215-232.
- Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
- N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides
Crossrefs
Programs
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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] -
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
Extensions
Escape clause added to definition by N. J. A. Sloane, Nov 14 2023
Comments