A230763 Length of segments delimited by the letter "e" when the sequence is spelled out in English.

6, 1, 8, 5, 2, 7, 4, 10, 1, 2, 2, 3, 7, 10, 2, 1, 2, 6, 19, 8, 10, 2, 2, 2, 2, 2, 20, 2, 10, 2, 4, 7, 11, 8, 2, 2, 2, 2, 5, 10, 2, 4, 7, 1, 2, 5, 2, 2, 2, 10, 2, 5, 10, 11, 2, 8, 2, 1, 10, 1, 6, 3, 10, 2, 11

Offset: 1

Author

Eric Angelini and M. F. Hasler, Oct 29 2013

Keywords

• nonn
• word

Comments

Translate the sequence S to English, to get English(S) = (SIX, ONE, EIGHT, FIVE, TWO, SEVEN, FOUR, TEN, ONE, TWO, TWO, THREE, SEVEN, TEN, TWO, ONE, TWO, SIX, NINETEEN, EIGHT, TEN, TWO, TWO, TWO, TWO, TWO, TWENTY, TWO, TEN, TWO, FOUR, SEVEN, ELEVEN, ...).

Then insert a stroke immediately after each letter "E" of S: "SIX, ONE|, E|IGHT, FIVE|, TWO, SE|VE|N, FOUR, TE|N, ONE|, TWO, TWO, THRE|E|, SE|VE|N, TE|N, TWO, ONE|, TWO, SIX, NINE|TE|E|N, E|IGHT, TE|N, TWO, TWO, TWO, TWO, TWO, TWE|NTY, TWO, TE|N, TWO, FOUR, SE|VE|N, E|LE|VE|N," ...

Now the length of each chunk is given by the sequence S itself. In case of several possible choices compatible with this property, we require the smallest one to be chosen.

Examples

			We see thus that the sequence cannot start with ONE, or TWO, or THREE, or FOUR or FIVE (because the presence or the absence of the letter "E" in those numbers leads to a contradiction). We could start with THIRTY, FORTY, FIFTY and SIXTY also -- but SIX is the smallest available integer. After SIX, we could also have chosen TWENTY, but again, we want the smallest integer fitting the pattern, which is ONE; therefore, the sequence starts: S = (SIX, ONE|, ...)
See the link to Angelini's page for further developments.
		

Links

• E. Angelini, Self-count of Letter-Chunks ending with E.
• E. Angelini, Self-count of Letter-Chunks ending with E [Cached copy, with permission]
• E. Angelini, Backtracking the letter E chunk, SeqFan List, Oct 22 2013

Crossrefs

Cf. A052360.

Programs

PARI

/* see A052360 for the function English(). We implement a function which "caches" the English words (as Vecsmall() of ASCII codes) in an array for faster treatment. */
VET=[];VE(n)={#VETL+a[j] || (( i==L+a[j] ) != (s[i] ==101))) && return; s[i] != 101 && next; L=i; j++)}
/* This extends the sequence 'a' by 'n' terms. Note, the last terms added may turn out to be not "valid", when the sequence is extended further. Could be programmed much more efficiently. */
extend(n,a=[],s=Vecsmall([]))={check(a,s)||return; n-- || return(a); my(t); for(k=1,99,(t=extend(n,concat(a,k),concat(s,VE(k))))&& return(t))}
/* To produce the sequence, use, e.g. */ extend(50)