Cumulative sum of matches used so far

This is the common typeface used hereunder for each of the 10 digits

Today, we want to produce a sequence S of integers a(n) where a(n) is the cumulative sum of the matches used so far in S.

S starts with 4 as 4 is the smallest available integer that uses 4 matches (S must be the lexicographically earliest sequence of distinct terms with this property). It follows:

S = 4, 7, 11, 19, 27.

S is impossible to extend after a(5) = 27.

Let's try to go further with a different start:

T = 5, 12, 17. End.

Backtracking for another start with a(1) = 5:

T = 5, 12, 22, 32. End.

Another start:

U = 6, 13, 23, 33, 42, 52, 63. End.

Well...

But our imagination is without limits! We had the hereunder (stupid and horrible) idea – which was to extend the above lexico-first S with a(6) = 42, like this:

S = 4, 7, 11, 19, 27. (45-3), ...

 (45 - 3 = 42)

Yes, I know, Goya was right, El sueño de la razon produce monstruos...

... this technique allows us to go further, and, probably, to extend S as far as we want (we will explain later how to submit this idea to the OEIS, using another dreadful technique!-)

S = 4, 7, 11, 19, 27. (45-3), 52, 63, (77-2), (95-3), 104, 113, 124, 138, 147, 157,...

 (77 - 2 = 75)

 (95 - 3 = 92)

If we forget the "cumulative" constraint, we can form self-describing subtractions:

14-2 (uses 12 matches)

15-2 (uses 13 matches)

16-2 (uses 14 matches)

18-3 (uses 15 matches)

19-5 (uses 14 matches)

20-3 (uses 17 matches)

23-6 (uses 17 matches)

27-10 (uses 17 matches)

28-8 (uses 20 matches)

...

But the number of such (a-b) subtractions is finite, I guess.

We could use additions instead of subtractions (the "plus" sign is made of two crossing matches) :

1+9 (uses 10 matches)

2+12 (uses 14 matches)

4+10 (uses 14 matches)

5+10 (uses 15 matches)

6+10 (uses 16 matches)

(If someone could compute all 2-term subtractions and additions, I would update this page – quoting him, of course.)

And now the "horrible" format of S mentioned above (S to be submitted to the OEIS?):

DATA

4, 7, 11, 19, 27, 45, -3, 52, 63, 77, -2, 95, -3, 104, 113, 124, 138, 147, 157,...

NAME

"a(n) is the cumulative sum of the matches used so far to form the sequence. In the Comments section we explain the negative terms and the typeface used to design the digits."

COMMENTS

The typeface used to design the digits is visible in Axxxxxx.

When the extension of the sequence is blocked [after a(5) = 27, no term is available for a(6)], we use the two-term subtraction (t - u) with "t" = a(k) and "u" = -a(k+1). The count of the matches involved in the subtraction takes into account the "flat" match used for the minus sign. This means that a negative term must not be seen as the cumulative sum of anything but only as a way to escape a blockade. Subtractions are used only if all other "natural" ways of extending the sequence have failed (including backtrackings).

EXAMPLE

a(1) = 4 as 4 uses 4 matches (in the selected typeface);

a(2) = 7 as we need 7 matches to form the succession a(1) and a(2);

a(3) = 11 as we need 11 matches to form the succession a(1), a(2) and a(3);

...

a(6) = 45 must be "corrected" as a(6) is followed by the negative term a(7);

a(7) = -3 "corrects" a(6);

a(8) = 52 as we need 52 matches to form the succession a(1), a(2), a(3)... a(8); etc.

____________________

Update

Scott Sh. wrote in private to me that there might be an infinite such sequence (without negative terms) in binary.