OFFSET
1,1
COMMENTS
The sum from one set of digits to the following set of digits equals the term. The first is the 5th triangular number: 15 = 1 + 2 + 3 + 4 + 5.
These are the positive integer solutions for the formula sum (x to y) = (10^k)*x + y, where 0 < x < y < 10^k for some k >= 1.
On the left hand side of this equation, the sum can be written as A000217(y) - A000217(x-1) = (x+y)*(1-x+y)/2, and the right hand side is the concatenation of the decimal digits of x and y.
The graph of the function is a hyperbola; the solutions are for positive x and y, where y does not "overlap" and add to x. The first 21 terms are all of the solutions for n = 1 to 4. n = 5 solutions add two 9-digit and six 10-digit terms.
Note the pattern 15 = sum (1 to 5); 1353 = sum (13 to 53); 133533 = sum (133 to 533); 13335333 = sum (1333 to 5333). This pattern continues: 1333353333 = sum (13333 to 53333); 133333533333 = sum (133333 to 533333); etc. These are not the next terms in the sequence, however (see A350994).
See A186076 for the case of a sum countdown from the more significant to less significant digits.
From Jinyuan Wang, Sep 14 2019: (Start)
All terms form the concatenation of x = (s+t+1)/2 - 10^k and y = (s-t+1)/2, where s*t = 100^k - 10^k, 10^(k-1) < (s-t+1)/2 < 10^k, and gcd(s, t) is an odd number.
Strictly speaking, 1301613 = 13 + 14 + ... + 1612 + 1613 does not meet the concatenation criterion. So 1301613 is not a term.
(End)
From Bernard Schott, Jan 26 2022: (Start)
Other subsequence pattern: 27, 1863, 178623, 17786223, 1777862223, ... where 17..78 +...+ 62..23 = 17..7862..23. (End)
LINKS
Jinyuan Wang, Table of n, a(n) for n = 1..18738 (first 142 terms from David A. Corneth)
Diophante, A1945 - Concaténations en tous genres (in French).
Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.
FORMULA
EXAMPLE
429 = 4 + 5 + 6 + ... + 28 + 29.
7119 = 7 + 8 + 9 + ... + 118 + 119.
3882813 = 388 + 389 + ... + 2812 + 2813.
MAPLE
# See "Astonishing Pairs of Numbers" article referenced above.
PROG
(PARI) do(s, t, k) = if(10^(k-1) < (s-t+1)/2 && (s-t+1)/2 < 10^k, (1-10^k+s)*(1+10^k-t)/2);
lista(nn) = {my(m, v=List()); for(k = 1, nn, fordiv(50^k - 5^k, s, t = (100^k-10^k)/s; if(m=do(s, t, k), listput(v, m)); if(m=do(2^k*s, t/2^k, k), listput(v, m)))); vecsort(Vec(v)); } \\ Jinyuan Wang, Aug 29 2019
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Matthew Goers, Feb 11 2011
EXTENSIONS
a(22)-a(29) from Matthew Goers, Apr 11 2013
STATUS
approved