A279085 Number of distinct residues of triangular numbers mod 10^n.

1, 6, 44, 424, 4176, 41696, 416704, 4166784, 41666816

Offset

0,2

Comments

Number of distinct n-digit endings of triangular numbers A000217 (written in base 10).

Links

Table of n, a(n) for n=0..8.

Formula

(Empirical) a(n) = (5*10^n + (9 - 2*(-1)^n)*2^n)/12.

Example

The last digit of a triangular number is one of 0, 1, 3, 5, 6, or 8, so a(1) = 6. (To verify that no number from {2, 4, 7, 9} can be the last digit of a triangular number T, note that 8*T+1, which must be a square, would end with 7, 3, 7, or 3, respectively, but no square ends with 3 or 7.)
The 44 two-digit combinations with which a triangular number may end are 00, 01, 03, 05, 06, 10, 11, 15, 16, 20, 21, 25, 26, 28, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51, 53, 55, 56, 60, 61, 65, 66, 70, 71, 75, 76, 78, 80, 81, 85, 86, 90, 91, 95, 96. Of these, there are ten combinations of each of the forms x0, x1, x5, and x6; the other four, which are the only ones that end with a digit other than 0, 1, 5, or 6, are 03, 28, 53, and 78 (i.e., numbers whose residue modulo 25 is 3):
.
                            last digit
                   0  1  2  3  4  5  6  7  8  9
                +------------------------------
              0 | 00 01    03    05 06
              1 | 10 11          15 16
              2 | 20 21          25 26    28
next-to-last  3 | 30 31          35 36
    digit     4 | 40 41          45 46
              5 | 50 51    53    55 56
              6 | 60 61          65 66
              7 | 70 71          75 76    78
              8 | 80 81          85 86
              9 | 90 91          95 96

Crossrefs

Cf. A000217, A000993.

Sequence in context: A361649 A084965 A203159 * A124996 A226045 A001852

Adjacent sequences: A279082 A279083 A279084 * A279086 A279087 A279088

Keyword

nonn,base,more

Author

Jon E. Schoenfield, Jan 14 2017

Status

approved