

A309507


Number of ways the nth triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.


7



0, 1, 1, 1, 3, 3, 1, 2, 5, 3, 3, 3, 3, 7, 3, 1, 5, 5, 3, 7, 7, 3, 3, 5, 5, 7, 7, 3, 7, 7, 1, 3, 7, 7, 11, 5, 3, 7, 7, 3, 7, 7, 3, 11, 11, 3, 3, 5, 8, 11, 7, 3, 7, 15, 7, 7, 7, 3, 7, 7, 3, 11, 5, 3, 15, 7, 3, 7, 15, 7, 5, 5, 3, 11, 11, 7, 15, 7, 3, 9, 9, 3, 7
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OFFSET

1,5


COMMENTS

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(n+1) + k*(k+1)  m*(m+1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A053141.  Bradley Klee, Mar 01 2020; edited by N. J. A. Sloane, Mar 31 2020


LINKS



FORMULA

a(n) = 1 <=> n in { A068194 } \ { 1 }.
a(n) is even <=> n in { A001108 } \ { 0 }.
a(n) = number of odd divisors of n*(n+1) (or, equally, of T(n)) that are greater than 1.  N. J. A. Sloane, Apr 03 2020


EXAMPLE

a(5) = 3: T(5) = T(6)T(3) = T(8)T(6) = T(15)T(14).
a(7) = 1: T(7) = T(28)T(27).
a(8) = 2: T(8) = T(13)T(10) = T(36)T(35).
a(9) = 5: T(9) = T(10)T(4) = T(11)T(6) = T(16)T(13) = T(23)T(21) = T(45)T(44).
a(49) = 8: T(49) = T(52)T(17) = T(61)T(36) = T(94)T(80) = T(127)T(117) = T(178)T(171) = T(247)T(242) = T(613)T(611) = T(1225)T(1224).
The triples with n <= 16 are:
2, 2, 3
3, 5, 6
4, 9, 10
5, 3, 6
5, 6, 8
5, 14, 15
6, 5, 8
6, 9, 11
6, 20, 21
7, 27, 28
8, 10, 13
8, 35, 36
9, 4, 10
9, 6, 11
9, 13, 16
9, 21, 23
9, 44, 45
10, 8, 13
10, 26, 28
10, 54, 55
11, 14, 18
11, 20, 23
11, 65, 66
12, 17, 21
12, 24, 27
12, 77, 78
13, 9, 16
13, 44, 46
13, 90, 91
14, 5, 15
14, 11, 18
14, 14, 20
14, 18, 23
14, 33, 36
14, 51, 53
14, 104, 105
15, 21, 26
15, 38, 41
15, 119, 120


MATHEMATICA

TriTriples[TNn_] := Sort[Select[{TNn, (TNn + TNn^2  #  #^2)/(2 #),
(TNn + TNn^2  # + #^2)/(2 #)} & /@
Complement[Divisors[TNn (TNn + 1)], {TNn}],
And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[TriTriples[#]] & /@ Range[100]


CROSSREFS

See also A053141. The monotonic triples [n,k,m] with n <= k <= m are counted in A333529.


KEYWORD

nonn


AUTHOR



STATUS

approved



