Editing Triangular number (section)

==Other properties==
Triangular numbers correspond to the first-degree case of [[Faulhaber's formula]].
	
{{annotated image
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|caption    =[[Proof without words]] that all hexagonal numbers are odd-sided triangular numbers
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Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers.

Every even [[perfect number]] is triangular (as well as hexagonal), given by the formula
<math display=block>M_p 2^{p-1} = \frac{M_p (M_p + 1)}2 = T_{M_p}</math>
where {{mvar|M<sub>p</sub>}} is a [[Mersenne prime]]. No odd perfect numbers are known; hence, all known perfect numbers are triangular.

For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.

The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.

In [[base 10]], the [[digital root]] of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:
{{Block indent|left=1.6|1={{Break lines|1=
0 = 9 × 0
1 = 9 × 0 + 1
3 = 9 × 0 + 3
6 = 9 × 0 + 6
10 = 9 × 1 + 1
15 = 9 × 1 + 6
21 = 9 × 2 + 3
28 = 9 × 3 + 1
36 = 9 × 4
45 = 9 × 5
55 = 9 × 6 + 1
66 = 9 × 7 + 3
78 = 9 × 8 + 6
91 = 9 × 10 + 1
...}}}}

The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".

The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.

If {{mvar|x}} is a triangular number, {{mvar|a}} is an odd square, and {{math|''b'' {{=}} {{sfrac|''a'' − 1|8}}}}, then {{math|''ax'' + ''b''}} is also a triangular number. Note that {{mvar|b}} will always be a triangular number, because {{math|1=8''T<sub>n</sub>'' + 1 = (2''n'' + 1)<sup>2</sup>}}, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for {{mvar|b}} given {{mvar|a}} is an odd square is the inverse of this operation.
The first several pairs of this form (not counting {{math|1''x'' + 0}}) are: {{math|9''x'' + 1}}, {{math|25''x'' + 3}}, {{math|49''x'' + 6}}, {{math|81''x'' + 10}}, {{math|121''x'' + 15}}, {{math|169''x'' + 21}}, ... etc. Given {{mvar|x}} is equal to {{mvar|T<sub>n</sub>}}, these formulas yield {{math|''T''<sub>3''n'' + 1</sub>}}, {{math|''T''<sub>5''n'' + 2</sub>}}, {{math|''T''<sub>7''n'' + 3</sub>}}, {{math|''T''<sub>9''n'' + 4</sub>}}, and so on.

The sum of the [[multiplicative inverse|reciprocals]] of all the nonzero triangular numbers is
<math display=block> \sum_{n=1}^\infty{1 \over {{n^2 + n} \over 2}} = 2\sum_{n=1}^\infty{1 \over {n^2 + n}} = 2 .</math>

This can be shown by using the basic sum of a [[telescoping series]]:
<math display=block> \sum_{n=1}^\infty{1 \over {n(n+1)}} = 1 .</math>

In addition, the ''n''th partial sum of this series can be written as {{sfrac|''2n''|''n'' + 1}}

Two other formulas regarding triangular numbers are
<math display=block>T_{a+b} = T_a + T_b + ab</math>
and
<math display=block>T_{ab} = T_aT_b + T_{a-1}T_{b-1},</math>
both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra. The first formula are relevant to [[multiplication algorithm#Quarter square multiplication]].

In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers, writing in his diary his famous words, "[[Eureka (word)|ΕΥΡΗΚΑ!]] {{nowrap|1=num = Δ + Δ + Δ}}". The three triangular numbers are not necessarily distinct, or nonzero; for example 20 = 10 + 10 + 0. This is a special case of the [[Fermat polygonal number theorem]].

The largest triangular number of the form {{math|2<sup>''k''</sup> −&thinsp;1}} is [[4000 (number)#4001 to 4099|4095]] (see [[Ramanujan–Nagell equation]]).

[[Wacław Sierpiński|Wacław Franciszek Sierpiński]] posed the question as to the existence of four distinct triangular numbers in [[geometric progression]]. It was conjectured by Polish mathematician [[Kazimierz Szymiczek]] to be impossible and was later proven by Fang and Chen in 2007.<ref>[http://www.emis.de/journals/INTEGERS/papers/h19/h19.pdf Chen, Fang: Triangular numbers in geometric progression]</ref><ref>[http://www.emis.de/journals/INTEGERS/papers/h57/h57.pdf Fang: Nonexistence of a geometric progression that contains four triangular numbers]</ref>

Formulas involving expressing an integer as the sum of triangular numbers are connected to [[theta function]]s, in particular the [[Ramanujan theta function]].<ref>{{Cite journal|last=Liu|first=Zhi-Guo|date=2003-12-01|title=An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers|journal=The Ramanujan Journal|language=en|volume=7|issue=4|pages=407–434|doi=10.1023/B:RAMA.0000012425.42327.ae|s2cid=122221070|issn=1382-4090}}</ref><ref>{{cite arXiv|last=Sun|first=Zhi-Hong|date=2016-01-24|title=Ramanujan's theta functions and sums of triangular numbers|eprint=1601.06378|class=math.NT}}</ref>

The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a [[recurrence relation]]:
<math display=block>L_n = 3 T_{n-1} = 3{n \choose 2};\qquad L_n = L_{n-1} + 3(n-1), ~L_1 = 0.</math>

In the [[Limit of a sequence|limit]], the ratio between the two numbers, dots and line segments is
<math display=block>\lim_{n\to\infty} \frac{T_n}{L_n} = \frac{1}{3}.</math>