0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in the OEIS)
Explicit formulasedit
Write Nk for the kth square triangular number, and write sk and tk for the sides of the corresponding square and triangle, so that
Define the triangular root of a triangular number N = n(n + 1)/2 to be n. From this definition and the quadratic formula,
Therefore, N is triangular (n is an integer) if and only if8N + 1 is square. Consequently, a square number M2 is also triangular if and only if 8M2 + 1 is square, that is, there are numbers x and y such that x2 − 8y2 = 1. This is an instance of the Pell equation with n = 8. All Pell equations have the trivial solution x = 1, y = 0 for any n; this is called the zeroth solution, and indexed as (x0, y0) = (1,0). If (xk, yk) denotes the kth nontrivial solution to any Pell equation for a particular n, it can be shown by the method of descent that
Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever n is not a square. The first non-trivial solution when n = 8 is easy to find: it is (3,1). A solution (xk, yk) to the Pell equation for n = 8 yields a square triangular number and its square and triangular roots as follows:
Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from 6 × (3,1) − (1,0) = (17,6), is 36.
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers:[7] If the nth triangular number n(n + 1)/2 is square, then so is the larger 4n(n + 1)th triangular number, since:
As the product of three squares, the right hand side is square. The triangular roots tk are alternately simultaneously one less than a square and twice a square if k is even, and simultaneously a square and one less than twice a square if k is odd. Thus,
49 = 72 = 2 × 52 − 1,
288 = 172 − 1 = 2 × 122, and
1681 = 412 = 2 × 292 − 1.
In each case, the two square roots involved multiply to give sk: 5 × 7 = 35, 12 × 17 = 204, and 29 × 41 = 1189.[citation needed]
Additionally:
36 − 1 = 35, 1225 − 36 = 1189, and 41616 − 1225 = 40391. In other words, the difference between two consecutive square triangular numbers is the square root of another square triangular number.[citation needed]
As k becomes larger, the ratio tk/sk approaches √2 ≈ 1.41421356, and the ratio of successive square triangular numbers approaches (1 + √2)4= 17 + 12√2 ≈ 33.970562748. The table below shows values of k between 0 and 11, which comprehend all square triangular numbers up to 1016.
k
Nk
sk
tk
tk/sk
Nk/Nk − 1
0
0
0
0
1
1
1
1
1
2
36
6
8
1.33333333
36
3
1225
35
49
1.4
34.027777778
4
41616
204
288
1.41176471
33.972244898
5
1413721
1189
1681
1.41379310
33.970612265
6
48024900
6930
9800
1.41414141
33.970564206
7
1631432881
40391
57121
1.41420118
33.970562791
8
55420693056
235416
332928
1.41421144
33.970562750
9
1882672131025
1372105
1940449
1.41421320
33.970562749
10
63955431761796
7997214
11309768
1.41421350
33.970562748
11
2172602007770041
46611179
65918161
1.41421355
33.970562748
See alsoedit
Cannonball problem, on numbers that are simultaneously square and square pyramidal
Sixth power, numbers that are simultaneously square and cubical
^ abcEuler, Leonhard (1813). "Regula facilis problemata Diophantea per numeros integros expedite resolvendi (An easy rule for Diophantine problems which are to be resolved quickly by integral numbers)". Mémoires de l'Académie des Sciences de St.-Pétersbourg (in Latin). 4: 3–17. Retrieved 2009-05-11. According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.
^
Barbeau, Edward (2003). Pell's Equation. Problem Books in Mathematics. New York: Springer. pp. 16–17. ISBN 978-0-387-95529-2. Retrieved 2009-05-10.
^Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press. p. 210. ISBN 0-19-853171-0. Theorem 244
^
Pietenpol, J. L.; Sylwester, A. V.; Just, Erwin; Warten, R. M. (February 1962). "Elementary Problems and Solutions: E 1473, Square Triangular Numbers". American Mathematical Monthly. Mathematical Association of America. 69 (2): 168–169. doi:10.2307/2312558. ISSN 0002-9890. JSTOR 2312558.
^Plouffe, Simon (August 1992). "1031 Generating Functions" (PDF). University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. Archived from the original (PDF) on 2012-08-20. Retrieved 2009-05-11.
External linksedit
Triangular numbers that are also square at cut-the-knot