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 Consecutive Triangles (Posted on 2009-11-29)
A, B, C and D are triangular numbers.

A, B and C are always consecutive while D is their sum.

Determine (and explain as best as possible1) how such sets of values are distributed across the number system.

1. This can be explained in terms of a single variable expression.

 See The Solution Submitted by brianjn No Rating

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 computer-aided solution | Comment 3 of 8 |

DECLARE FUNCTION isTri# (t#)
DECLARE FUNCTION tr# (n#)
DEFDBL A-Z
tr1 = 1
tr2 = 3
tr3 = 6
sum = tr1 + tr2 + tr3
DO
IF isTri(sum) THEN PRINT tr1; tr2; tr3, isTri(tr1); isTri(tr2); isTri(tr3), sum; isTri(sum): PRINT
sum = sum - tr1 + tr4
tr1 = tr2: tr2 = tr3: tr3 = tr4
LOOP

FUNCTION isTri (t)
n = INT(SQR(t * 2))
np = n + 1
IF n * np = 2 * t THEN isTri = n:  ELSE isTri = 0
END FUNCTION

FUNCTION tr (n)
tr = n * (n + 1) / 2
END FUNCTION

`finds                                           Among triangular numbers                                                    Ordinal ofA  B  C                                               A  B  C                          D  Ord of D`
`1  3  6                                               1  2  3                         10  4`
`36  45  55                                            8  9  10                       136  16`
`595  630  666                                        34  35  36                     1891  61                                                                            8646  8778  8911                                    131  132  133                  26335  229`
`121771  122265  122760                              493  494  495                 366796  856`
`1701090  1702935  1704781                          1844  1845  1846              5108806  3196`
`23711941  23718828  23725716                       6886  6887  6888             71156485  11929`
`330334956  330360660  330386365                   25703  25704  25705          991081981  44521`
`4601234485  4601330415  4601426346                95929  95930  95931        13803991246  166156`
`64087907136  64088265153  64088623171            358016  358017  358018     192264795460  620104`
`892633045591  892634381730  892635717870        1336138  1336139 1336140   2677903145191  2314261`
`12432788092530  12432793079070  12432798065611  4986539  4986540 4986541  37298379237211  8636941`

The ordinal triangular number of A, follows Sloane's A082840, which has a comment by our own brianjn, based upon this problem. The ordinal position of D is given by A133161.

The formula given in Sloane for the ordinal of A (i.e., the nth triangular number), is:

With a=2+sqrt(3), b=2-sqrt(3)
a(n)=-3/2+(1/12)(a-2b+5)a^n+(1/12)(b-2a+5)b^n.

To test that out:

DEFDBL A-Z
CLS
a = 2 + SQR(3): b = 2 - SQR(3)
FOR n = 1 TO 10
ans = -3 / 2 + (1 / 12) * (a - 2 * b + 5) * a ^ n + (1 / 12) * (b - 2 * a + 5) * b ^ n
PRINT n, ans; TAB(38); INT(ans + .000000001#)
NEXT

finds:

`1             .9999999999999999      12             7.999999999999999      83             34                     344             131                    1315             492.9999999999999      4936             1844                   18447             6885.999999999998      68868             25702.99999999999      257039             95928.99999999997      9592910            358015.9999999999      358016`

The successive 9's show the rounded value at the right is the actual value.

 Posted by Charlie on 2009-11-29 19:04:46

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