triangular numbers

Triangular numbers are defined as $T_n=n(n+1)/2$

and are among the simplest figurate numbers (see picture aside).

Gauss proved that every number is the sum of at most 3 triangular numbers. Ming showed that the only triangular numbers which are also Fibonacci numbers are 1, 3, 21 and 55.

N.Tzanahis B.M.M. de Weger proved that there are six $T_n$ which are equal to the product of 3 consecutive integers: $T_3$ , $T_{15}$ , $T_{20}$ , $T_{44}$ , $T_{608}$ and

$T_{22736}=258474216=636\cdot 637\cdot638.$

The first nontrivial palindromic $T_n$

with palindromic index are $T_{11}=66$ , $T_{77}=3003$ , $T_{1111}=617716$ and $T_{2662}=3544453$ . The largest know such number, found by P.De Geest, is $T_{3654345456545434563}$ .

There are several interesting formulas involving triangular numbers, for example:

$\begin{array}{lcl} T_n+T_{n-1}=n^2\,, & & T_{2n}=3\cdot T_n+T_{n+1}\,,\\ T_n^2-T_{n-1}^2=n^3\,, & & T_{3n+1}=9\cdot T_n+1\,,\\ T_n^2=\sum_{k=1}^nk^3\,, && \int_0^1\int_0^1|x-y|^n\,dx\,dy=\frac{1}{T_{n+1}}\\ T_{k^2}=T_{k}^2+T_{k-1}^2\,,& \end{array}$

The sum of the reciprocals of triangular number is 2.

By solving the diophantine equation $x^2=y(y+1)/2$ , it is easy to find infinite triangular numbers which are also square. The first are 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056,...

The 3 smallest primes which concatenated with the previous prime gives a triangular number are 1812341, 624403308264975451 and 48127684695939820823. For this 3rd value we have

$\underline{48127684695939820823}\,\underline{48127684695939820753}=T_{98109820809070710302}.$

Below, the spiral pattern of triangular numbers up to $256^2$ . See the page on prime numbers for an explanation and links to similar pictures.

The first triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many triangular numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Triangular Number
Wikipedia, Triangular number
OEIS, Sequence A000217

Triangular numbers can also be... (you may click on names or numbers and on + to get more values)

aban 10 15 21 + 999949000753 abundant 36 66 78 + 49995000 Achilles 29403 74024028 27665282700 + 1665678432600 admirable 66 78 120 + 523776 alternating 10 21 36 + 985036305 amenable 21 28 36 + 999961560 amicable 122265 apocalyptic 528 666 820 + 29890 arithmetic 15 21 45 + 9997156 astonishing 15 Bell 15 binomial 10 15 21 + 10000002437316 brilliant 10 15 21 + 998977951 c.decagonal 1711 2467531 3558178261 5130890585101 c.heptagonal 253 64261 16322041 + 267457893082381 c.nonagonal 10 28 55 + 999999953042761 c.octagonal 1225 1413721 1631432881 1882672131025 c.pentagonal 276 1891 88831 + 954928097357851 c.triangular 10 136 1891 + 519499406175760 cake 15 378 Carmichael 561 8911 10585 + 921323712961 congruent 15 21 28 + 9992685 constructible 10 15 120 + 2147516416 Cunningham 10 15 28 + 937385441796000 Curzon 21 78 105 + 199970001 cyclic 15 91 435 + 9992685 D-number 15 21 1953 + 5572791 d-powerful 153 378 1326 + 9792525 de Polignac 46971 79003 93961 + 99934453 decagonal 10 1540 11935 + 18342198104230 deceptive 91 703 8911 + 99851537521 deficient 10 15 21 + 9992685 dig.balanced 10 15 21 + 199850028 double fact. 15 105 Duffinian 21 36 55 + 9956953 eban 36 66 36046 + 32064036004000 economical 10 15 21 + 19961721 emirpimes 15 1891 3403 + 97531561 equidigital 10 15 21 + 19961721 eRAP 29938677951 412156185486 esthetic 10 21 45 + 5676765 Eulerian 66 120 evil 10 15 36 + 999916840 factorial 120 fibodiv 28 Fibonacci 21 55 Friedman 153 46665 94395 + 857395 frugal 117855 118341 130816 + 957053125 gapful 105 120 190 + 99994143600 Gilda 78 990 happy 10 28 91 + 9988215 harmonic 28 496 8128 + 137438691328 Harshad 10 21 36 + 9999171820 heptagonal 55 121771 5720653 + 593128762435 hex 91 8911 873181 + 80505887094361 hexagonal 15 28 45 + 999999997764120 highly composite 36 120 25200 2162160 hoax 136 861 4656 + 99793128 Hogben 21 91 703 + 160829915470003 house 78 7759830 hyperperfect 21 28 325 + 137438691328 iban 10 21 120 + 742371 idoneal 10 15 21 + 253 inconsummate 276 630 861 + 980700 interprime 15 21 45 + 99906180 Jacobsthal 21 171 Jordan-Polya 36 120 junction 105 210 406 + 99892045 Kaprekar 45 55 703 + 50000005000000 katadrome 10 21 91 + 986310 Lehmer 15 91 435 + 999406030321 lonely 120 Lucas 5778 lucky 15 21 105 + 9988215 Lynch-Bell 15 36 1326 + 416328 magic 15 magnanimous 21 136 6824665 metadrome 15 28 36 + 2346 modest 406 666 3081 + 1984027528 Moran 21 45 153 + 666 Motzkin 21 narcissistic 153 nialpdrome 10 21 55 + 22222221111111 nonagonal 325 82621 20985481 + 343874433963061 nude 15 36 55 + 488234376 O'Halloran 36 oban 10 15 28 + 990 octagonal 21 11781 203841 + 18793835590881 odious 21 28 55 + 999961560 palindromic 55 66 171 + 684866959668486 panconsummate 10 15 21 + 231 pandigital 15 21 78 + 9654871320 partition 15 231 pentagonal 210 40755 7906276 + 57722156241751 perfect 28 496 8128 + 137438691328 pernicious 10 21 28 + 9939111 Perrin 10 persistent 1520843976 2718609453 3574816290 + 97486512903 plaindrome 15 28 36 + 355555577777778 Poulet 561 2701 4371 + 999961448323753 power 36 1225 41616 + 1882672131025 powerful 36 1225 29403 + 194838725161125 practical 28 36 66 + 9965880 prim.abundant 66 78 4095 + 57571815 primorial 210 pronic 210 7140 242556 + 372759573255306 pseudoperfect 28 36 66 + 33550336 repdigit 55 66 666 repfigit 28 repunit 15 21 91 + 231627523606480 Rhonda 244856171115 Ruth-Aaron 15 78 105 + 921236838753 Saint-Exupery 780 25743900 Sastry 528 self 378 435 703 + 999246160 semiprime 10 15 21 + 99214741 Smith 378 666 861 + 99906180 sphenic 66 78 105 + 99934453 square 36 1225 41616 + 63955431761796 star 253 49141 9533161 + 69599723176621 straight-line 210 630 666 741 super Niven 10 36 120 + 20000100000 super-d 105 190 231 + 9916831 superabundant 36 120 25200 2162160 tau 36 136 276 + 999246160 taxicab 61249825000 174882006936 tetrahedral 10 120 1540 7140 tetranacci 15 uban 10 15 21 + 98000091000021 Ulam 28 36 253 + 9956953 undulating 171 595 5050 5151 unprimeable 325 2485 3240 + 9997156 untouchable 120 210 276 + 990528 upside-down 28 55 91 + 485628284526 vampire 14340690 1465813440 1658332845 + 7589181600 wasteful 28 36 45 + 9997156 Wieferich 6161805 Zeisel 105 Zuckerman 15 36 1176 + 911111328 Zumkeller 28 66 78 + 98790 zygodrome 55 66 666 + 558822550004400