centered triangular numbers

Centered triangular numbers are figurate numbers (see picture aside) defined as $T'_n=3n(n-1)/2+1$

Two formulas involving centered triangular numbers :

$\sum_{n=1}^{\infty}\frac{1}{T'_n}= \frac{2\pi}{\sqrt{15}}\tanh\left( \frac{\pi}{2}\sqrt{\frac{5}{3}} \right)\,\quad\mbox{and}\quad\sum_{n=1}^{\infty}\frac{T'_n}{2^n}=7\,.$

The first centered triangular numbers are 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694 more terms

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

remainders of centered triangular numbers

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

divisibility of centered triangular numbers

Useful links Mathworld, Centered triangular Number
Wikipedia, Centered triangular number
OEIS, Sequence A000217

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

a-pointer 631 108139 145549 + 9890281801 ABA 64 aban 10 19 31 + 998188000939 abundant 460 760 2584 + 49654144 Achilles 78528691 admirable 1652176 8139856 12776464 + 86423536 alt.fact. 19 alternating 10 85 109 + 989232121 amenable 64 85 109 + 999814960 apocalyptic 361 1135 1219 + 29611 arithmetic 19 31 46 + 9996214 balanced p. 412651 1282051 4670191 + 9787991431 binomial 10 136 1891 + 2677903145191 brilliant 10 361 901 + 993551149 c.decagonal 31 361 14851 + 801550501782031 c.heptagonal 316 7246 3818431 + 558871623400861 c.nonagonal 10 136 1891 + 519499406175760 c.octagonal 361 6241 3463321 + 319312633001041 c.pentagonal 31 1891 117181 + 107213302675081 c.square 85 16381 3177721 + 23199907725541 cake 64 1004914 Carmichael 512461 3858853681 Chen 19 31 109 + 99988591 congruent 31 46 85 + 9996214 constructible 10 64 85 + 514 cube 64 Cunningham 10 31 901 + 796926124814401 cyclic 19 31 85 + 9988471 d-powerful 1306 2224 3385 + 9734635 de Polignac 50509 111385 176989 + 99866161 decagonal 10 85 976 + 864014364059701 deceptive 512461 1179740971 3858853681 deficient 10 19 31 + 9996214 dig.balanced 10 19 166 + 199843960 Duffinian 64 85 235 + 9988471 eban 46 64 30034 economical 10 19 31 + 19989226 emirp 31 199 9721 + 199878589 emirpimes 85 514 1135 + 99792739 equidigital 10 19 31 + 19989226 eRAP 197110 esthetic 10 45676 evil 10 46 85 + 999892414 fibodiv 19 199 1999 Fibonacci 2584 Friedman 16381 24385 54436 + 457885 frugal 6241 27136 212629 + 651177091 gapful 4960 9964 10210 + 99991177735 good prime 4621 1173511 1358029 + 166368739 happy 10 19 31 + 9988471 Harshad 10 460 2584 + 9973768960 heptagonal 235 2839 902101 + 160992923703385 hex 19 631 21421 + 32919051946411 hexagonal 1891 26335 71156485 + 37298379237211 hoax 85 136 166 + 99621526 Hogben 31 2971 291061 + 26835295698121 Honaker 10711 265231 867541 + 955548541 iban 10 274 2110 + 772210 iccanobiF 514 idoneal 10 85 760 impolite 64 inconsummate 30460 108139 237010 + 736051 interprime 64 274 12835 + 99230734 Jacobsthal 85 Jordan-Polya 64 junction 109 1219 1306 + 99890641 katadrome 10 31 64 + 7543210 Lehmer 85 1891 26335 + 501279470749 Lucas 199 lucky 31 235 361 + 9957529 magnanimous 85 136 316 31756 metadrome 19 46 136 + 4789 modest 19 46 109 + 1798076260 Moran 2710 22510 135451 + 95676274 nialpdrome 10 31 64 + 886666652110 nonagonal 46 228736 12750211 + 33478154327026 nude 2224 11224 22144 + 421421824 oban 10 19 85 + 976 octagonal 97921 1039585 130402572385 1384429704481 odious 19 31 64 + 999969871 Ormiston 623071 6744781 15858379 + 1966148731 palindromic 15151 45154 66466 + 479573060375974 palprime 98689 pancake 46 631 8779 + 173166468725254 panconsummate 10 31 85 + 571 pandigital 19 694 1054 + 377047756 pernicious 10 19 31 + 9988471 Perrin 10 persistent 10751862349 12467543089 13018597624 + 98017361254 Pierpont 19 109 plaindrome 19 46 136 + 1344555888889 Poulet 30889 512461 9371251 + 597565005525001 power 64 361 6241 + 3258631508224 powerful 64 361 6241 + 319312633001041 practical 64 460 760 + 9696460 prim.abundant 2584 22144 1652176 + 96060010 prime 19 31 109 + 999980205949 pseudoperfect 460 760 2584 + 975664 repfigit 19 repunit 31 85 2971 + 26835295698121 Ruth-Aaron 1754464 4086226 6556293985 + 953269039936 self 31 64 514 + 999272866 semiprime 10 46 85 + 99964099 Smith 85 166 274 + 99621526 sphenic 1054 1786 2110 + 99743806 square 64 361 6241 + 319312633001041 strong prime 631 4621 7039 + 99988591 super Niven 10 super-d 19 31 631 + 9988471 tau 136 4456 8104 + 999814960 tetrahedral 10 4960 428536 triangular 10 136 1891 + 519499406175760 tribonacci 274 truncatable prime 31 twin 19 31 109 + 998730919 uban 10 19 31 + 96000060000010 Ulam 316 409 1489 + 9934354 undulating 15151 unprimeable 514 1135 1684 + 9965260 untouchable 976 8326 10210 + 995116 upside-down 19 46 64 vampire 7036628359 wasteful 46 85 136 + 9996214 weak prime 19 31 109 + 99401611 weakly prime 5830255021 32411956519 Zuckerman 22144 Zumkeller 460 760 2584 + 99460 zygodrome 6644778888844 8882229992266 55777555332244 + 110055449944444