centered square numbers

Centered square numbers are figurate numbers (see picture aside) defined as $Q'_n=n^2+(n-1)^2$

Two formulas involving centered square numbers :

$\sum_{n=1}^{\infty}\frac{1}{Q'_n}= \frac{1}{2}\pi\tanh\left(\frac{\pi}{2}\right)\quad\mbox{and}\quad\sum_{n=1}^{\infty}\frac{Q'_n}{2^n}=9\,.$

The first centered square numbers are 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013 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 centered square numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Centered square Number
Wikipedia, Centered square number
OEIS, Sequence A001844

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

a-pointer 13 181 2521 + 9699319921 aban 13 25 41 + 995778000313 Achilles 9870125 alternating 25 41 61 + 987412361 amenable 13 25 41 + 999983921 apocalyptic 841 925 1105 + 29525 arithmetic 13 41 61 + 9994921 automorphic 25 balanced p. 15313 20201 156241 + 9992597081 brilliant 25 221 481 + 999894481 c.decagonal 61 1201 87781 + 263205560604901 c.heptagonal 841 43513 54236113 + 181102250526121 c.octagonal 25 841 28561 + 43892069261881 c.pentagonal 181 58141 18721081 + 625008553923781 c.triangular 85 16381 3177721 + 23199907725541 Carmichael 1105 Chen 13 41 113 + 99531941 congruent 13 41 61 + 9985981 constructible 85 Cullen 25 Cunningham 145 4901 166465 + 255821727047185 Curzon 41 113 221 + 199540265 cyclic 13 41 61 + 9994921 d-powerful 2245 9385 24865 + 9843485 de Polignac 1985 6161 7813 + 99899113 decagonal 85 1105 98125 + 150798234220045 deceptive 481 4141 27613 + 59030823601 deficient 13 25 41 + 9994921 dig.balanced 41 613 841 + 199460365 Duffinian 25 85 221 + 9994921 economical 13 25 41 + 19964881 emirp 13 113 761 + 198622381 emirpimes 85 221 265 + 99701321 equidigital 13 25 41 + 19964881 esthetic 545 Eulerian 1013 evil 85 113 221 + 999983921 fibodiv 61 1301 8845 Fibonacci 13 Friedman 25 3281 16381 + 683281 frugal 28561 325625 1041125 + 950698013 gapful 4141 6161 21425 + 89971547405 good prime 41 1861 2521 + 140868113 happy 13 313 365 + 9985981 Harshad 481 20605 26221 + 9968014013 heptagonal 4141 85285 429332905 + 916808615026525 hex 61 5941 582121 + 53670591396241 hoax 85 265 2965 + 97482685 Hogben 13 421 14281 + 745519146510613 Honaker 1301 86113 97241 + 959176201 house 1582421 hyperperfect 96361 iban 41 221 421 + 770041 iccanobiF 13 idoneal 13 25 85 inconsummate 761 3281 4705 + 989825 interprime 5513 6613 20605 + 99870845 Jacobsthal 85 junction 113 313 925 + 99870845 katadrome 41 61 85 + 8321 Lehmer 85 481 1105 + 964634109181 Leyland 145 lucky 13 25 421 + 9994921 m-pointer 61 magic 1105 magnanimous 25 41 61 + 88621 metadrome 13 25 145 modest 13 545 39481 + 950174825 Moran 481 26221 82825 + 98434481 nialpdrome 41 61 85 + 755221 nonagonal 3961 58825 255538225 + 244832844124441 oban 13 25 85 + 925 octagonal 481 8321 4617761 + 425750177334721 odious 13 25 41 + 999894481 Ormiston 19013 37813 277513 + 1986075313 palindromic 181 313 545 + 582818040818285 palprime 181 313 3187813 panconsummate 61 85 481 pandigital 21425 27145 42925 pentagonal 145 925 28085 + 254088532692545 pernicious 13 25 41 + 9968113 persistent 12530869741 13285967041 13978421605 + 97738142065 Pierpont 13 plaindrome 13 25 113 + 22222224444445 Poulet 1105 8321 107185 + 444733131907121 power 25 841 28561 + 43892069261881 powerful 25 841 28561 + 43892069261881 prime 13 41 61 + 999982234201 primeval 13 113 1013 Proth 13 25 41 + 549756862465 repfigit 61 repunit 13 85 421 + 745519146510613 Rhonda 836197345261 Ruth-Aaron 25 572471285 35401677961 51193920181 self 613 1513 1985 + 999805045 semiprime 25 85 145 + 99983941 sliding 25 925 3445 Smith 85 265 2965 + 97482685 Sophie Germain 41 113 761 + 9997687013 sphenic 1105 2665 3445 + 99955661 square 25 841 28561 + 43892069261881 star 13 181 2521 + 692665874901013 strobogrammatic 181 strong prime 41 613 1301 + 99757813 subfactorial 265 super-d 181 481 2381 + 9985981 tribonacci 13 trimorphic 25 truncatable prime 13 113 313 + 5661613 twin 13 41 61 + 999983921 uban 13 25 41 + 98000098000025 Ulam 13 145 221 + 9790313 undulating 181 313 545 + 6161 unprimeable 3785 5725 9385 + 9959185 upside-down 7928552813 98941355796121 wasteful 85 221 481 + 9994921 weak prime 13 61 113 + 99531941 weakly prime 18539484241