# »  Solutions to puzzles in my National Review Online Diary

## December 2008

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As usual at year end, your task in the December Math Corner was to find something interesting to say about the number 2009.

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Solutions

Several readers resorted to the Online Encyclopedia of Integer Sequences. This never occurs to me: which is odd, as I own the original (1995) book, which I am pretty sure predated the online version.

Here is the most comprehensive of those reader responses. The "Annnnnn" preceding each entry is the number of the relevant sequence in the encyclopedia.

• Miscellany:

A009720:   2009 is the 7th derivative of tan(tanh(x)*cos(x)) at x=0

A046735:   2009 does not divide any Tribonacci number

A051336:   There are 2009 arithmetic progressions in {1,2,3,...,35}

A067593:   There are 2009 partitions of 39 into Lucas numbers

• Possible Puzzles:

A025405, A025409 etc:
Trivial partitioning of 2009 into 4 cubes: 13 + 23 + 103 + 103.  Knowing the Ramanujan-Hardy taxicab story, do it in two distinctly different ways.

A056745:   Show 2009 divides 62009 + 52009 + 42009 + 32009 + 22009 + 12009 [over 3K digits]

• Mysterious(?) Coincidences:

A006768:   There are 2009 5-dimensional polyominoes with 8 cells
A057865:   There are 2009 simple Hamilton-connected graphs on 8 nodes [And Mathworld says, "every 8-connected claw-free graph is Hamilton-connected (Hu et al. 2005)."]
A008764:   There are 2009 nonisomorphic symmetric 3 × 3 matrices over N0 with row and column sums equal to 49 [Note that 49 divides 2009; is this the largest such case?]

• Boundary Connections:

A039768:   gcd(phi(2009),2008) = tau(2008)
A125680:   31 Dec 2009 is the next Blue Moon