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The most surprising of these is that the sum of the numbers in the triangles that point upwards is the same as the sum of those in triangles that point downwards (no matter how large the T-hexagon). In the above example, 17 + 20 + 22 + 21 + 2 + 6 + 10 + 14 + 3 + 16 + 12 + 7 = 5 + 11 + 19 + 9 + 8 + 13 + 4 + 1 + 24 + 15 + 23 + 18 = 150
obtained their generating function and studied their asymptotics at large n. Clearly, G n = G n (1). These numbers are strictly alternating G n (k) = (-1) n-1 |G n (k)| and involved in various expansions for the zeta-functions, Euler's constant and polygamma functions. A different generalization of the same kind was also proposed by Komatsu [31]
One may show by induction that F(n) counts the number of ways that a n × 1 strip of squares may be covered by 2 × 1 and 1 × 1 tiles. On the other hand, if such a tiling uses exactly k of the 2 × 1 tiles, then it uses n − 2 k of the 1 × 1 tiles, and so uses n − k tiles total.
L14 Sinus Iridum. The Lunar 100 (L100) is a list of one hundred of the most interesting features to observe on the Moon.The list was first described by Charles A. Wood in the article The Lunar 100 in Sky & Telescope magazine, April 2004.
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a ...
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
[0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …] [OEIS 100] Computed up to 1 011 597 392 terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property. [Mw 85]
In mathematics, a dodecagonal number is a figurate number that represents a dodecagon.The dodecagonal number for n is given by the formula = The first few dodecagonal numbers are: