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The other direction follows from the fact that the Koszul functors are quasi-inverse to each other and preserve quasi-isomorphisms because a split and extended weak module is
These denominators generate an elliptic divisibility sequence, a genus-1 analogue of more classical sequences such as Fibonacci or Mersenne, and the hypothesis, which they
r -bonacci numbers; there are sequences in both families that grow at different rates (the Fibonacci sequence, the Tanny sequence, and an eventually constant sequence can all occur as
These numbers are also called the Bernoulli numbers of the second kind, and are represented by sequences A006232 and A006233 in Sloane’s Encyclopedia.. The Cauchy numbers play
Our argument also shows that the set of augmented amicable numbers has density zero (see sequences A007992 , A015630 ).. Najar, More reduced amicable pairs,
Deutsch, An involution on Dyck paths and its consequences, Discrete Math.. E˘gecio˘glu, The parity of the Catalan numbers via lattice paths,
Upon combining these two entries, he also noted that it was not officially a theorem (that the product of the Fibonacci and Pell numbers was always equal to the number of domino
Hence, second order Lucas numbers are also “hidden” in a triangle like the second order Fibonacci numbers, the difference is that now instead of the Pascal triangle we have a
