[math-fun] an almost trivial telescopy
(Spun off from the Sierpinski triangle-squarefree string business.) n /===\ | | [ f(k) g(k) ] | | [ ] = | | [ g(k) f(k) ] k = 1 n n /===\ /===\ [ 1 1 ] | | [ 1 - 1 ] | | [ ] | | (g(k) + f(k)) + [ ] | | (f(k) - g(k)) [ 1 1 ] | | [ - 1 1 ] | | k = 1 k = 1 ---------------------------------------------------------------. 2 ("Proof": Replace the lhs matrix with the prod from k to k of the rhs, then note how the crossterms vanish.) E.g., [ k ] [ 1 q ] [ inf inf ] inf [ -------- -------- ] [ ==== ==== ] /===\ [ 2 k 2 k ] [ \ n \ n ] | | [ 1 - q 1 - q ] [ > p (n) q > p (n) q ] | | [ ] = [ / even / odd ], | | [ k ] [ ==== ==== ] k = 1 [ q 1 ] [ n = 0 n = 0 ] [ -------- -------- ] [ ] [ 2 k 2 k ] [ vice versa ] [ 1 - q 1 - q ] where p_even(n) and p_odd(n) are the partitions of n into an even and odd number of parts: inf inf /===\ /===\ | | 1 | | 1 inf inf | | ------ + | | ------ ==== ==== 2 n | | k | | k \ n \ q k = 1 1 - q k = 1 1 + q
p (n) q = > --------- = ---------------------------, / even / (q; q) 2 ==== ==== 2 n n = 0 n = 0
2 3 4 5 6 7 8 9 = 1 + q + q + 3 q + 3 q + 6 q + 7 q + 12 q + 14 q + . . . (A027187) and inf inf /===\ /===\ | | 1 | | 1 inf inf | | ------ - | | ------ ==== ==== 2 n + 1 | | k | | k \ n \ q k = 1 1 - q k = 1 1 + q
p (n) q = > ------------- = ---------------------------. / odd / (q; q) 2 ==== ==== 2 n + 1 n = 0 n = 0
2 3 4 5 6 7 8 9 = q + q + 2 q + 2 q + 4 q + 5 q + 8 q + 10 q + 16 q + . . . (A027193). --rwg PS, RKG mentioned the open problem of packing the 1/n by 1/(n+1) rectangles, n>0, into the unit square. What's the greatest n that have been crammed?
I said
n /===\ | | [ f(k) g(k) ] | | [ ] = | | [ g(k) f(k) ] k = 1 n n /===\ /===\ [ 1 1 ] | | [ 1 - 1 ] | | [ ] | | (g(k) + f(k)) + [ ] | | (f(k) - g(k)) [ 1 1 ] | | [ - 1 1 ] | | k = 1 k = 1 ---------------------------------------------------------------. 2 [...] E.g., Simpler: [ inf inf ] inf [ ==== ==== ] /===\ [ k ] [ \ n \ n ] | | [ 1 q ] [ > Q (n) q > Q (n) q ] | | [ ] = [ / even / odd ] | | [ k ] [ ==== ==== ] k = 1 [ q 1 ] [ n = 0 n = 0 ] [ ] [ vice versa ]
where Q_even(n) and Q_odd(n) are the partitions of n into an even and odd number of distinct parts: inf inf ==== ==== n (2 n + 1) \ n \ q
Q (n) q = > ---------------- / even / qpoch(q, q, 2 n) ==== ==== n = 0 n = 0 3 4 5 6 7 8 9 10 11 = 1 + q + q + 2 q + 2 q + 3 q + 3 q + 4 q + 5 q + 6 q + . . . (A067661),
inf inf ==== ==== (n + 1) (2 n + 1) \ n \ q
Q (n) q = > -------------------- / odd / qpoch(q, q, 2 n + 1) ==== ==== n = 0 n = 0
2 3 4 5 6 7 8 9 10 11 = q + q + q + q + q + 2 q + 2 q + 3 q + 4 q + 5 q + 6 q + . . . (A067659) Neil, you might want to add these G.f.s (and yesterday's A027193,A027193) to the EIS. Also, you might ammend A000041, G.f.: Product_{k>0} 1/(1-x^k)=1+Sum_{k>0} x^(k^2)/(Product_{i=1..k} (1-x^i))^2. to include a simpler sum: G.f.: Product_{k>0} 1/(1-x^k)=Sum_{k>=0} x^k Product_{i=1..k} 1/(1-x^i) =1+Sum_{k>0} x^(k^2)/(Product_{i=1..k} (1-x^i))^2. You could also add a sum G.f. to A00009: G.f.: Product ... = Product ... =Sum(k>=0) Product_{i=1..k} x^i/(1-x^i) --rwg
participants (1)
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R. William Gosper