"Counting in their heads"  1895 oil painting

08112020, 05:07 PM
(This post was last modified: 08112020 08:51 PM by Albert Chan.)
Post: #18




RE: "Counting in their heads"  1895 oil painting
(08112020 12:43 PM)Gerson W. Barbosa Wrote: sumofnsquares = n(c² + d²(n²  1)/12) Perhaps this explain the similarity, for S_{p} = a^{p} + (a+1)^{p} + ... + b^{p}, b = a+n1 With center c = (a+b)/2, g = (n1)/2 = bc = ca, we have S(a,n) = S(c,bc+1)  S(c,ac) = S(c,g+1)  S(c,g) = FiniteDifferenceDiagonals • \(\left( [\binom{g+1}{1},\binom{g+1}{2},\cdots]  [\binom{g}{1},\binom{g}{2},\cdots] \right)\) Let's see what the combinatorial coefficients look like: XCas> g := (n1)/2 XCas> coef := makelist(r > simplify(comb(g+1,r)  comb(g,r)),1,6) → \(\Large [n,\,0,\,\frac{(n^{3}n)}{24},\,\frac{(n^{3}+n)}{24},\,\frac{(n^{5}+70\cdot n^{3}71\cdot n)}{1920},\,\frac{(n^{5}30\cdot n^{3}+31\cdot n)}{960}]\) coef(1) = n, which make first term of S_{p} = c^p*n coef(2) = 0, which make the central element based formula compact (Δf * 0 = 0) coef(3) = coef(4), which meant dotproducts have this common factor. Finite Difference table for (c+x)² and (c+x)³ // note: [1,0,0] ≡ c², [1,2,1] ≡ c²+2c+1 Code: x (c+x)^2 If we let k = coef(3) = (n³n)/24: sumofnsquares = c^2*n + 2k sumofncubes = c^3*n + (6c+6)*k + 6*(k) = c*(c^2*n + 6k) coef(4) and coef(5) are not close, which meant similarity ends after cubes XCas> sumpow(p) := expand(simplify(sum(x^p, x=cg .. c+g))) XCas> for(p=1; p<6; p++) {print(p, sumpow(p))} 1, c*n 2, (n^3)/12+c^2*nn/12 3, (c*n^3)/4+c^3*n(c*n)/4 4, (n^5)/80+(c^2*n^3)/2+c^4*n(n^3)/24(c^2*n)/2+(7*n)/240 5, (c*n^5)/16+(5*c^3*n^3)/6+c^5*n(5*c*n^3)/24(5*c^3*n)/6+(7*c*n)/48 

« Next Oldest  Next Newest »

User(s) browsing this thread: 1 Guest(s)