[ProjectEuler] Problem #242

沒有什麼特別想法，先暴力算算看 f(n, k) 好了。

[ProjectEuler] Notes on Problem #435

Lemma: F_n(x) = (f_{n+1} x^{n+1} + f_{n} x^{n+2} - x) / (x^2 + x - 1)

F_n(x) 

= sum_{0 <= i <= n} f_i x^i 

= sum_{1 <= i <= n} f_i x^i ...... (A)

x F_n(x) 

= sum_{0 <= i <= n} f_{i} x^{i+1} 

= sum_{1 <= i+1 <= n+1} f_{i+1 - 1} x^{i+1}

= sum_{1 <= j <= n+1} f_{j-1} x^{j}

= sum_{1 <= i <= n} f_{i-1} x^{i} + f_n x^{n+1} ...... (B)

                     

(A) + (B) implies that (x+1) F_n(x) = sum_{1 <= i <= n} f_{i+1} x^i + f_n x^{n+1}

Multiply x on both sides:

x(x+1) F_n(x) = sum_{1 <= i <= n} f_{i+1} x^{i+1} + f_n x^{n+2}

= sum_{2 <= i+1 <= n+1} f_{i+1} x^{i+1} + f_n x^{n+2}

= sum_{2 <= i <= n+1} f_i x^i + f_n x^{n+2}

= sum_{1 <= i <= n} f_i x^i - f_1 x + f_{n+1} x^{n+1} + f_n x^{n+2}

= F_n(x) + f_{n+1} x^{n+1} + f_n x^{n+2} - x

(x^2 + x - 1) F_n(x) = f_{n+1} x^{n+1} + f_n x^{n+2} - x

So, F_n(x) = (f_{n+1} x^{n+1} + f_n x^{n+2} - x) / (x^2 + x - 1)

不過這離解題還有一大段的距離。

[ProjectEuler] Problem #320

結果我還是用筆電硬幹 ((當然有一些技巧))

跑了五小時 Macbook Pro 完全性的在發燙呀！

還好答案是對的，有時候很懷疑電腦跑太久答案會算錯。

[FWD] Factorials

http://www.cut-the-knot.org/blue/LegendresTheorem.shtml

[FWD] Collaborative filtering

http://www.ideal.ece.utexas.edu/seminar/LatentFactorModels.pdf

這個是自己人寫的（飄）

[ProjectEuler] TODO: Dynamic programming

Problem #217

Apply dynamic programming on n, not on 10^n ((Solved))



Problem #427

Apply dynamic programming on n-sequence (?)



Problem #413

Apply dynamic programming (?)



應該吧，總之就是再多想想好了。

[ProjectEuler] Problem #379

不知道怎麼解。

參考資料：

http://2000clicks.com/mathhelp/NumberTh05DivisorsTau.aspx

但我要計算的是 Σ Tau(k^2) where k = 1 to N.  N = 10^6 or 10^12.