Solve 684. Little proud of the math I did to figure this one out.

python/e684.py

@@ -0,0 +1,54 @@
+def r_modulo_closed_form(n, m):
+    # From e132.py
+    assert n > 0 and m > 0
+    return ((pow(10, n, 9 * m) - 1) // 9) % m
+
+
+def s(n, mod):
+    a = n % 9
+    b = n // 9
+    r = (a + 1) * pow(10, b, mod) - 1
+    return r % mod
+
+
+def s_big(n, mod):
+    # s = (n % 9 + 1) * 10^(n/9) - 1
+    # S = sum(k=0, n/9) (1 * 10^k + 2 * 10^k + ... + 9 * 10^k) - n
+    #   = 45 * sum(k=0, n/9) 10^k - n
+    #   = 45 * r(n/9) - n
+    # Plus some extra math to make n % 9 == 8 so that the above works.
+
+    r = -1
+    while n % 9 != 8:
+        n += 1
+        r -= s(n, mod)
+
+    r += (r_modulo_closed_form(n // 9 + 1, mod) * 45) % mod
+    r -= n
+    return r % mod
+
+
+def s_big_naiv(n, mod):
+    r = 0
+    for i in range(1, n + 1):
+        r += s(i, mod)
+        if mod is not None:
+            r %= mod
+    return r
+
+
+def euler_684():
+    assert s_big_naiv(20, 11) == s_big(20, 11)
+    r = 0
+    mod = 1_000_000_007
+    a, b = 0, 1
+    for _ in range(2, 91):
+        a, b = b, a + b
+        r += s_big(b, mod)
+    return r % mod
+
+
+if __name__ == "__main__":
+    solution = euler_684()
+    print("e684.py: " + str(solution))
+    assert(solution == 922058210)