Solve problem 297

python/e297.py

@@ -0,0 +1,116 @@
+from functools import cache
+from typing import Optional
+
+
+def fib(max_n: int) -> list[int]:
+    a, b = 1, 2
+    xs = [a]
+    while b <= max_n:
+        xs.append(b)
+        a, b = b, a + b
+    return xs
+
+
+FIBS = fib(10**18)
+
+
+@cache
+def zgo(rem: int, i: int, count: int) -> Optional[int]:
+    if rem == 0:
+        return count
+    if rem < 0 or rem < FIBS[i]:
+        return None
+
+    r1 = zgo(rem - FIBS[i], i + 2, count + 1)
+    if r1:
+        return r1
+    return zgo(rem, i + 1, count)
+
+
+@cache
+def z(n: int) -> int:
+    return zgo(n, 0, 0) or 0
+
+
+def zsum(n: int) -> int:
+    return sum(z(i) for i in range(1, n))
+
+
+def zxsgo(rem: int, i: int, xs: list[int]) -> Optional[list]:
+    if rem == 0:
+        return xs
+    if rem < 0 or rem < FIBS[i]:
+        return None
+
+    r1 = zxsgo(rem - FIBS[i], i + 2, xs + [FIBS[i]])
+    if r1:
+        return r1
+    return zxsgo(rem, i + 1, xs)
+
+
+def zxs(n: int) -> Optional[list]:
+    return zxsgo(n, 0, [])
+
+
+@cache
+def zsumr(n: int) -> int:
+    if n == 1:
+        return 0
+    if n == 2:
+        return 1
+    if n in FIBS:
+        fib_i = FIBS.index(n)
+        a, b = FIBS[fib_i - 2], FIBS[fib_i - 1]
+        # print(f"zsumr({a})={zsumr(a)} zsumr({b})={zsumr(b)} {b - a}")
+        res = 0
+
+        # Zeckendorf sum up the previous fibonacci
+        res += zsumr(b)
+
+        # Zeckendorf sum up to the pervious previous fibonacci. This works because
+        # if we look at the Zeckendorf representations between two fibonaccis it is
+        # always a combination of the previous fibonacci plus the representations
+        # starting from zero. Since we cannot use the previous fibonacci because of
+        # the *nonconsecutive* requirement, it has to be the second previous.
+        res += zsumr(a)
+
+        # However, then for all the zsumr(a) terms we considered, that fibonacci
+        # a was not part of the count, so we have to add it:
+        res += n - b
+
+        return res
+
+    for fib_i, fib in enumerate(FIBS):
+        if fib > n:
+            break
+    else:
+        assert False
+    fib_i -= 1
+    b = FIBS[fib_i]  # fib before n
+
+    res = 0
+    res += zsumr(b)  # Zeckendorf up to previous fib
+    res += zsumr(n - b)  # Zeckendorf from b to n without last term
+    res += n - b  # account for last term
+    return res
+
+
+def euler_297():
+    assert zsumr(3) == 2
+    assert zsumr(5) == 5
+    assert zsumr(13) == 20
+    assert zsumr(89) == 235
+    assert zsumr(100) == 261
+    assert zsumr(10**6) == 7894453
+
+    assert z(1) == 1
+    assert z(5) == 1
+    assert z(14) == 2
+    assert z(100) == 3
+    return zsumr(10**17)
+
+
+if __name__ == "__main__":
+    solution = euler_297()
+    print(f"e297.py: {solution}")
+    assert solution == 2252639041804718029