# Project Euler 554
# Centaurs on a Chess Board
# C(n) = 8*C(2n,n) - 3n^2 - 2n - 7; sum C(F_i) for i=2..90 mod 10^8+7 via Lucas.
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const MOD: i64 = 100000007
const NEED_CAP: i64 = 4096
let mut TARGETS: ptr<i64> = null
let mut FACT: ptr<i64> = null
let mut INVFACT: ptr<i64> = null
let mut NT: i64 = 0
function modpow(base0: i64, exp0: i64, mod: i64) -> i64 {
let mut result: i64 = 1
let mut base: i64 = base0 % mod
let mut exp: i64 = exp0
while exp > 0 {
if (exp & 1) != 0 {
result = ((result as i128) * (base as i128) % (mod as i128)) as i64
}
base = ((base as i128) * (base as i128) % (mod as i128)) as i64
exp = exp >> 1
}
return result
}
function add_need(x: i64) -> void {
if x < 0 || x >= MOD { return }
let mut i: i64 = 0
while i < NT {
if TARGETS[i] == x { return }
i = i + 1
}
TARGETS[NT] = x
NT = NT + 1
}
function mark_binom_needs(n: i64) -> void {
let mut N: i64 = 2 * n
let mut K: i64 = n
while N > 0 || K > 0 {
let ni: i64 = N % MOD
let ki: i64 = K % MOD
if ki <= ni {
add_need(ni)
add_need(ki)
add_need(ni - ki)
}
N = N / MOD
K = K / MOD
}
}
function sort_targets() -> void {
let mut i: i64 = 1
while i < NT {
let key: i64 = TARGETS[i]
let mut j: i64 = i - 1
while j >= 0 && TARGETS[j] > key {
TARGETS[j + 1] = TARGETS[j]
j = j - 1
}
TARGETS[j + 1] = key
i = i + 1
}
}
function index_of(x: i64) -> i64 {
let mut lo: i64 = 0
let mut hi: i64 = NT - 1
while lo <= hi {
let mid: i64 = (lo + hi) / 2
let v: i64 = TARGETS[mid]
if v == x { return mid }
if v < x { lo = mid + 1 } else { hi = mid - 1 }
}
return -1
}
function small_binom(n: i64, k: i64) -> i64 {
if k < 0 || k > n { return 0 }
let idx_n: i64 = index_of(n)
let idx_k: i64 = index_of(k)
let idx_nk: i64 = index_of(n - k)
if idx_n < 0 || idx_k < 0 || idx_nk < 0 { return 0 }
return ((FACT[idx_n] as i128) * (INVFACT[idx_k] as i128) % (MOD as i128) * (INVFACT[idx_nk] as i128) % (MOD as i128)) as i64
}
function lucas_binom(N0: i64, K0: i64) -> i64 {
let mut N: i64 = N0
let mut K: i64 = K0
let mut res: i64 = 1
while N > 0 || K > 0 {
let ni: i64 = N % MOD
let ki: i64 = K % MOD
if ki > ni { return 0 }
res = ((res as i128) * (small_binom(ni, ki) as i128) % (MOD as i128)) as i64
N = N / MOD
K = K / MOD
}
return res
}
function C_mod(n: i64) -> i64 {
let b: i64 = lucas_binom(2 * n, n)
let nn: i64 = n % MOD
let poly: i64 = ((3 * nn % MOD) * nn + 2 * nn + 7) % MOD
let mut r: i64 = (8 * b - poly) % MOD
if r < 0 { r = r + MOD }
return r
}
function main() -> i32 {
let F: ptr<i64> = calloc(91, 8)
if F == null { return 1 }
F[0] = 0
F[1] = 1
let mut i: i64 = 2
while i <= 90 {
F[i] = F[i - 1] + F[i - 2]
i = i + 1
}
TARGETS = calloc(NEED_CAP, 8)
FACT = calloc(NEED_CAP, 8)
INVFACT = calloc(NEED_CAP, 8)
if TARGETS == null || FACT == null || INVFACT == null { return 1 }
add_need(0)
add_need(1)
i = 2
while i <= 90 {
mark_binom_needs(F[i])
i = i + 1
}
mark_binom_needs(1)
mark_binom_needs(2)
mark_binom_needs(10)
sort_targets()
let max_idx: i64 = TARGETS[NT - 1]
let mut f: i64 = 1
let mut ti: i64 = 0
FACT[0] = 1
INVFACT[0] = 1
ti = 1
i = 1
while i <= max_idx {
f = ((f as i128) * (i as i128) % (MOD as i128)) as i64
if ti < NT && TARGETS[ti] == i {
FACT[ti] = f
INVFACT[ti] = modpow(f, MOD - 2, MOD)
ti = ti + 1
}
i = i + 1
}
let mut ans: i64 = 0
i = 2
while i <= 90 {
ans = ans + C_mod(F[i])
if ans >= MOD { ans = ans - MOD }
i = i + 1
}
printf("%lld\n", ans)
free(F)
free(TARGETS)
free(FACT)
free(INVFACT)
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int64_t modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod);
void add_need_i64(int64_t x);
void mark_binom_needs_i64(int64_t n);
void sort_targets(void);
int64_t index_of_i64(int64_t x);
int64_t small_binom_i64_i64(int64_t n, int64_t k);
int64_t lucas_binom_i64_i64(int64_t N0, int64_t K0);
int64_t C_mod_i64(int64_t n);
int32_t main(void);
static const int64_t MOD = 100000007;
static const int64_t NEED_CAP = 4096;
/* Module statics */
static int64_t* TARGETS = NULL;
static int64_t* FACT = NULL;
static int64_t* INVFACT = NULL;
static int64_t NT = 0;
int64_t modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod) {
int64_t result = 1;
int64_t base = FLOW_CHECKED_MOD((base0), (mod));
int64_t exp = exp0;
while (exp > 0) {
if ((exp & 1) != 0) {
result = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(result)) * ((__int128)(base)))), (((__int128)(mod))))));
}
base = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(base)) * ((__int128)(base)))), (((__int128)(mod))))));
exp = FLOW_CHECKED_SHR((exp), (1));
}
return result;
}
void add_need_i64(int64_t x) {
if ((x < 0 || x >= MOD)) {
return;
}
int64_t i = 0;
while (i < NT) {
if (TARGETS[i] == x) {
return;
}
i = (i + 1);
}
TARGETS[NT] = x;
NT = (NT + 1);
}
void mark_binom_needs_i64(int64_t n) {
int64_t N = (2 * n);
int64_t K = n;
while ((N > 0 || K > 0)) {
int64_t ni = FLOW_CHECKED_MOD((N), (MOD));
int64_t ki = FLOW_CHECKED_MOD((K), (MOD));
if (ki <= ni) {
add_need_i64(ni);
add_need_i64(ki);
add_need_i64((ni - ki));
}
N = FLOW_CHECKED_DIV((N), (MOD));
K = FLOW_CHECKED_DIV((K), (MOD));
}
}
void sort_targets(void) {
int64_t i = 1;
while (i < NT) {
int64_t key = TARGETS[i];
int64_t j = (i - 1);
while ((j >= 0 && TARGETS[j] > key)) {
TARGETS[(j + 1)] = TARGETS[j];
j = (j - 1);
}
TARGETS[(j + 1)] = key;
i = (i + 1);
}
}
int64_t index_of_i64(int64_t x) {
int64_t lo = 0;
int64_t hi = (NT - 1);
while (lo <= hi) {
int64_t mid = FLOW_CHECKED_DIV(((lo + hi)), (2));
int64_t v = TARGETS[mid];
if (v == x) {
return mid;
}
if (v < x) {
lo = (mid + 1);
} else {
hi = (mid - 1);
}
}
return (-1);
}
int64_t small_binom_i64_i64(int64_t n, int64_t k) {
if ((k < 0 || k > n)) {
return 0;
}
int64_t idx_n = index_of_i64(n);
int64_t idx_k = index_of_i64(k);
int64_t idx_nk = index_of_i64((n - k));
if (((idx_n < 0 || idx_k < 0) || idx_nk < 0)) {
return 0;
}
return ((int64_t)(FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((((__int128)(FACT[idx_n])) * ((__int128)(INVFACT[idx_k])))), (((__int128)(MOD)))) * ((__int128)(INVFACT[idx_nk])))), (((__int128)(MOD))))));
}
int64_t lucas_binom_i64_i64(int64_t N0, int64_t K0) {
int64_t N = N0;
int64_t K = K0;
int64_t res = 1;
while ((N > 0 || K > 0)) {
int64_t ni = FLOW_CHECKED_MOD((N), (MOD));
int64_t ki = FLOW_CHECKED_MOD((K), (MOD));
if (ki > ni) {
return 0;
}
res = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(res)) * ((__int128)(small_binom_i64_i64(ni, ki))))), (((__int128)(MOD))))));
N = FLOW_CHECKED_DIV((N), (MOD));
K = FLOW_CHECKED_DIV((K), (MOD));
}
return res;
}
int64_t C_mod_i64(int64_t n) {
int64_t b = lucas_binom_i64_i64((2 * n), n);
int64_t nn = FLOW_CHECKED_MOD((n), (MOD));
int64_t poly = FLOW_CHECKED_MOD(((((FLOW_CHECKED_MOD(((3 * nn)), (MOD)) * nn) + (2 * nn)) + 7)), (MOD));
int64_t r = FLOW_CHECKED_MOD((((8 * b) - poly)), (MOD));
if (r < 0) {
r = (r + MOD);
}
return r;
}
int32_t main(void) {
int64_t* F = (int64_t*)(calloc(91, 8));
if (F == NULL) {
return 1;
}
F[0] = 0;
F[1] = 1;
int64_t i = 2;
while (i <= 90) {
F[i] = (F[(i - 1)] + F[(i - 2)]);
i = (i + 1);
}
TARGETS = calloc(NEED_CAP, 8);
FACT = calloc(NEED_CAP, 8);
INVFACT = calloc(NEED_CAP, 8);
if (((TARGETS == NULL || FACT == NULL) || INVFACT == NULL)) {
return 1;
}
add_need_i64(0);
add_need_i64(1);
i = 2;
while (i <= 90) {
mark_binom_needs_i64(F[i]);
i = (i + 1);
}
mark_binom_needs_i64(1);
mark_binom_needs_i64(2);
mark_binom_needs_i64(10);
sort_targets();
int64_t max_idx = TARGETS[(NT - 1)];
int64_t f = 1;
int64_t ti = 0;
FACT[0] = 1;
INVFACT[0] = 1;
ti = 1;
i = 1;
while (i <= max_idx) {
f = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(f)) * ((__int128)(i)))), (((__int128)(MOD))))));
if ((ti < NT && TARGETS[ti] == i)) {
FACT[ti] = f;
INVFACT[ti] = modpow_i64_i64_i64(f, (MOD - 2), MOD);
ti = (ti + 1);
}
i = (i + 1);
}
int64_t ans = 0;
i = 2;
while (i <= 90) {
ans = (ans + C_mod_i64(F[i]));
if (ans >= MOD) {
ans = (ans - MOD);
}
i = (i + 1);
}
printf("%lld\n", ans);
free(F);
free(TARGETS);
free(FACT);
free(INVFACT);
return 0;
}