Faulhaber's formula: Difference between revisions
Rename Perl 6 -> Raku, alphabetize, minor clean-up
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9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2</pre>
=={{header|C
{{trans|Java}}
<lang csharp>using System;
Line 568 ⟶ 383:
}
}
}</lang>
{{out}}
<pre>0 : n
1 : 1/2n^2 + 1/2n
2 : 1/3n^3 + 1/2n^2 + 1/6n
3 : 1/4n^4 + 1/2n^3 + 1/4n^2
4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n
5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2
6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n
7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2
8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n
9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2</pre>
=={{header|C++}}==
{{trans|D}}
Uses C++17
<lang cpp>#include <iostream>
#include <numeric>
#include <sstream>
#include <vector>
class Frac {
public:
Frac(long n, long d) {
if (d == 0) {
throw new std::runtime_error("d must not be zero");
}
long nn = n;
long dd = d;
if (nn == 0) {
dd = 1;
} else if (dd < 0) {
nn = -nn;
dd = -dd;
}
long g = abs(std::gcd(nn, dd));
if (g > 1) {
nn /= g;
dd /= g;
}
num = nn;
denom = dd;
}
Frac operator-() const {
return Frac(-num, denom);
}
Frac operator+(const Frac& rhs) const {
return Frac(num*rhs.denom + denom * rhs.num, rhs.denom*denom);
}
Frac operator-(const Frac& rhs) const {
return Frac(num*rhs.denom - denom * rhs.num, rhs.denom*denom);
}
Frac operator*(const Frac& rhs) const {
return Frac(num*rhs.num, denom*rhs.denom);
}
bool operator==(const Frac& rhs) const {
return num == rhs.num && denom == rhs.denom;
}
bool operator!=(const Frac& rhs) const {
return num != rhs.num || denom != rhs.denom;
}
bool operator<(const Frac& rhs) const {
if (denom == rhs.denom) {
return num < rhs.num;
}
return num * rhs.denom < rhs.num * denom;
}
friend std::ostream& operator<<(std::ostream&, const Frac&);
static Frac ZERO() {
return Frac(0, 1);
}
static Frac ONE() {
return Frac(1, 1);
}
private:
long num;
long denom;
};
std::ostream & operator<<(std::ostream & os, const Frac &f) {
if (f.num == 0 || f.denom == 1) {
return os << f.num;
}
std::stringstream ss;
ss << f.num << "/" << f.denom;
return os << ss.str();
}
Frac bernoulli(int n) {
if (n < 0) {
throw new std::runtime_error("n may not be negative or zero");
}
std::vector<Frac> a;
for (int m = 0; m <= n; m++) {
a.push_back(Frac(1, m + 1));
for (int j = m; j >= 1; j--) {
a[j - 1] = (a[j - 1] - a[j]) * Frac(j, 1);
}
}
// returns 'first' Bernoulli number
if (n != 1) return a[0];
return -a[0];
}
int binomial(int n, int k) {
if (n < 0 || k < 0 || n < k) {
throw new std::runtime_error("parameters are invalid");
}
if (n == 0 || k == 0) return 1;
int num = 1;
for (int i = k + 1; i <= n; i++) {
num *= i;
}
int denom = 1;
for (int i = 2; i <= n - k; i++) {
denom *= i;
}
return num / denom;
}
void faulhaber(int p) {
using namespace std;
cout << p << " : ";
auto q = Frac(1, p + 1);
int sign = -1;
for (int j = 0; j < p + 1; j++) {
sign *= -1;
auto coeff = q * Frac(sign, 1) * Frac(binomial(p + 1, j), 1) * bernoulli(j);
if (coeff == Frac::ZERO()) {
continue;
}
if (j == 0) {
if (coeff == -Frac::ONE()) {
cout << "-";
} else if (coeff != Frac::ONE()) {
cout << coeff;
}
} else {
if (coeff == Frac::ONE()) {
cout << " + ";
} else if (coeff == -Frac::ONE()) {
cout << " - ";
} else if (coeff < Frac::ZERO()) {
cout << " - " << -coeff;
} else {
cout << " + " << coeff;
}
}
int pwr = p + 1 - j;
if (pwr > 1) {
cout << "n^" << pwr;
} else {
cout << "n";
}
}
cout << endl;
}
int main() {
for (int i = 0; i < 10; i++) {
faulhaber(i);
}
return 0;
}</lang>
{{out}}
Line 1,970:
8: -1/30 n+2/9 n^3-7/15 n^5+2/3 n^7+1/2 n^8+1/9 n^9
9: -3/20 n^2+1/2 n^4-7/10 n^6+3/4 n^8+1/2 n^9+1/10 n^10
</pre>
Line 2,265 ⟶ 2,210:
f(8) = 1/9 * (n^9 + (9/2 * n^8) + (6 * n^7) + (-21/5 * n^5) + (2 * n^3) + (-3/10 * n))
f(9) = 1/10 * (n^10 + (5 * n^9) + (15/2 * n^8) + (-7 * n^6) + (5 * n^4) + (-3/2 * n^2))
</pre>
=={{header|Raku}}==
(formerly Perl 6)
{{works with|Rakudo|2018.04.01}}
<lang perl6>sub bernoulli_number($n) {
return 1/2 if $n == 1;
return 0/1 if $n % 2;
my @A;
for 0..$n -> $m {
@A[$m] = 1 / ($m + 1);
for $m, $m-1 ... 1 -> $j {
@A[$j - 1] = $j * (@A[$j - 1] - @A[$j]);
}
}
return @A[0];
}
sub binomial($n, $k) {
$k == 0 || $n == $k ?? 1 !! binomial($n-1, $k-1) + binomial($n-1, $k);
}
sub faulhaber_s_formula($p) {
my @formula = gather for 0..$p -> $j {
take '('
~ join('/', (binomial($p+1, $j) * bernoulli_number($j)).Rat.nude)
~ ")*n^{$p+1 - $j}";
}
my $formula = join(' + ', @formula.grep({!m{'(0/1)*'}}));
$formula .= subst(rx{ '(1/1)*' }, '', :g);
$formula .= subst(rx{ '^1'» }, '', :g);
"1/{$p+1} * ($formula)";
}
for 0..9 -> $p {
say "f($p) = ", faulhaber_s_formula($p);
}</lang>
{{out}}
<pre>
f(0) = 1/1 * (n)
f(1) = 1/2 * (n^2 + n)
f(2) = 1/3 * (n^3 + (3/2)*n^2 + (1/2)*n)
f(3) = 1/4 * (n^4 + (2/1)*n^3 + n^2)
f(4) = 1/5 * (n^5 + (5/2)*n^4 + (5/3)*n^3 + (-1/6)*n)
f(5) = 1/6 * (n^6 + (3/1)*n^5 + (5/2)*n^4 + (-1/2)*n^2)
f(6) = 1/7 * (n^7 + (7/2)*n^6 + (7/2)*n^5 + (-7/6)*n^3 + (1/6)*n)
f(7) = 1/8 * (n^8 + (4/1)*n^7 + (14/3)*n^6 + (-7/3)*n^4 + (2/3)*n^2)
f(8) = 1/9 * (n^9 + (9/2)*n^8 + (6/1)*n^7 + (-21/5)*n^5 + (2/1)*n^3 + (-3/10)*n)
f(9) = 1/10 * (n^10 + (5/1)*n^9 + (15/2)*n^8 + (-7/1)*n^6 + (5/1)*n^4 + (-3/2)*n^2)
</pre>
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