C++

C++ port of @Anders Kaseorg's Rust answer. Try it online!

Use __builtin_bitreverse32 from clang, or you can implement one.

ref Rust code

#include <bits/stdc++.h>
using namespace std;


typedef uint32_t T;
static const unsigned BITS = 8 * sizeof(T);


static inline uint32_t reverse_bits(uint32_t x) {
    // This is a common bitwise trick to reverse bits:
    x = ((x & 0xFFFF0000u) >> 16) | ((x & 0x0000FFFFu) << 16);
    x = ((x & 0xFF00FF00u) >> 8)  | ((x & 0x00FF00FFu) << 8);
    x = ((x & 0xF0F0F0F0u) >> 4)  | ((x & 0x0F0F0F0Fu) << 4);
    x = ((x & 0xCCCCCCCCu) >> 2)  | ((x & 0x33333333u) << 2);
    x = ((x & 0xAAAAAAAAu) >> 1)  | ((x & 0x55555555u) << 1);
    return x;
}

struct VectorHash {
    size_t operator()(const vector<T> &vec) const {
        // A simple combination hash approach
        std::size_t h = 0;
        for (auto &x : vec) {
            // Combine from boost::hash_combine-like approach
            h ^= std::hash<T>()(x) + 0x9e3779b97f4a7c15ULL + (h << 6) + (h >> 2);
        }
        return h;
    }
};

struct VectorEq {
    bool operator()(const vector<T>& a, const vector<T>& b) const {
        return a == b;
    }
};

int main(int argc, char *argv[]) {
    if (argc != 2) {
        cerr << "Please provide exactly one integer argument.\n";
        return 1;
    }
    unsigned n;
    try {
        n = static_cast<unsigned>(stoul(argv[1]));
    } catch (...) {
        cerr << "Failed to parse argument as integer.\n";
        return 1;
    }
    if (n == 0 || n > BITS) {
        cerr << "n must be between 1 and " << BITS << ".\n";
        return 1;
    }

    vector<unordered_map<vector<T>, bool, VectorHash, VectorEq>> unique(9 - 2 + 1);
    vector<T> sums(n, 0);
    T max_val = (static_cast<T>(-1)) >> (BITS - n); // (1<<n)-1

    for (T a = 0; a <= max_val; ++a) {
        T rev = __builtin_bitreverse32(a) >> (BITS - n);
        if (a > rev) {
            continue;
        }

        // sums[...] = 0 each iteration
        fill(sums.begin(), sums.end(), 0);
        for (unsigned i = 0; i < n; i++) {
            T bit = 1;
            for (unsigned j = i; j < n; j++) {
                bit <<= ((a >> j) & 1);
                sums[j - i] |= bit;
            }
        }

        for (unsigned s = 2; s <= 9; s++) {
            unsigned length = (n + (n - 1) * s);
            unsigned arraySize = length / BITS + 1;

            vector<T> sums_s(arraySize, 0);
            unsigned pos = 0;
            unsigned shift = 0;
            T lo = 0;
            T hi = 0;

            for (auto &v : sums) {
                lo |= (v << shift);
                if (BITS - shift < n) {
                    hi |= (v >> (BITS - shift));
                }
                shift += s;

                if (shift >= BITS) {
                    shift -= BITS;
                    sums_s[pos] = lo;
                    pos++;
                    lo = hi;
                    hi = 0;
                }
            }

            if (lo != 0 || hi != 0) {
                sums_s[pos] = lo;
                // pos++ only if we need to use hi
                if (hi != 0) {
                    pos++;
                    sums_s[pos - 1] = hi;
                }
            }

            auto &umap = unique[s - 2];
            auto it = umap.find(sums_s);
            if (it != umap.end()) {
                it->second = false;
            } else {
                umap.insert({sums_s, true});
            }
        }
    }

    vector<T> counts;
    counts.push_back(static_cast<T>(n + 1));

    for (auto &m : unique) {
        T sum_val = 0;
        for (auto &kv : m) {
            if (kv.second) {
                sum_val += 1; // each true counts as 1
            }
        }
        counts.push_back(sum_val);
    }

    // Print out contents of counts in a Rust-like "[..]" format
    cout << "[";
    for (size_t i = 0; i < counts.size(); i++) {
        cout << counts[i];
        if (i + 1 < counts.size()) {
            cout << ", ";
        }
    }
    cout << "]\n";

    return 0; // process::exit(0) in Rust
}

Clean

Certainly not the most efficient approach, but I'm interested in seeing how well a naive by-value filter does.

That said, there's still a bit of improvement to be made using this method.

module main
import StdEnv, Data.List, System.CommandLine

f l = sort (nub [sum t \\ i <- inits l, t <- tails i])

Start w
	# ([_:args], w) = getCommandLine w
	= case map toInt args of
		[n] = map (flip countUniques n) [1..9]
		_ = abort "Wrong number of arguments!"

countUniques 1 n = inc n
countUniques s n = length uniques
where
	lists = [[s + ((i >> p) bitand 1) \\ p <- [0..dec n]] \\ i <- [0..2^n-1]]
	pairs = sortBy (\(a,_) (b,_) = a < b) (zip (map f lists, lists))
	groups = map (snd o unzip) (groupBy (\(a,_) (b,_) = a == b) pairs)
	uniques = filter (\section = case section of [a, b] = a == reverse b; [_] = True; _ = False) groups

Place in a file named main.icl, or change the top line to module <your_file_name_here>.

Compile with clm -h 1500m -s 50m -fusion -t -IL Dynamics -IL StdEnv -IL Platform main.

You can get the version TIO (and myself) use from the link in the heading, or a more recent one from here.

Common Lisp SBCL, N = 14

call function (goahead n s)

    (defun sub-lists(l m &optional(x 0)(y 0))
  (cond; ((and(= y (length l))(= x (length l)))nil)
        ((= y (length l))m)
        ((= x (length l))(sub-lists l m 0(1+ y)))
    (t (sub-lists l (cons(loop for a from x to (+ x y)

             when (and(nth (+ x y)l)(nth a l)(< (+ x y)(length l)))
                ;   while (nth a l)
             ;while(and(< (+ x y)(length l))(nth a l))
                    collect (nth a l))m) (1+ x)y))
    ))
(defun permutations(size elements)
  (if (zerop size)'(())
 (mapcan (lambda (p)
                    (map 'list (lambda (e)
                           (cons e p))
                         elements))
     (permutations (1- size) elements))))
(defun remove-reverse(l m)
  (cond ((endp l)m)
    ((member (reverse (first l))(rest l) :test #'equal)(remove-reverse (rest l)m))
    (t (remove-reverse (rest l)(cons (first l)m)))))
(defun main(n s)
  (let((l (remove-reverse (permutations n `(,s ,(1+ s)))nil)))

  (loop for x in l
     for j = (remove 'nil (sub-lists x nil))
       collect(sort (make-set(loop for y in j
        collect (apply '+ y))nil)#'<)
     )
  ))
(defun remove-dups(l m n)
  (cond ((endp l)n)
        ((member (first l) (rest l) :test #'equal)(remove-dups(rest l)(cons (first l) m) n))
    ((member (first l) m :test #'equal)(remove-dups(rest l)m n))
    (t(remove-dups (rest l) m (cons (first l) n))))

  )
(defun goahead(n s)
  (loop for a from 1 to s
  collect(length (remove-dups(main n a)nil nil))))
(defun make-set (L m)
  "Returns a set from a list. Duplicate elements are removed."
  (cond ((endp L) m)
    ((member (first L) (rest L)) (make-set (rest L)m))
    ( t (make-set (rest L)(cons (first l)m)))))

here is the run times

CL-USER> (time (goahead 14 9))
Evaluation took:
  34.342 seconds of real time
  34.295000 seconds of total run time (34.103012 user, 0.191988 system)
  [ Run times consist of 0.263 seconds GC time, and 34.032 seconds non-GC time. ]
  99.86% CPU
  103,024,254,028 processor cycles
  1,473,099,744 bytes consed

(15 1047 4893 6864 7270 7324 7328 7328 7328)
CL-USER> (time (goahead 15 9))
Evaluation took:
  138.639 seconds of real time
  138.511089 seconds of total run time (137.923824 user, 0.587265 system)
  [ Run times consist of 0.630 seconds GC time, and 137.882 seconds non-GC time. ]
  99.91% CPU
  415,915,271,830 processor cycles
  3,453,394,576 bytes consed

(16 1502 8848 13336 14418 14578 14594 14594 14594)

Rust, n ≈ 24

Requires nightly Rust for the convenient reverse_bits feature. Compile with rustc -O unique.rs and run with (e.g.) ./unique 24.

#![feature(reverse_bits)]
use std::{collections::HashMap, env, mem, process};

type T = u32;
const BITS: u32 = mem::size_of::<T>() as u32 * 8;

fn main() {
    let args = env::args().collect::<Vec<_>>();
    assert!(args.len() == 2);
    let n: u32 = args[1].parse().unwrap();
    assert!(n > 0);
    assert!(n <= BITS);
    let mut unique = (2..=9).map(|_| HashMap::new()).collect::<Vec<_>>();
    let mut sums = vec![0 as T; n as usize];
    for a in 0 as T..=!0 >> (BITS - n) {
        if a <= a.reverse_bits() >> (BITS - n) {
            for v in &mut sums {
                *v = 0;
            }
            for i in 0..n {
                let mut bit = 1;
                for j in i..n {
                    bit <<= a >> j & 1;
                    sums[(j - i) as usize] |= bit;
                }
            }
            for s in 2..=9 {
                let mut sums_s =
                    vec![0 as T; ((n + (n - 1) * s) / BITS + 1) as usize].into_boxed_slice();
                let mut pos = 0;
                let mut shift = 0;
                let mut lo = 0;
                let mut hi = 0;
                for &v in &sums {
                    lo |= v << shift;
                    if BITS - shift < n {
                        hi |= v >> (BITS - shift);
                    }
                    shift += s;
                    if shift >= BITS {
                        shift -= BITS;
                        sums_s[pos] = lo;
                        pos += 1;
                        lo = hi;
                        hi = 0;
                    }
                }
                if lo != 0 || hi != 0 {
                    sums_s[pos] = lo;
                    pos += 1;
                    if hi != 0 {
                        sums_s[pos] = hi;
                    }
                }
                unique[s as usize - 2]
                    .entry(sums_s)
                    .and_modify(|u| *u = false)
                    .or_insert(true);
            }
        }
    }
    let mut counts = vec![n + 1];
    counts.extend(
        unique
            .iter()
            .map(|m| m.values().map(|&u| u as T).sum::<T>())
            .collect::<Vec<_>>(),
    );
    println!("{:?}", counts);
    process::exit(0); // Avoid running destructors.
}