PreserveFirstHalf

function generateFirstHalfPreservedSequence(num){
    const halfsParity = [];
    let length = num;
    
    while(length > 1){
        halfsParity.push((length % 2) !== 0);
        length = Math.floor(length / 2) + length % 2;
    }
    
    const stack = [1];
    let runningNum = 1;
    while(halfsParity.length > 0){
        const isThisHalfOdd = halfsParity.pop();
        let stackLength = stack.length;
        
        if(isThisHalfOdd) stackLength--;
        
        while(stackLength--){
            const frontElement = stack.shift();
            stack.push(frontElement);
            stack.push(++runningNum);
        }
        
        if(isThisHalfOdd){
            const frontElement = stack.shift();
            stack.push(frontElement);
        }
        
    }
    return stack;
}

console.log(generateFirstHalfPreservedSequence(10));