Code to use SP2013 search REST API from .NET CSOM.

SearchResults.aspx.cs

using System;
using System.Collections.Generic;
using System.Linq;
using System.Net;
using System.Web;
using System.Web.UI;
using System.Web.UI.WebControls;
using System.Xml.Linq;

namespace COB.SharePointAutohostedAppWeb.Pages
{
    public partial class SearchResults : System.Web.UI.Page
    {
        SharePointContextToken contextToken;
        string accessToken;
        Uri sharepointHostWebUrl;

        protected void Page_Load(object sender, EventArgs e)
        {
            string queryText = "timesheets";
            
            TokenHelper.TrustAllCertificates();
            string contextTokenString = TokenHelper.GetContextTokenFromRequest(Request);

            if (contextTokenString != null)
            {
                // get context token from TokenHelper..
                contextToken = TokenHelper.ReadAndValidateContextToken(contextTokenString, Request.Url.Authority);

                // use it to get the access token..
                sharepointHostWebUrl = new Uri(Request.QueryString["SPHostUrl"]);
                accessToken = TokenHelper.GetAccessToken(contextToken, sharepointHostWebUrl.Authority).AccessToken;

                // pass the access token - N.B. if this was a control event (rather than page load), we'd need to persist 
                // this somewhere it could be accessed in the event handler..
                runSearch(accessToken, queryText);
            }
        }

        private void runSearch(string accessToken, string queryText)
        {
            if (IsPostBack)
            {
                sharepointHostWebUrl = new Uri(Request.QueryString["SPHostUrl"]);
            }

            string searchRestUrl = string.Format("/_api/search/query?querytext='{0}'", queryText);

            HttpWebRequest request = (HttpWebRequest)HttpWebRequest.Create(sharepointHostWebUrl.ToString() + searchRestUrl);
            request.Method = "GET";
            request.Accept = "application/atom+xml";
            request.ContentType = "application/atom+xml;type=entry";
            request.Headers.Add("Authorization", "Bearer " + accessToken);
            HttpWebResponse response = (HttpWebResponse)request.GetResponse();

            // process response..
            XDocument oDataXML = XDocument.Load(response.GetResponseStream(), LoadOptions.None);
            XNamespace atom = "http://www.w3.org/2005/Atom";
            XNamespace d = "http://schemas.microsoft.com/ado/2007/08/dataservices";
            XNamespace m = "http://schemas.microsoft.com/ado/2007/08/dataservices/metadata";

            List<XElement> items = oDataXML.Descendants(d + "query")
                                     .Elements(d + "PrimaryQueryResult")
                                     .Elements(d + "RelevantResults")
                                     .Elements(d + "Table")
                                     .Elements(d + "Rows")
                                     .Elements(d + "element")
                                     .ToList();

            // N.B. there might be a more elegant/efficient way of extracting the values from the (slightly awkward) XML than this.. 
            var searchResults = from item in items
                                select new
                                {
                                    Title = item.Element(d + "Cells").Descendants(d + "Key").First(a => a.Value == "Title").Parent.Element(d + "Value").Value,
                                    Author = item.Element(d + "Cells").Descendants(d + "Key").First(a => a.Value == "Author").Parent.Element(d + "Value").Value,
                                    HitHighlightedSummary = item.Element(d + "Cells").Descendants(d + "Key").First(a => a.Value == "HitHighlightedSummary").Parent.Element(d + "Value").Value,
                                    Path = item.Element(d + "Cells").Descendants(d + "Key").First(a => a.Value == "Path").Parent.Element(d + "Value").Value
                                };

            // data-bind to ListView..
            lvSearchResults.DataSource = searchResults;
            lvSearchResults.DataBind();
        }
    }
}