def A(n,stack=1):##I ran this program with pypy3 which ran much faster than python3 and it took over a day to compute n=4, so n=5 would take months
	#stack=0 produces stepping stones sequence (A337663), stack=1 Stacking stones sequence (A340000)

	##ENTER YOUR FILE NAMES FIRST ON LINES 90+91!!!##

	gridsize=26 ##gridsize to test with
	steppingdist=5 ##distance between 2 different 1's which can't be connected by adding numbers in the cells between them. I think 5 is safe.
	if n<2:n=2
	oo,nx,kk,mm=2,(2**n)*8+2,2,2 ##oo=counter for the 3 configurations of starting 1's, nx=(over) estimate of a(n) to initialise lists, kk=numerate maximum solutions, mm=maximum a(n) so far
	while n<=255:
		g=[[0 for xp in range(gridsize)] for yp in range(gridsize)] 
		dim=[[len(g)//2-2,len(g)//2,len(g)//2-2,len(g)//2-2+oo] for xp in range(nx+n-4)] 
		p=[[0,0] for yp in range(nx+n-4)] 
		c=[[-1,-1] for xp in range(nx+n-4)] 
		r=[0 for xp in range(nx+n-4)] 
		g[len(g)//2-2][len(g)//2-2],g[len(g)//2][len(g)//2-2+oo],i,xx,yy=1,1,4-n,1,1
		while i>3-n:
			ii,ip,io=i+n-4,i+n-5,i+n-3
			if i<2:
				xx=len(g)
				if c[ii][0]>-1: xx=c[ii][0]
				while xx>0:
					x=(len(g)+(len(g)//2-xx))%len(g)
					if abs(x-(len(g)//2-1))>steppingdist+1:
						iz=ii-1
						while iz>=0:
							if abs(x-p[iz][0])<=steppingdist:break
							iz-=1
						if iz>-1:
							xx-=1
							continue
					if (ii!=0 or x>=len(g)//2-1) and x<=len(g)-(5+ii*2) and x>=(5+ii*2):
						yy=len(g)
						if c[ii][1]>-1 and xx==c[ii][0]: yy=c[ii][1]
						if yy<1:
							xx-=1
							continue
						while yy>0:
							y=(len(g)+(len(g)//2-yy))%len(g)
							if abs(y-(len(g)//2-1))>steppingdist+1:
								iz=ii-1
								while iz>=0:
									if abs(y-p[iz][1])<=steppingdist:break
									iz-=1
								if iz>-1:
									yy-=1
									continue
							if y<=len(g)-(5+ii*2) and y>=(5+ii*2):
								if g[x][y]+g[x-1][y-1]+g[x-1][y]+g[x-1][y+1]+g[x][y-1]+g[x][y+1]+g[x+1][y-1]+g[x+1][y]+g[x+1][y+1]==0:
									g[x][y],dim[io][0],dim[io][1],dim[io][2],dim[io][3]=1,min(dim[ii][0],x),max(dim[ii][1],x),min(dim[ii][2],y),max(dim[ii][3],y)
									p[ii],c[ii]=[x,y],[xx,yy-1]
									xx=-1
									break
							yy-=1
					xx-=1
			else:
				xx=dim[ii][1]+1
				if c[ii][0]>-1: xx=c[ii][0]
				while xx>=dim[ii][0]-1:
					y=dim[ii][3]+1
					if c[ii][1]>-1 and xx==c[ii][0]: y=c[ii][1]
					if y<dim[ii][2]-1:
						xx-=1
						continue
					while y>=dim[ii][2]-1:
						if ((xx==len(g)//2-1 and y>=len(g)//2-3 and y<=len(g)//2-1) or i>2) and (stack==1 or g[xx][y]==0):
							if g[xx-1][y-1]+g[xx-1][y]+g[xx-1][y+1]+g[xx][y-1]+g[xx][y+1]+g[xx+1][y-1]+g[xx+1][y]+g[xx+1][y+1]==i:
								r[ii]=g[xx][y]
								g[xx][y]=i
								dim[io][0],dim[io][1],dim[io][2],dim[io][3]=min(dim[ii][0],xx),max(dim[ii][1],xx),min(dim[ii][2],y),max(dim[ii][3],y)
								p[ii],c[ii]=[xx,y],[xx,y-1]
								xx=-1
								break
						y-=1
					xx-=1
			if xx>-1:
				i-=1
				if i>(n-1)*(10+stack)+4: ##to print only solutions close to maximum (16,27,38,49...)
					if i==mm:
						kk=kk+1
					elif i>mm:
						kk=1
						mm=i
					print(str(n)+"__"+str(i)+"/"+str(kk))
					f=0

					##!!! THIS PROGRAM SAVES TO FILES !!!
					##!!! CREATE 2 EMPTY FILES FIRST AND ENTER THEIR PATHS HERE !!!

					if i==mm: f = open('/path/file.txt', 'a') ##file for solutions that are new records or equal to it
					else: f = open('/path/file2.txt', 'a') ##file for all other big solutions
					f.write(str(n)+"__"+str(i)+"/"+str(kk)+"\n")
					y=dim[ii][2]
					while y<=dim[ii][3]:
						x=dim[ii][0]
						while x<=dim[ii][1]:
							if g[x][y]==0:
								print("   ",end='')
								f.write("   ")
							else:
								if g[x][y]<10:
									print(" ",end='')
									f.write(" ")
								print(" "+str(g[x][y]),end='')
								f.write(" "+str(g[x][y]))
							x+=1
						print("")
						f.write("\n")
						y+=1
					if stack==1:
						jj,st,rr=ip,"",r[:]
						while jj>n-2:
							if rr[jj]>0:
								if st=="":st=str(jj-n+4)+" -> "+str(rr[jj])
								else: st+=", "+str(jj-n+4)+" -> "+str(rr[jj])
								jjj=rr[jj]+n-4
								while rr[jjj]>0:
									rj,st=rr[jjj],st+" -> "+str(rr[jjj])
									rr[jjj]=0
									jjj=rj+n-4
							jj-=1
						if st!="":
							print("")
							f.write("\n")
							print(st)
							f.write(st+"\n")
							print("")
							f.write("\n")
					f.close()
				g[p[ip][0]][p[ip][1]],c[ii]=r[ip],[-1,-1]
			else: i+=1
		oo-=1
		if oo<0:
			oo,n=2,n+1
			nx,kk,mm=(2**n)*8+2,2,2