A135737 Ulam type (1-additive) sequences u[1]=2, u[2]=2n+1, u[k+1] is least unique sum u[i]+u[j]>u[k], 1<=i

2, 3, 2, 5, 5, 2, 7, 7, 7, 2, 8, 9, 9, 9, 2, 9, 11, 11, 11, 11, 2, 13, 12, 13, 13, 13, 13, 2, 14, 13, 15, 15, 15, 15, 15, 2, 18, 15, 16, 17, 17, 17, 17, 17, 2, 19, 19, 17, 19, 19, 19, 19, 19, 19, 2, 24, 23, 19, 20, 21, 21, 21, 21, 21, 21, 2, 25, 27, 21, 21, 23, 23, 23, 23, 23, 23, 23

Offset: 1

Author

M. F. Hasler, Nov 26 2007

Keywords

• nice
• nonn
• tabl

Comments

Any of the sequences u=U(2,2n+1) has u[1]=2 and u[n+4]=4n+4; in between these there are the odd numbers 2n+1,...,4n-3. For n>1 there are no other even terms and the sequence of first differences becomes periodic for k>=t (transient phase), such that u[k] = u[k-floor((k-t)/p)*p] + floor((k-t)/p)*d, where p is the period (cf. A100729) and d the fundamental difference (cf. A100730). See the cross-references, especially A002858, for more information.

Examples

			The sequence contains the terms of the table T[n,k] = U(2,2n+1)[k], read by antidiagonals: a[1]=T[1,1]=2, a[2]=T[1,2]=3, a[3]=T[2,1]=2, a[4]=T[1,3]=5,...
n=1: U(2,3)= 2, 3, 5, 7, 8, 9,13,14...
n=2: U(2,5)= 2, 5, 7, 9,11,12,...
n=3: U(2,7)= 2, 7, 9,11,13,...
n=4: U(2,9)= 2, 9,11,...
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000, Apr 10 2025
• Project Euler, Problem 167: Investigating Ulam sequences

Crossrefs

Cf. A001857 = U(2, 3) = row 1, A007300 = U(2, 5) = row 2, A003668 = U(2, 7) = row 3; A100729-A100730 (period).

Cf. A002858 = U(1, 2): this would be row 0, with u[1], u[2] exchanged.

See also: A002859 = U(1, 3), A003666 = U(1, 4), A003667 = U(1, 5).

Programs

PARI

ulam(a,b,Nmax=30,i)=a=[a,b]; b=[a[1]+b]; for( k=3,Nmax, i=1; while(( i<#b && b[i]==b[i+1] && i+=2 ) || ( i>1 && b[i]==b[i-1] && i++),); a=concat(a,b[i]); b=vecsort(concat(vecextract(b,Str("^..",i)),vector(k-1,j,a[k]+a[j]))); i=0; for(j=1,#b-2, if( b[j]==b[j+2], i+=1<A135737(Nmax=100)=local(T=vector(sqrtint(Nmax*2)+1,n, ulam(2,2*n+1, sqrtint(Nmax*2)+2-n)),i,j); vector(Nmax,k,if(j>1,T[i++ ][j-- ],j=i+1;T[i=1][j]))