

A135737


Ulam type (1additive) sequences u[1]=2, u[2]=2n+1, u[k+1] is least unique sum u[i]+u[j]>u[k], 1<=i<j<=k; formatted as a table and read by antidiagonals.


9



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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,...,4n3. 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[kfloor((kt)/p)*p] + floor((kt)/p)*d, where p is the period (cf. A100729) and d the fundamental difference (cf. A100730). See the crossreferences, especially A002858, for more information.


LINKS



EXAMPLE

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,...


PROG

(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[i1] && i++), ); a=concat(a, b[i]); b=vecsort(concat(vecextract(b, Str("^..", i)), vector(k1, j, a[k]+a[j]))); i=0; for(j=1, #b2, if( b[j]==b[j+2], i+=1<<j)); if(i, b=vecextract(b, 2^#b1i))); a
/* now this sequence */
A135737(Nmax=100)=local(T=vector(sqrtint(Nmax*2)+1, n, ulam(2, 2*n+1, sqrtint(Nmax*2)+2n)), i, j); vector(Nmax, k, if(j>1, T[i++ ][j ], j=i+1; T[i=1][j]))


CROSSREFS

Cf. A001857 = U(2, 3), A007300 = U(2, 5), A003668 = U(2, 7); A100729A100730 (period); A002858 = U(1, 2), A002859 = U(1, 3), A003666 = U(1, 4), A003667 = U(1, 5).


KEYWORD



AUTHOR



STATUS

approved



