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Search: id:A135737
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| A135737 |
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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<j<=k; formatted as a table and read by antidiagonals. |
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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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.
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LINKS
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Project Euler, Problem 167: Investigating Ulam sequences
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EXAMPLE
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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,...
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PROGRAM
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(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<<j)); if(i, b=vecextract(b, 2^#b-1-i))); a} /* now this sequence */ 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]))}
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CROSSREFS
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Cf. A001857 = U(2, 3), A007300 = U(2, 5), A003668 = U(2, 7); A100729-A100730 (period); A002858 = U(1, 2), A002859 = U(1, 3), A003666 = U(1, 4), A003667 = U(1, 5).
Sequence in context: A139712 A075365 A075274 this_sequence A125179 A035361 A137851
Adjacent sequences: A135734 A135735 A135736 this_sequence A135738 A135739 A135740
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KEYWORD
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nice,nonn,tabl
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AUTHOR
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M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Nov 26 2007
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