A003668 -id:A003668 - cp's OEIS frontend

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

M. F. Hasler, Nov 26 2007

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.

			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,...

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

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).

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]))

A100729 Period of the first difference of Ulam 1-additive sequence U(2,2n+1).

Original entry on oeis.org

32, 26, 444, 1628, 5906, 80, 126960, 380882, 2097152, 1047588, 148814, 8951040, 5406720, 242, 127842440, 11419626400, 12885001946, 160159528116, 687195466408, 6390911336402, 11728121233408, 20104735604736

Offset: 2

Ralf Stephan, Dec 03 2004

nonn

It was proved by Akeran that a(2^k-1) = 3^(k+1) - 1.

Note that a(n)=2^(2n+1) as soon as A100730(n)=2^(2n+3)-2, that happens for n=(m-2)/2 with m>=6 being an even element of A073639.

			For k=2, we have a(3)=3^3-1=26.

Max Alekseyev, Table of n, a(n) for n = 2..31
M. Akeran, On some 1-additive sequences
J. Cassaigne and S. R. Finch, A class of 1-additive sequences and additive recurrences
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.

Cf. A100730 for the fundamental difference, A001857 for U(2, 3), A007300 for U(2, 5), A003668 for U(2, 7).

Cf. also A006844.

a(3) corrected from 25 to 26 by Hugo van der Sanden and Bertram Felgenhauer (int-e(AT)gmx.de), Nov 11 2007
More terms from Balakrishnan V (balaji.iitm1(AT)gmail.com), Nov 15 2007
a(21..31) and b-file from Max Alekseyev, Dec 01 2007

A100730 Fundamental difference of Ulam 1-additive sequence starting U(2,2n+1).

Original entry on oeis.org

126, 126, 1778, 6510, 23622, 510, 507842, 1523526, 8388606, 4194302, 597870, 35791394, 21691754, 2046, 511305630, 45678505642, 51539607546, 640638112422, 2748779069430, 25563645345606, 46912496118442, 80418967640942

Offset: 2

Ralf Stephan, Dec 03 2004

nonn

Max Alekseyev, Table of n, a(n) for n = 2..610
M. Akeran, On some 1-additive sequences
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.

Cf. A100729 for the period, A001857 for U(2, 3), A007300 for U(2, 5), A003668 for U(2, 7).

a(n) = 2 * A046932(2*n+2)

2 more terms from Balakrishnan V (balaji.iitm1(AT)gmail.com), Nov 15 2007
Further new terms and b-file from Max Alekseyev, Dec 01 2007
b-file extended by Max Alekseyev, Aug 17 2015

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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

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A100729 Period of the first difference of Ulam 1-additive sequence U(2,2n+1).

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A100730 Fundamental difference of Ulam 1-additive sequence starting U(2,2n+1).

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