1,1

An Ulam-type sequence - see A002858 for many further references, comments, etc.

I have a note saying that this is periodic mod 126. Is that correct? - N. J. A. Sloane (njas(AT)research.att.com), Apr 29 2006.

Comments from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 24 2006: "Concerning the conjecture about periodicity mod 126. Out of the first 300 terms, only the 2 and 12 are even. But if you neglect those first 6 terms, mod 2 they're all odd, mod 9 it goes: 0 4 6 1 7 4 6 8 7 2 4 6 8 5 0 8 1 2 5 7 0 2 4 6 1 5 0 2 8 1 5 7 which appears to repeat indefinitely and mod 7 it goes: 0 2 6 1 3 0 2 6 5 4 6 1 2 6 1 3 5 4 1 2 4 0 5 0 2 4 6 1 5 2 6 1 which also appears to repeat indefinitely.

"So it seems as though neglecting the first few terms, it is indeed periodic mod 126 with period 32. In fact it appears that after the first few terms, a(n+32) = a(n) + 126. But this is only based on the first few hundred terms and is not proved!

"The Mathworld link cites a proof that sequences of this type (2,n) have only two even terms and another proof that sequences with only finitely many even terms must eventually have periodic first differences. So I think the period 32 difference of 126 conjecture may be proved in those references."

Given that the sequence of first differences is periodic with period 32 after the first 6 terms (3,2,2,2,1,1), the repeating digits being p=(2,4,4,4,2,6,2,4,2,2,4,2,4,6,6,2,2,8,4,2,2,2,6,4,8,2,10,12,2,2,2,2), one can calculate the n-th term (n>6) as a(n)=13+floor((n-7)/32)*S(32)+S(n-7 mod 32) where S(k)=sum(p(i),i=1..k): (S(k);k=0..32)=(0, 2, 6, 10, 14, 16, 22, 24, 28, 30, 32, 36, 38, 42, 48, 54, 56, 58, 66, 70, 72, 74, 76, 82, 86, 94, 96, 106, 118, 120, 122, 124, 126). - M. F. Hasler (maximilian.hasler(AT)gmail.com), Nov 25 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. R. Finch, On the regularity of certain 1-additive sequences, J. Combin. Theory, A60 (1992), 123-130.

R. K. Guy, Unsolved Problems in Number Theory, Section C4.

R. Queneau, Sur les suites s-additives, J. Combin. Theory, A12 (1972), 31-71.

T. D. Noe, Table of n, a(n) for n=1..1000

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.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Project Euler, Problem 167: Investigating Ulam Sequences.

For n>6, a(n+32)=a(n)+126. - T. D. Noe, Jan 21 2008

A007300:=n->if n<7 then [2, 5, 7, 9, 11, 12][n] else floor((n-7)/32)*126+[13, 15, 19, 23, 27, 29, 35, 37, 41, 43, 45, 49, 51, 55, 61, 67, 69, 71, 79, 83, 85, 87, 89, 95, 99, 107, 109, 119, 131, 133, 135, 137][modp(n-7, 32)+1] fi; - M. F. Hasler (maximilian.hasler(AT)gmail.com), Nov 25 2007

theList = {2, 5}; Print[2]; Print[5]; For[i=1, i <= 500, i++, count=0; For[j=1, j <= Length[theList]-1, j++, For[k=j+1, k <= Length[theList], k++, If[theList[[j]]+theList[[k]] == i, count++ ]; ]; ]; If[count == 1, Print[i]; theList = Append[theList, i]; ]; ]; - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 08 2006

Cf. A100729, A003666, A003667, A006844.

Sequence in context: A047482 A062288 A077059 this_sequence A007069 A140139 A130773

Adjacent sequences: A007297 A007298 A007299 this_sequence A007301 A007302 A007303

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)

More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 24 2006

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 08 2006