

A035313


(Largest) diagonal of the Zorach additive triangle A035312.


11



1, 3, 9, 26, 66, 154, 346, 771, 1726, 3887, 8768, 19700, 43890, 96717, 210665, 453893, 968903, 2053260, 4328489, 9093971, 19068611, 39943689, 83628399, 175018523, 366081209, 765102907, 1597315656, 3330380593, 6933810145
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OFFSET

0,2


COMMENTS

From Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 22 2007: (Start)
Starting with 1, smallest sequence for which:
all its terms a1(n).............................. 1,3,9,26,66
all terms of first differences a2(n)=a1(n+1)a1(n) 2,6,17,40
all terms of second differences a3(n)=a2(n+1)a2(n) 4,11,23
...
all terms of (1+i)th differences ai(n)=ai1(n+1)ai1(n)
are different for any n and any i (End)
Which is to say, this sequence is the lexicographically earliest sequence of positive integers such that the sequence itself and its nth differences for n >= 1 are pairwise disjoint.  David W. Wilson, Feb 26 2012
Conjecturally, every positive integer occurs in the sequence or one of its nth differences, which would imply that the sequence and its nth differences partition the positive integers.  David W. Wilson, Feb 26 2012
Conjecture: lim(n>infinity, a(n+1)/a(n)) = 2.  David W. Wilson, Feb 26 2012
Note that the nth differences yield the nth subdiagonals (parallels to the right edge) in the triangle A035312. Therefore Lallouet's statement and Wilson's 1st comment above are just rephrasing the definition of that triangle.  M. F. Hasler, May 09 2013
Binomial transform of A035311. Hence, from the observed asymptotic equality A035311(n) ~ 2*n, a stronger statement than the one given above follows: a(n) ~ n*2^n.  Andrey Zabolotskiy, Feb 08 2017


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. C. Zorach, Additive triangle
A. C. Zorach, Haskell programs for sequences in connection with Zorach additive triangle


EXAMPLE

Start with 1; 2 is the next, then add 1+2 to get 3, then 4 is next, then 4+2=6 and 6+3 is 9, then 5 is not next because 5+4=9 and 9 was already used, so 7 is next...which ultimately generates 26 in the final column...


MATHEMATICA

(* Assuming n <= t(n, 1) <= 3n *) rows = 29; uniqueQ[t1_, n_] := (t[n, 1] = t1; Do[t[n, k] = t[n, k1] + t[n1, k1], {k, 2, n}]; n*(n+1)/2 == Length[ Union[ Flatten[ Table[t[m, k], {m, 1, n}, {k, 1, m}]]]]); t[n_, 1] := t[n, 1] = Select[ Complement[Range[n, 3 n], Flatten[ Table[t[m, k], {m, 1, n1}, {k, 1, m}]]], uniqueQ[#, n] &, 1][[1]]; Last /@ Table[t[n, k], {n, 1, rows}, {k, 1, n}] (* JeanFrançois Alcover, Jun 05 2012 *)


PROG

See link for Haskell program.


CROSSREFS

Cf. A035311, A035312, A189713, A035358.
Sequence in context: A235538 A218916 A037260 * A055293 A034531 A258097
Adjacent sequences: A035310 A035311 A035312 * A035314 A035315 A035316


KEYWORD

nonn,easy,nice


AUTHOR

Alex Zorach


EXTENSIONS

More terms from Christian G. Bower and Dean Hickerson


STATUS

approved



