A035313 (Largest) diagonal of the Zorach additive triangle A035312.

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

Offset: 0

Author

Alex Zorach

Keywords

• nonn
• easy
• nice

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)=ai-1(n+1)-ai-1(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 n-th 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 n-th differences, which would imply that the sequence and its n-th 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 n-th differences yield the n-th 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

Examples

			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...
		

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

Crossrefs

Cf. A035311, A035312, A189713, A035358.

Programs

Haskell

-- See link for Haskell program.

Mathematica

(* Assuming n <= t(n,1) <= 3n *) rows = 29; uniqueQ[t1_, n_] := (t[n, 1] = t1; Do[t[n, k] = t[n, k-1] + t[n-1, k-1], {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, n-1}, {k, 1, m}]]], uniqueQ[#, n] &, 1][[1]]; Last /@ Table[t[n, k], {n, 1, rows}, {k, 1, n}] (* Jean-François Alcover, Jun 05 2012 *)

Extensions

More terms from Christian G. Bower and Dean Hickerson