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%I #47 Feb 16 2017 03:16:10
%S 1,3,9,26,66,154,346,771,1726,3887,8768,19700,43890,96717,210665,
%T 453893,968903,2053260,4328489,9093971,19068611,39943689,83628399,
%U 175018523,366081209,765102907,1597315656,3330380593,6933810145
%N (Largest) diagonal of the Zorach additive triangle A035312.
%C From Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 22 2007: (Start)
%C Starting with 1, smallest sequence for which:
%C all its terms a1(n).............................. 1,3,9,26,66
%C all terms of first differences a2(n)=a1(n+1)-a1(n) 2,6,17,40
%C all terms of second differences a3(n)=a2(n+1)-a2(n) 4,11,23
%C ...
%C all terms of (1+i)th differences ai(n)=ai-1(n+1)-ai-1(n)
%C are different for any n and any i (End)
%C 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
%C 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
%C Conjecture: lim(n->infinity, a(n+1)/a(n)) = 2. - _David W. Wilson_, Feb 26 2012
%C 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
%C 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
%H Reinhard Zumkeller, <a href="/A035313/b035313.txt">Table of n, a(n) for n = 0..1000</a>
%H A. C. Zorach, <a href="http://www.cazort.net/static/triangle.php">Additive triangle</a>
%H A. C. Zorach, <a href="/A035312/a035312_2.hs.txt">Haskell programs for sequences in connection with Zorach additive triangle</a>
%e 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...
%t (* 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 *)
%o See link for Haskell program.
%Y Cf. A035311, A035312, A189713, A035358.
%K nonn,easy,nice
%O 0,2
%A _Alex Zorach_
%E More terms from _Christian G. Bower_ and _Dean Hickerson_
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