A035311 First column of the Zorach additive triangle A035312.

1, 2, 4, 7, 5, 8, 10, 14, 15, 19, 16, 20, 21, 22, 27, 30, 28, 33, 32, 38, 37, 39, 44, 45, 51, 46, 47, 54, 55, 52, 59, 62, 60, 64, 50, 67, 72, 78, 74, 68, 81, 82, 80, 85, 79, 87, 86, 91, 90, 94, 102, 98, 103, 99, 105, 100, 108, 112, 110, 113, 127, 128, 125, 133, 118, 123

Offset: 0

Author

Alex Zorach

Keywords

• nonn
• easy
• nice

Comments

One can see that (least unused number in rows 1 through n of A035312) ~ A035311(n) ~ 2n. (Asymptotic equality, and the first quantity does not exceed either of the two others.) - M. F. Hasler, May 09 2013

Links

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

Crossrefs

Cf. A035312, A035313, A189713, A035358.

Programs

Haskell

-- See link for Haskell program.

Mathematica

(* Assuming n <= t(n,1) <= 3n *) 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]]; Table[t[n, 1], {n, 1, 66}](* Jean-François Alcover, Dec 02 2011 *)

Extensions

More terms from Christian G. Bower