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A337277 Stern's triangle read by rows.

%I A337277 #31 Nov 19 2020 19:33:54
%S A337277 1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,3,2,3,1,3,2,3,1,2,1,1,1,1,2,1,3,2,3,1,
%T A337277 4,3,5,2,5,3,4,1,4,3,5,2,5,3,4,1,3,2,3,1,2,1,1,1,1,2,1,3,2,3,1,4,3,5,
%U A337277 2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,4,3,5,2,5,3,4,1,3,2,3,1,2,1,1
%N A337277 Stern's triangle read by rows.
%C A337277 The first two rows are 1, then 1,1,1. To get row n, copy row n-1, and insert c+d between every pair of adjacent terms c,d, and finally insert a 1 at the beginning and end of the row.
%C A337277 The maximum value in row n is A000045(n+1). - _Alois P. Heinz_, Sep 09 2020
%D A337277 Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.
%H A337277 Alois P. Heinz, <a href="/A337277/b337277.txt">Rows n = 0..14, flattened</a>
%H A337277 Richard P. Stanley, <a href="https://arxiv.org/abs/1901.04647">Some Linear Recurrences Motivated by Stern's Diatomic Array</a>, arXiv:1901.04647 [math.CO], 2019.
%F A337277 T(n,n) = A002487(n+1). - _Alois P. Heinz_, Sep 09 2020
%e A337277 Triangle begins:
%e A337277   1;
%e A337277   1, 1, 1;
%e A337277   1, 1, 2, 1, 2, 1, 1;
%e A337277   1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1;
%e A337277   1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1;
%e A337277   ...
%p A337277 T:= proc(n) option remember; `if`(n=0, 1, (L-> [1, L[1], seq(
%p A337277       [L[i-1]+L[i], L[i]][], i=2..nops(L)), 1][])([T(n-1)]))
%p A337277     end:
%p A337277 seq(T(n), n=0..6);  # _Alois P. Heinz_, Sep 09 2020
%t A337277 Nest[Append[#, Flatten@ Join[{1}, If[Length@ # > 1, Map[{#1, #1 + #2} & @@ # &, Partition[#[[-1]], 2, 1] ], {}], {#[[-1, -1]]}, {1}]] &, {{1}}, 5] // Flatten (* _Michael De Vlieger_, Sep 09 2020 *)
%Y A337277 Cf. A000045, A002487, A049456, A334627.
%Y A337277 Row sums give A000244.
%Y A337277 Row lengths give A126646.
%K A337277 nonn,tabf
%O A337277 0,7
%A A337277 _N. J. A. Sloane_, Sep 09 2020