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A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.

%I A046937 #20 Mar 27 2022 10:54:23
%S A046937 1,2,3,3,5,8,8,11,16,24,24,32,43,59,83,83,107,139,182,241,324,324,407,
%T A046937 514,653,835,1076,1400,1400,1724,2131,2645,3298,4133,5209,6609,6609,
%U A046937 8009,9733,11864,14509,17807,21940,27149,33758
%N A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.
%H A046937 Reinhard Zumkeller, <a href="/A046937/b046937.txt">Rows n = 0..120 of triangle, flattened</a>
%e A046937 Triangle starts:
%e A046937 [0] [   1]
%e A046937 [1] [   2,    3]
%e A046937 [2] [   3,    5,    8]
%e A046937 [3] [   8,   11,   16,    24]
%e A046937 [4] [  24,   32,   43,    59,    83]
%e A046937 [5] [  83,  107,  139,   182,   241,   324]
%e A046937 [6] [ 324,  407,  514,   653,   835,  1076,  1400]
%e A046937 [7] [1400, 1724, 2131,  2645,  3298,  4133,  5209,  6609]
%e A046937 [8] [6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758]
%p A046937 # Compare the analogue algorithm for the Catalan triangle in A350584.
%p A046937 A046937Triangle := proc(len) local A, P, T, n; A := [2]; P := [1]; T := [[1]];
%p A046937 for n from 1 to len-1 do P := ListTools:-PartialSums([A[-1], op(P)]);
%p A046937 A := P; T := [op(T), P] od; T end:
%p A046937 A046937Triangle(9): ListTools:-Flatten(%); # _Peter Luschny_, Mar 27 2022
%t A046937 a[0, 0] = 1; a[1, 0] = 2; a[n_, 0] := a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 06 2013 *)
%o A046937 (Haskell)
%o A046937 a046937 n k = a046937_tabl !! n !! k
%o A046937 a046937_row n = a046937_tabl !! n
%o A046937 a046937_tabl = [1] : iterate (\row -> scanl (+) (last row) row) [2,3]
%o A046937 -- _Reinhard Zumkeller_, Jan 13 2013
%Y A046937 Borders give A038561.
%Y A046937 Cf. A011971.
%K A046937 tabl,easy,nice,nonn
%O A046937 0,2
%A A046937 _N. J. A. Sloane_, _R. K. Guy_