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Table T(n,k), n>=0 and k>=0: Stern's diatomic array read by antidiagonals (version 5).
0

%I #9 Dec 16 2023 13:10:54

%S 1,0,1,-1,1,1,-3,0,2,1,-2,-1,1,3,1,-7,-1,1,2,4,1,-5,-4,0,3,3,5,1,-8,

%T -3,-1,1,5,4,6,1,-3,-5,-1,2,2,7,5,7,1,-13,-2,-2,1,5,3,9,6,8,1,-10,-9,

%U -1,1,3,8,4,11,7,9,1,-17,-7,-5,0,4,5,11,5,13,8,10,1,-7,-12,-4,-1,1,7,7,14,6,15,9,11,1,-18,-5,-7,-1,3,2,10,9,17,7

%N Table T(n,k), n>=0 and k>=0: Stern's diatomic array read by antidiagonals (version 5).

%F Each row is obtained by copying the previous row but interpolating the sum of pairs of adjacent terms.

%F T(n, 2*k) = T(n-1, k) = T(n, k) - A002487(k).

%F T(n, 2*k+1) = T(n, 2*k) + T(n, 2*k+2); T(0, 0)=1, T(0, 1)=0.

%F The k-th column is an arithmetic progression with : T(n, k) = T(0, k) + n* A002487(k).

%e row n=0 : 1, 0, -1, -3, -2, -7, -5, -8, -3, -13, -10, -17, -7, -18, -11, ...

%e row n=1 : 1, 1, 0, -1, -1, -4, -3, -5, -2, -9, -7, -12, -5, -13, ...

%e row n=2 : 1, 2, 1, 1, 0, -1, -1, -2, -1, -5, -4, -7, -3, ...

%e row n=3 : 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, ...

%e row n=4 : 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, ...

%Y Cf. A049456, A070878, A070879, A049455, A002487.

%K sign,tabl

%O 0,7

%A _Philippe Deléham_, Dec 30 2003