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%I #12 Oct 09 2021 16:00:27
%S 1,1,1,1,2,2,1,3,3,5,1,4,4,8,12,1,5,5,11,19,29,1,6,6,14,26,46,70,1,7,
%T 7,17,33,63,111,169,1,8,8,20,40,80,152,268,408,1,9,9,23,47,97,193,367,
%U 647,985
%N Triangle, row sums = Pell numbers, A000129.
%C Deleting the right border gives triangle A117895.
%H G. C. Greubel, <a href="/A117894/b117894.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Rows are composed of difference terms of triangle A117584.
%F Rows sum to Pell numbers, A000129.
%F From _G. C. Greubel_, Sep 27 2021: (Start)
%F T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1.
%F T(n, k) = A117584(n+1, k+1) - A117584(n+1, k) with T(n, 0) = 1.
%F T(n, 1) = n for n >= 1.
%F T(n, 2) = n for n >= 2.
%F T(n, n) = [n=0] + A000129(n).
%F T(n, n-1) = 2*[n=0] + A078343(n). (End)
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 1, 2, 2;
%e 1, 3, 3, 5;
%e 1, 4, 4, 8, 12;
%e 1, 5, 5, 11, 19, 29;
%e 1, 6, 6, 14, 26, 46, 70;
%e 1, 7, 7, 17, 33, 63, 111, 169;
%e ...
%e Row 4 of A117584 = (1, 4, 7, 12). Difference terms (1, 3, 3, 5) = row 4 of A117894.
%t T[n_, k_]:= T[n, k]= If[k==0, 1, (k-n)*Fibonacci[k+1, 2] + (3*n-3*k+1)*Fibonacci[k, 2]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 27 2021 *)
%o (Magma) Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
%o [k eq 0 select 1 else (k-n)*Pell(k+1) + (3*n-3*k+1)*Pell(k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 27 2021
%o (Sage)
%o def P(n): return lucas_number1(n, 2, -1)
%o def A117894(n,k): return 1 if (k==0) else (k-n)*P(k+1) + (3*n-3*k+1)*P(k)
%o flatten([[A117894(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 27 2021
%Y Cf. A000129, A078343, A117584, A117895.
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, Mar 30 2006
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