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A117895 Triangle T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.

%I A117895 #12 Sep 30 2021 18:15:40
%S A117895 1,1,2,1,3,3,1,4,4,8,1,5,5,11,19,1,6,6,14,26,46,1,7,7,17,33,63,111,1,
%T A117895 8,8,20,40,80,152,268,1,9,9,23,47,97,193,367,647,1,10,10,26,54,114,
%U A117895 234,466,886,1562,1,11,11,29,61,131,275,565,1125,2139,3771,1,12,12,32,68,148,316,664,1364,2716,5164,9104
%N A117895 Triangle T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.
%C A117895 Successive deletions of the right borders of triangle A117894 produces triangles whose row sums = generalized Pell sequences starting (1, 2...), (1, 3...), (1, 4...); etc. Row sums of A117894 = A000129: (1, 2, 5...). Row sums of A117895 = A001333: (1, 3, 7...). Deletion of the border of A117895 would produce a triangle with row sums of the Pell sequence A048654 (1, 4, 9...); and so on.
%H A117895 G. C. Greubel, <a href="/A117895/b117895.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117895 Delete right border of triangle A117894. Alternatively, let row 1 = 1 and using the heading 0, 1, 1, 3, 7, 17, 41, 99, 239...(i.e. A001333 starting with 0, 1, 1, 3...); add the first n terms of the heading to n-th row of triangle A117894.
%F A117895 From _G. C. Greubel_, Sep 27 2021: (Start)
%F A117895 T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1.
%F A117895 T(n, 1) = n+1 for n >= 1.
%F A117895 T(n, 2) = n+1 for n >= 2.
%F A117895 T(n, n) = 2*[n=0] + A078343(n). (End)
%e A117895 First few rows of the triangle are:
%e A117895   1;
%e A117895   1, 2;
%e A117895   1, 3, 3;
%e A117895   1, 4, 4,  8;
%e A117895   1, 5, 5, 11, 19;
%e A117895   1, 6, 6, 14, 26, 46;
%e A117895   1, 7, 7, 17, 33, 63, 111;
%e A117895   1, 8, 8, 20, 40, 80, 152, 268;
%e A117895 ...
%e A117895 Row 4, (1, 4, 4, 8) is produced by adding (0, 1, 1, 3) to row 4 of A117894: (1, 3, 3, 5).
%t A117895 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-1}]//Flatten (* _G. C. Greubel_, Sep 27 2021 *)
%o A117895 (Magma) Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
%o A117895 [k eq 0 select 1 else (k-n)*Pell(k+1) + (3*n-3*k+1)*Pell(k): k in [0..n-1], n in [0..12]]; // _G. C. Greubel_, Sep 27 2021
%o A117895 (Sage)
%o A117895 def P(n): return lucas_number1(n, 2, -1)
%o A117895 def A117895(n,k): return 1 if (k==0) else (k-n)*P(k+1) + (3*n-3*k+1)*P(k)
%o A117895 flatten([[A117895(n,k) for k in (0..n-1)] for n in (0..12)]) # _G. C. Greubel_, Sep 27 2021
%Y A117895 Cf. A000129, A001333, A048654, A048655, A048693, A117894.
%K A117895 nonn,tabl
%O A117895 0,3
%A A117895 _Gary W. Adamson_, Mar 30 2006
%E A117895 New name and more terms added by _G. C. Greubel_, Sep 27 2021