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A117894
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Triangle, row sums = Pell numbers, A000129.
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2
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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, 7, 17, 33, 63, 111, 169, 1, 8, 8, 20, 40, 80, 152, 268, 408, 1, 9, 9, 23, 47, 97, 193, 367, 647, 985
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OFFSET
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0,5
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COMMENTS
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Deleting the right border gives triangle A117895.
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LINKS
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FORMULA
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Rows are composed of difference terms of triangle A117584.
T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1.
T(n, 1) = n for n >= 1.
T(n, 2) = n for n >= 2.
T(n, n-1) = 2*[n=0] + A078343(n). (End)
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EXAMPLE
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First few rows of the triangle are:
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, 7, 17, 33, 63, 111, 169;
...
Row 4 of A117584 = (1, 4, 7, 12). Difference terms (1, 3, 3, 5) = row 4 of A117894.
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MATHEMATICA
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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 *)
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PROG
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(Magma) Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
[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
(Sage)
def P(n): return lucas_number1(n, 2, -1)
def A117894(n, k): return 1 if (k==0) else (k-n)*P(k+1) + (3*n-3*k+1)*P(k)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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