A117584 Generalized Pellian triangle.
1, 1, 2, 1, 3, 5, 1, 4, 7, 12, 1, 5, 9, 17, 29, 1, 6, 11, 22, 41, 70, 1, 7, 13, 27, 53, 99, 169, 1, 8, 15, 32, 65, 128, 239, 408, 1, 9, 17, 37, 77, 157, 309, 577, 985, 1, 10, 19, 42, 89, 186, 379, 746, 1393, 2378
Offset: 1
Examples
First few rows of the triangle are: 1; 1, 2; 1, 3, 5; 1, 4, 7, 12; 1, 5, 9, 17, 29; 1, 6, 11, 22, 41, 70; 1, 7, 13, 27, 53, 99, 169; ... The triangle rows are antidiagonals of the generalized Pellian array: 1, 2, 5, 12, 29, ... 1, 3, 7, 17, 41, ... 1, 4, 9, 22, 53, ... 1, 5, 11, 27, 65, ... ... For example, in the row (1, 5, 11, 27, 65, ...), 65 = 2*27 + 11.
Links
- G. C. Greubel, Rows n = 1..100, flattened
Programs
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Magma
P:= func< n | Round( ((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2)) ) >; T:= func< n,k | P(k) + (n-1)*P(k-1) >; [T(n-k+1, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 05 2021
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Mathematica
T[n_, k_]:= Fibonacci[k, 2] + (n-1)*Fibonacci[k-1, 2]; Table[T[n-k+1, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2021 *) -
Sage
def T(n,k): return lucas_number1(k,2,-1) + (n-1)*lucas_number1(k-1,2,-1) flatten([[T(n-k+1, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 05 2021
Formula
Antidiagonals of the generalized Pellian array. First row of the array = A000129: (1, 2, 5, 12, ...). n-th row of the array starts (1, n+1, ...); as a Pellian sequence.
From G. C. Greubel, Jul 05 2021: (Start)
T(n, k) = P(k) + (n-1)*P(k-1), where P(n) = A000129(n) (square array).
Sum_{k=1..n} T(n-k+1, k) = A117185(n). (End)