A158753 Triangle T(n, k) = A000032(2*(n-k) + 1), read by rows.
1, 4, 1, 11, 4, 1, 29, 11, 4, 1, 76, 29, 11, 4, 1, 199, 76, 29, 11, 4, 1, 521, 199, 76, 29, 11, 4, 1, 1364, 521, 199, 76, 29, 11, 4, 1, 3571, 1364, 521, 199, 76, 29, 11, 4, 1, 9349, 3571, 1364, 521, 199, 76, 29, 11, 4, 1, 24476, 9349, 3571, 1364, 521, 199, 76, 29, 11, 4, 1
Offset: 2
Examples
Triangle begins as:
1;
4, 1;
11, 4, 1;
29, 11, 4, 1;
76, 29, 11, 4, 1;
199, 76, 29, 11, 4, 1;
521, 199, 76, 29, 11, 4, 1;
1364, 521, 199, 76, 29, 11, 4, 1;
References
- H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp. 159-162.
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
-
Magma
[Lucas(2*n-2*k+1): k in [2..n], n in [2..16]]; // G. C. Greubel, Dec 06 2021
-
Mathematica
Table[LucasL[2*(n-k) + 1], {n, 2, 16}, {k, 2, n}]//Flatten (* G. C. Greubel, Dec 06 2021 *) -
Sage
flatten([[lucas_number2(2*(n-k)+1, 1, -1) for k in (2..n)] for n in (2..16)]) # G. C. Greubel, Dec 06 2021
Formula
T(n, k) = 5*e(n, k), where e(n,k) = (e(n-1, k)*e(n, k-1) + 1)/e(n-1, k-1), and e(n, 0) = GoldenRatio^(n) + GoldenRatio^(-n).
Sum_{k=0..n} T(n, k) = A004146(n-1).
T(n, k) = A000032(2*(n-k) + 1). - G. C. Greubel, Dec 06 2021
Extensions
Edited by G. C. Greubel, Dec 06 2021