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

Author

Roger L. Bagula and Gary W. Adamson, Mar 25 2009

Keywords

• nonn
• tabl

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

Crossrefs

Cf. A000032, A002878, A004146, A158786.

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