A158786 Irregular triangle T(n, k) = A000032(n-2*k+1) if (n-2*k) mod 2 = 0, otherwise 25*A000032(n-2*k), read by rows.

1, 25, 4, 1, 100, 25, 11, 4, 1, 275, 100, 25, 29, 11, 4, 1, 725, 275, 100, 25, 76, 29, 11, 4, 1, 1900, 725, 275, 100, 25, 199, 76, 29, 11, 4, 1, 4975, 1900, 725, 275, 100, 25, 521, 199, 76, 29, 11, 4, 1, 13025, 4975, 1900, 725, 275, 100, 25, 1364, 521, 199, 76, 29, 11, 4, 1, 34100, 13025, 4975, 1900, 725, 275, 100, 25

Offset: 2

Author

Roger L. Bagula and Gary W. Adamson, Mar 26 2009

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins as:
     1;
    25;
     4,    1;
   100,   25;
    11,    4,   1;
   275,  100,  25;
    29,   11,   4,   1;
   725,  275, 100,  25;
    76,   29,  11,   4,  1;
  1900,  725, 275, 100, 25;
   199,   76,  29,  11,  4,   1;
  4975, 1900, 725, 275, 100, 25;
		

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 irregular triangle, flattened

Crossrefs

Cf. A000032, A002878, A004146, A158753.

Programs

Mathematica

(* First program *)
e[n_, 0]:= Sqrt[5]*(GoldenRatio^(n) + GoldenRatio^(-n));
e[n_, k_]:= If[k>n-1, 0, (e[n-1, k]*e[n, k-1] +1)/e[n-1, k-1]];
T[n_,k_]:= 5*Rationalize[N[e[n, k]]];
Table[T[n, k], {n, 2, 16}, {k, Mod[n, 2] +1, n-1,2}]//Flatten
(* Second program *)
f[n_]:= f[n]= If[EvenQ[n], LucasL[n-1], 25*LucasL[n-2]];
T[n_, k_]:= f[n-2*k];
Table[T[n, k], {n, 2, 16}, {k, 0, (n-2)/2}]//Flatten (* G. C. Greubel, Dec 06 2021 *)

Sage

def A158786(n,k): return lucas_number2(n-2*k-1,1,-1) if ((n-2*k)%2==0) else 25*lucas_number2(n-2*k-2,1,-1)
flatten([[A158786(n,k) for k in (0..((n-2)//2))] 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) = sqrt(5)*(GoldenRatio^(n) + GoldenRatio^(-n)).

From G. C. Greubel, Dec 06 2021: (Start)

T(n, k) = A000032(n-2*k+1) if (n-2*k) mod 2 = 0, otherwise 25*A000032(n-2*k).

Sum_{k=0..floor(n/2)} T(n, k) = A000032(n) - 2 if (n mod 2 = 0), otherwise 25*(A000032(n-1) - 2). (End)

Extensions

Edited by G. C. Greubel, Dec 06 2021