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

2,2

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A000032, A002878, A004146, A158753.

Sequence in context: A040615 A317325 A040610 * A040611 A236176 A040608

Adjacent sequences: A158783 A158784 A158785 * A158787 A158788 A158789

Keyword

nonn,tabf,changed

Author

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

Extensions

Edited by G. C. Greubel, Dec 06 2021

Status

approved