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%I #9 Dec 08 2021 07:34:45
%S 1,25,4,1,100,25,11,4,1,275,100,25,29,11,4,1,725,275,100,25,76,29,11,
%T 4,1,1900,725,275,100,25,199,76,29,11,4,1,4975,1900,725,275,100,25,
%U 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
%N 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.
%D H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp. 159-162.
%H G. C. Greubel, <a href="/A158786/b158786.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F 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)).
%F From _G. C. Greubel_, Dec 06 2021: (Star)
%F T(n, k) = A000032(n-2*k+1) if (n-2*k) mod 2 = 0, otherwise 25*A000032(n-2*k).
%F Sum_{k=0..floor(n/2)} T(n, k) = A000032(n) - 2 if (n mod 2 = 0), otherwise 25*(A000032(n-1) - 2). (End)
%e Irregular triangle begins as:
%e 1;
%e 25;
%e 4, 1;
%e 100, 25;
%e 11, 4, 1;
%e 275, 100, 25;
%e 29, 11, 4, 1;
%e 725, 275, 100, 25;
%e 76, 29, 11, 4, 1;
%e 1900, 725, 275, 100, 25;
%e 199, 76, 29, 11, 4, 1;
%e 4975, 1900, 725, 275, 100, 25;
%t (* First program *)
%t e[n_, 0]:= Sqrt[5]*(GoldenRatio^(n) + GoldenRatio^(-n));
%t e[n_, k_]:= If[k>n-1, 0, (e[n-1, k]*e[n, k-1] +1)/e[n-1, k-1]];
%t T[n_,k_]:= 5*Rationalize[N[e[n, k]]];
%t Table[T[n, k], {n, 2, 16}, {k, Mod[n, 2] +1, n-1,2}]//Flatten
%t (* Second program *)
%t f[n_]:= f[n]= If[EvenQ[n], LucasL[n-1], 25*LucasL[n-2]];
%t T[n_, k_]:= f[n-2*k];
%t Table[T[n, k], {n, 2, 16}, {k, 0, (n-2)/2}]//Flatten (* _G. C. Greubel_, Dec 06 2021 *)
%o (Sage)
%o 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)
%o flatten([[A158786(n,k) for k in (0..((n-2)//2))] for n in (2..16)]) # _G. C. Greubel_, Dec 06 2021
%Y Cf. A000032, A002878, A004146, A158753.
%K nonn,tabf
%O 2,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 26 2009
%E Edited by _G. C. Greubel_, Dec 06 2021
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