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A158786
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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.
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2
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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
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OFFSET
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2,2
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REFERENCES
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H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp. 159-162.
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LINKS
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FORMULA
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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)).
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)
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EXAMPLE
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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;
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MATHEMATICA
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(* 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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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