A070825 -id:A070825 - cp's OEIS frontend

A385733 Triangle read by rows: the denominators of the Lucas triangle.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2, 14, 2, 3, 1, 1, 1, 1, 3, 3, 7, 7, 3, 3, 1, 1, 1, 1, 1, 1, 7, 77, 7, 1, 1, 1, 1, 1, 1, 1, 1, 7, 77, 77, 7, 1, 1, 1, 1, 1, 1, 3, 2, 1, 11, 99, 11, 1, 2, 3, 1, 1

Offset: 0

Peter Luschny, Jul 08 2025

			Triangle begins:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1, 1;
  [3] 1, 1, 1, 1;
  [4] 1, 1, 3, 1,  1;
  [5] 1, 1, 3, 3,  1, 1;
  [6] 1, 1, 1, 2,  1, 1, 1;
  [7] 1, 1, 1, 2,  2, 1, 1, 1;
  [8] 1, 1, 3, 2, 14, 2, 3, 1, 1;
  [9] 1, 1, 3, 3,  7, 7, 3, 3, 1, 1;

Diana L. Wells, The Fibonacci and Lucas triangles modulo 2, Fibonacci Quart. 32, no. 2 (1994), p. 112.

Cf. A385732 (numerators), A070825 (Lucanorial), A003266 (Fibonorial), A010048 (Fibonomial).

c := arccsch(2) - I*Pi/2:
LT := (n, k) -> mul(I^j*cosh(c*j), j = k + 1..n) / mul(I^j*cosh(c*j), j = 1..n - k):
T := (n, k) -> denom(simplify(LT(n, k))): seq(seq(T(n, k), k = 0..n), n = 0..12);

T[n_, k_] := With[{c = ArcCsch[2] - I Pi/2}, Product[I^j Cosh[c j], {j, k + 1, n}] / Product[I^j Cosh[c j], {j, 1, n - k}]];
Table[Simplify[T[n, k]], {n, 0, 8}, {k, 0, n}] // Flatten // Denominator

LT(n, k) = Product_{j=k+1..n} i^j*cosh(c*j) / Product_{j=1..n-k} i^j*cosh(c*j) where c = arccsch(2) - i*Pi/2 and i is the imaginary unit.

T(n, k) = denominator(LT(n, k)).

A203010 (n-1)-st elementary symmetric function of first n Lucas numbers, starting with L(1)=1.

Original entry on oeis.org

1, 4, 19, 145, 1679, 31146, 919866, 43716030, 3345087696, 413168662224, 82432477483344, 26585428576089600, 13864587294260493504, 11694921751248976025856, 15957837208927564640940096, 35227081534568618432596098240

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

From R. J. Mathar, Oct 01 2016 (Start):
The k-th elementary symmetric functions of the A000204(j), j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:
1
1 1
1 4 3
1 8 19 12
1 15 75 145 84
1 26 240 970 1679 924
1 44 708 5290 19139 31146 16632
1 73 1984 25822 172549 586177 919866 482328
1 120 5415 119070 1386183 8695980 28470185 43716030 22669416
This here is the first subdiagonal. The diagonal is A070825. The 2nd column is A027961. (End)

Cf. A203009.

f[k_] := LucasL[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203010 *)

A217473 Product of the first n+1 odd-indexed Lucas numbers A000032.

Original entry on oeis.org

1, 4, 44, 1276, 96976, 19298224, 10054374704, 13714167096256, 48973290700730176, 457851294761126415424, 11206368290573330143917824, 718092873691648422292110244096, 120467978583384630972146706659789056

Offset: 0

Wolfdieter Lang, Oct 12 2012

This sequence is used in A217472 to produce the integer instead of rational Ozeki-Prodinger triangle, and the corresponding sequence A217474.

Cf. A000032, A001622 (phi), 2*A070825 (product of the first n+1 Lucas numbers).

Cf. A217472, A217474.

FoldList[Times, LucasL[Range[1, 26, 2]]] (* Amiram Eldar, Jul 09 2025 *)

a(n) = Product_{i=0..n} L(2*i+1), n >= 0, with the Lucas numbers L=A000032.

a(n) ~ c * phi^(n*(n+2)), where c = Product_{k>=1} (1 - 1/phi^(4*k+2)) = 0.93529381211738925891... . - Amiram Eldar, Jul 09 2025

A103845 Product of first n Lucas numbers, plus one.

Original entry on oeis.org

1, 2, 4, 13, 85, 925, 16633, 482329, 22669417, 1722875617, 211913700769, 42170826452833, 13579006117811905, 7074662187380001985, 5963940223961341672513, 8134814465483270041306369

Offset: 0

Jonathan Vos Post, Mar 30 2005

Prime for n = 1, 3, 6; semiprime for n = 2, 4, 7, 8, 9, 10, 13, 14, 15, 23, 26. a(n) asymptotic to Phi^A000217(n).

			a(10) = 1 * 3 * 4 * 7 * 11 * 18 * 29 * 47 * 76 * 123 + 1 = 211913700769 = 349 * 607202581 which is semiprime.

Cf. A000045, A000204, A000217, A052449, A103815.

Join[{1},Rest[FoldList[Times,1,LucasL[Range[20]]]]+1] (* Harvey P. Dale, Oct 25 2011 *)

a(n) = Product[Lucasi[k], {k, 1, n}]+1. a(n) = Product[A000217(k), {k, 1, n}]+1.

a(n) = A070825(n)+1, n>0. - R. J. Mathar, Dec 22 2010

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A385733 Triangle read by rows: the denominators of the Lucas triangle.

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A203010 (n-1)-st elementary symmetric function of first n Lucas numbers, starting with L(1)=1.

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A217473 Product of the first n+1 odd-indexed Lucas numbers A000032.

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A103845 Product of first n Lucas numbers, plus one.

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