A378277 Denominators in a harmonic triangle, based on products of Fibonacci numbers.

1, 2, 2, 2, 3, 6, 2, 3, 10, 15, 2, 3, 10, 24, 40, 2, 3, 10, 24, 65, 104, 2, 3, 10, 24, 65, 168, 273, 2, 3, 10, 24, 65, 168, 442, 714, 2, 3, 10, 24, 65, 168, 442, 1155, 1870, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 4895, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 7920, 12816

Offset

1,2

Comments

The harmonic triangle uses the terms of this sequence as denominators, numerators = 1.

The inverse of the harmonic triangle has entries -(Fibonacci(k+1))^2 for 1<=k<n (subdiagonals) and Fibonacci(n) * Fibonacci(n+1) (main diagonal).

Row sums of the harmonic triangle are 1.

Conjecture: Alt. row sums of the harmonic triangle are Fibonacci(n-2) / Fibonacci(n+1), where Fibonacci(-1) = 1.

Links

Table of n, a(n) for n=1..66.

Formula

T(n, k) = Fibonacci(n) * Fibonacci(n+1) if k = n, and Fibonacci(k) * Fibonacci(k+2) if 1 <= k < n.

Row sums are A110035(n) - 1 = -A110034(n+1).

G.f.: A(t, x) = x*t*(1 + t - x*t^2) / ((1 - t) * (1 + x*t) * (1 - 3*x*t + x^2*t^2)).

Example

Triangle T(n, k) for 1 <= k <= n starts:
n\ k :  1  2   3   4   5    6    7     8     9    10     11
===========================================================
   1 :  1
   2 :  2  2
   3 :  2  3   6
   4 :  2  3  10  15
   5 :  2  3  10  24  40
   6 :  2  3  10  24  65  104
   7 :  2  3  10  24  65  168  273
   8 :  2  3  10  24  65  168  442   714
   9 :  2  3  10  24  65  168  442  1155  1870
  10 :  2  3  10  24  65  168  442  1155  3026  4895
  11 :  2  3  10  24  65  168  442  1155  3026  7920  12816
  etc.

Prog

(PARI) T(n, k)=if(k==n, Fibonacci(n)*Fibonacci(n+1), Fibonacci(k)*Fibonacci(k+2))

Crossrefs

Cf. A000045, A110034, A110035, A001654 (main diagonal), A059929 (subdiagonals).

Sequence in context: A268595 A299019 A347879 * A070610 A156820 A104346

Adjacent sequences: A378274 A378275 A378276 * A378278 A378279 A378280

Keyword

nonn,easy,tabl,frac,new

Author

Werner Schulte, Nov 21 2024

Status

approved