A110035 -id:A110035 - cp's OEIS frontend

A110033 A characteristic triangle for the Fibonacci numbers.

1, -1, 1, 1, -3, 1, 0, 3, -8, 1, 0, 0, 9, -21, 1, 0, 0, 0, 24, -55, 1, 0, 0, 0, 0, 64, -144, 1, 0, 0, 0, 0, 0, 168, -377, 1, 0, 0, 0, 0, 0, 0, 441, -987, 1, 0, 0, 0, 0, 0, 0, 0, 1155, -2584, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3025, -6765, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7920, -17711, 1

Offset: 0

Paul Barry, Jul 08 2005

			Rows begin
1;
-1,1;
1,-3,1;
0,3,-8,1;
0,0,9,-21,1;
0,0,0,24,-55,1;
0,0,0,0,64,-144,1;
0,0,0,0,0,168,-377,1;

Form the n X n Hankel matrices F(i+j-1), 1<=i, j<=n for the Fibonacci numbers and take the characteristic polynomials of these matrices. Triangle rows give coefficients of these characteristic polynomials. (Construction described by Michael Somos in A064831). Diagonal is (-1)^n*F(2n+2). Subdiagonal is A064831. Row sums are A110034. The unsigned matrix has row sums A110035.

A206282 a(n) = ( a(n-1) * a(n-3) + a(n-2) ) / a(n-4), a(1) = a(2) = 1, a(3) = -1, a(4) = -4.

Original entry on oeis.org

1, 1, -1, -4, -5, 1, 9, 11, -4, -25, -31, 9, 64, 79, -25, -169, -209, 64, 441, 545, -169, -1156, -1429, 441, 3025, 3739, -1156, -7921, -9791, 3025, 20736, 25631, -7921, -54289, -67105, 20736, 142129, 175681, -54289, -372100, -459941, 142129, 974169, 1204139

Offset: 1

Michael Somos, Feb 05 2012

This satisfies the same recurrence as Dana Scott's sequence A048736.

			G.f. = x + x^2 - x^3 - 4*x^4 - 5*x^5 + x^6 + 9*x^7 + 11*x^8 - 4*x^9 - 25*x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,-2,0,0,2,0,0,1).

Cf. A000045, A048736, A110034, A110035, A126116.

a206282 n = a206282_list !! (n-1)
a206282_list = 1 : 1 : -1 : -4 :
   zipWith div
     (zipWith (+)
       (zipWith (*) (drop 3 a206282_list)
                    (drop 1 a206282_list))
       (drop 2 a206282_list))
     a206282_list
-- Same program as in A048736, see comment.
-- Reinhard Zumkeller, Feb 08 2012

I:=[1,1,-1,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2))/Self(n-4): n in [1..30]]; // G. C. Greubel, Aug 12 2018

CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x,0,50}], x] (* or *) RecurrenceTable[{a[n] == ( a[n-1]*a[n-3] + a[n-2] )/a[n-4], a[1] == a[2] == 1, a[3] == -1, a[4] == -4}, a, {n,1,50}] (* G. C. Greubel, Aug 12 2018 *)

{a(n) = my(k = n\3); (-1)^k * if( n%3 == 0, fibonacci( k )^2, n%3 == 1, fibonacci( k+2 )^2, fibonacci( k ) * fibonacci( k+3 ) + fibonacci( k+1 ) * fibonacci( k+2 ))};

x='x+O('x^30); Vec(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4 -2*x^6 +x^8 )) \\ G. C. Greubel, Aug 12 2018

G.f.: x * (1 + x - x^2 - 2*x^3 - 3*x^4 - x^5 - x^6 - x^7) / (1 + 2*x^3 - 2*x^6 - x^9).
a(n) = a(-5 - n) = a(n+2) * a(n-2) - a(n+1) * a(n-1) for all n in Z.
a(3*n) = (-1)^n * F(n)^2, a(3*n + 1) = (-1)^n * F(n + 2)^2 where F = Fibonacci A000045.
a(6*n - 4) = - A110034(2*n), a(6*n - 1) = - A110035(2*n), a(3*n + 2) = (-1)^n * A126116(2*n + 3).

A240836 Numbers n such that n^3 = xyz where 2 <= x <= y <= z , n^3+1 = (x-1)(y+1)(z+1).

Original entry on oeis.org

2, 12, 80, 546, 3740, 25632, 175682, 1204140, 8253296, 56568930, 387729212, 2657535552, 18215019650, 124847601996, 855718194320, 5865179758242, 40200540113372, 275538601035360, 1888569667134146, 12944449068903660, 88722573815191472, 608113567637436642

Offset: 1

Naohiro Nomoto, Apr 12 2014

Also, z/y approx = y/x approx = golden ratio.

			546^3 = 338 * 546 * 882, 546^3 + 1 = 337 * 547 * 883.
25632^3 = 15842 * 25632 * 41472, 25632^3 + 1 = 15841 * 25633 * 41473.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A079472, A081016, A110035, A126116.

F:=Fibonacci;; List([1..30], n-> 2*F(2*n)*F(2*n-1) ); # G. C. Greubel, Jul 15 2019

F:=Fibonacci; [2*F(2*n)*F(2*n-1): n in [1..30]]; // G. C. Greubel, Jul 15 2019

with(combinat); A240836:=n->2*fibonacci(2*n)*fibonacci(2*n-1); seq(A240836(n), n=1..30); # Wesley Ivan Hurt, Apr 13 2014

Table[2Fibonacci[2n]Fibonacci[2n-1], {n, 30}] (* Wesley Ivan Hurt, Apr 13 2014 *)

vector(30, n, f=fibonacci; 2*f(2*n)*f(2*n-1)) \\ G. C. Greubel, Jul 15 2019

f=fibonacci; [2*f(2*n)*f(2*n-1) for n in (1..30)] # G. C. Greubel, Jul 15 2019

a(n) = 2*F(2n)*F(2n-1) where F(n) are the Fibonacci numbers (A000045).
G.f.: 2*x*(1-2*x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Apr 13 2014
a(n) = 2 * A081016(n-1). - Wesley Ivan Hurt, Apr 13 2014

More terms from Colin Barker, Apr 13 2014

A176117 Primes p such that T(p) is prime in Juricevic conjecture on classification of Lehmer triples.

Original entry on oeis.org

2, 5, 809

Offset: 1

Jonathan Vos Post, Apr 08 2010

Primes p such that A110035(2p) is prime. The value after 809 is > 2741. - R. J. Mathar, Jul 22 2010

a(4) > 15661, if it exists. - D. S. McNeil, Nov 27 2010

Robert Juricevic, Classifying Lehmer triples, Acta Arith. 137 (2009), no. 3, 207-232. MR MR2496461, doi:10.4064/aa137-3-3
Yu Tsumura, On compositeness of special types of integers, April 8, 2010.

Cf. A000040.

p such that T(p) is prime, where T(p) = (1/5)*(((1+sqrt(5))*((3+sqrt(5))/2)^(2*p)) + ((1-sqrt(5))*((3-sqrt(5))/2)^(2*p)) + 3).

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

Original entry on oeis.org

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

Werner Schulte, Nov 21 2024

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

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.

			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.

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

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

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)).

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A110033 A characteristic triangle for the Fibonacci numbers.

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A206282 a(n) = ( a(n-1) * a(n-3) + a(n-2) ) / a(n-4), a(1) = a(2) = 1, a(3) = -1, a(4) = -4.

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A240836 Numbers n such that n^3 = x*y*z where 2 <= x <= y <= z , n^3+1 = (x-1)*(y+1)*(z+1).

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A176117 Primes p such that T(p) is prime in Juricevic conjecture on classification of Lehmer triples.

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

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PARI

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A240836 Numbers n such that n^3 = xyz where 2 <= x <= y <= z , n^3+1 = (x-1)(y+1)(z+1).