A374149 -id:A374149 - cp's OEIS frontend

A049661 a(n) = (Fibonacci(6*n+1) - 1)/4.

0, 3, 58, 1045, 18756, 336567, 6039454, 108373609, 1944685512, 34895965611, 626182695490, 11236392553213, 201628883262348, 3618083506169055, 64923874227780646, 1165011652593882577, 20905285872462105744

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..750 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Cf. A000045, A019863, A374149.

[(Fibonacci(6*n+1)-1)/4: n in [0..20] ]; // Vincenzo Librandi, Aug 23 2011

Table[(Fibonacci[6n+1]-1)/4,{n,0,20}] (* or *) LinearRecurrence[ {19,-19,1},{0,3,58},20] (* Harvey P. Dale, Aug 22 2011 *)

a(n)=fibonacci(6*n+1)>>2 \\ Charles R Greathouse IV, Aug 23 2011

From R. J. Mathar, Nov 04 2008: (Start)
G.f.: x*(3+x)/((1-x)*(1-18*x+x^2)).
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). (End)
a(n) = (-1/4+1/40*(9+4*sqrt(5))^(-n)*(5-sqrt(5)+(5+sqrt(5))*(9+4*sqrt(5))^(2*n))). - Colin Barker, Mar 03 2016
Product_{n>=1} (1 - 1/a(n)) = (sqrt(5)+3)/8 = phi^2/4 = cos(Pi/5)^2 = A019863^2 = (A374149 + 1)/10. - Amiram Eldar, Nov 28 2024

A374883 Decimal expansion of phi(2phi + 1) (i.e., (7 + 3*sqrt(5))/2), where phi is the golden ratio.

Original entry on oeis.org

6, 8, 5, 4, 1, 0, 1, 9, 6, 6, 2, 4, 9, 6, 8, 4, 5, 4, 4, 6, 1, 3, 7, 6, 0, 5, 0, 3, 0, 9, 6, 9, 1, 4, 3, 5, 3, 1, 6, 0, 9, 2, 7, 5, 3, 9, 4, 1, 7, 2, 8, 8, 5, 8, 6, 4, 0, 6, 3, 4, 5, 8, 6, 8, 1, 1, 5, 7, 8, 1, 3, 8, 8, 4, 5, 6, 7, 0, 7, 3, 4, 9, 1, 2, 1, 6, 2

Offset: 1

Marco Ripà, Jul 22 2024

The author conjectures that this is the minimum volume of an axis-aligned bounding box which includes the shortest minimum-link circuit joining all the vertices of the cube {0,1}^3 (i.e., the closed polygonal chains consisting of exactly 6 edges visiting all the points of the set {(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)}).
In detail, such a circuit of 6 links is given by (1/2,1+phi,1/2)-((1-phi)/2,0,(1+phi)/2)-((phi+1)/2,0, (1-phi)/2)-(1/2,1+phi,1/2)-((phi+1)/2,0,(phi+1)/2)-((1-phi)/2,0,(1-phi)/2(1/2,1+phi,1/2), where phi := (1+sqrt(5))/2 (see A001622).
Then, phi*(2*phi + 1) = phi^2*(phi + 1) since phi - 1 = 1/phi.

			6.8541019662496845446137605030969...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.

Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Marco Ripà, General uncrossing covering paths inside the Axis-Aligned Bounding Box, Journal of Fundamental Mathematics and Applications, Volume 4, 2021, Number 2, Pages 154-166.

Cf. A001622, A090550, A104457, A134973, A205769, A225227, A261547, A363755, A373537, A374149, A374260.

RealDigits[3*GoldenRatio + 2, 10, 120][[1]] (* Amiram Eldar, Jul 23 2024 *)

Equals (7 + 3*sqrt(5))/2.
Equals phi^2*(phi + 1), where phi = (1 + sqrt(5))/2.
Equals A104457^2 = 2*A205769. - Hugo Pfoertner, Jul 22 2024
Equals A090550 + 1 = A134973 + 5. - Amiram Eldar, Jul 23 2024
Equals phi^4. - Stefano Spezia, May 28 2025

A375803 a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(2n-1) Fibonacci(2*n+1).

Original entry on oeis.org

0, 20, 195, 4420, 72624, 1347905, 23877840, 430583140, 7712000835, 138485573876, 2484341814240, 44584372180225, 800002107309600, 14355674602647860, 257600625681170499, 4622465972012379940, 82946715695078486160, 1488418904383171787585, 26708590219470770907120

Offset: 1

Amiram Eldar, Aug 29 2024

Amiram Eldar, Table of n, a(n) for n = 1..798
Hideyuki Ohtskua, proposer, Problem H-944, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 3 (2024), p. 266.
Index entries for linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1).

Cf. A000045, A059929, A064170, A134944, A374149, A375804.

a[n_] := Fibonacci[n-1] * Fibonacci[n+1] * Fibonacci[2*n-1] * Fibonacci[2*n+1]; Array[a, 20]

a(n) = fibonacci(n-1) * fibonacci(n+1) * fibonacci(2*n-1) * fibonacci(2*n+1);

a(n) = A059929(n-1) * A059929(2*n-1) = A059929(n-1) * A064170(n+2).
Sum_{n>=2} (-1)^n/a(n) = (5*sqrt(5) - 11)/4 = A374149 - 11/2 = 10 * A134944 - 4 (Ohtskua, 2024).
G.f.: -x^2*(-20+65*x+195*x^2-84*x^3-13*x^4+x^5) / ( (1+x)*(x^2-3*x+1)*(x^2-18*x+1)*(x^2+7*x+1) ). - R. J. Mathar, Aug 30 2024

cp's OEIS Frontend

A049661 a(n) = (Fibonacci(6*n+1) - 1)/4.

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A374883 Decimal expansion of phi(2phi + 1) (i.e., (7 + 3*sqrt(5))/2), where phi is the golden ratio.

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A375803 a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(2n-1) Fibonacci(2*n+1).

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cp's OEIS Frontend

A049661 a(n) = (Fibonacci(6*n+1) - 1)/4.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A374883 Decimal expansion of phi*(2*phi + 1) (i.e., (7 + 3*sqrt(5))/2), where phi is the golden ratio.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A375803 a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(2*n-1) * Fibonacci(2*n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A374883 Decimal expansion of phi(2phi + 1) (i.e., (7 + 3*sqrt(5))/2), where phi is the golden ratio.

A375803 a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(2n-1) Fibonacci(2*n+1).