A098317 -id:A098317 - cp's OEIS frontend

A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			-0.04722009625435983376687869404879456549554899472734...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139349.

Join[{0}, RealDigits[Cos[GoldenRatio], 10, 100][[1]]] (* Amiram Eldar, Feb 07 2022 *)

-cos((1+sqrt(5))/2) \\ Charles R Greathouse IV, May 13 2019

Equals 1/A139349. - Amiram Eldar, Feb 07 2022

Edited by N. J. A. Sloane, Dec 11 2008

A144749 Decimal expansion of the golden ratio powered to itself.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 20 2008

See A092134 for the continued fraction of this value, phi^phi, where phi = (sqrt(5)+1)/2 = A001622. - M. F. Hasler, Oct 08 2014

			Equals 2.178457567937599147372545702871245851807043301693254611347781924...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers.

Cf. A001622, A104457, A098317, A094214, A139339, A139340, A144713.

RealDigits[N[GoldenRatio^GoldenRatio,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

(t=(sqrt(5)+1)/2)^t \\ Use \p99 to get 99 digits; digits(%\.1^99) for the sequence of digits. - M. F. Hasler, Oct 08 2014

numerical_approx(golden_ratio^golden_ratio, digits=120) # G. C. Greubel, Jun 16 2022

Equals A001622^A001622.

A374755 Decimal expansion of the surface area of a regular dodecahedron having unit inradius.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 20 2024

Bezdek's strong dodecahedral conjecture (proved by Hales, see links) states that, in any packing of unit spheres in the Euclidean 3-space, the surface area of every bounded Voronoi cell is at least this value.

			16.6508730855465308072112963409855177222127946386...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Károly Bezdek, On a stronger form of Rogers' lemma and the minimum surface area of Voronoi cells in unit ball packings, Journal für die reine und angewandte Mathematik, No. 518, 2000, pp. 131-143.
Thomas C. Hales, The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve, arXiv:1110.0402 [math.MG], 2012.
Wikipedia, Regular dodecahedron.

Cf. A374753 (dodecahedral conjecture), A374772, A374837, A374838.

Cf. A001622, A098317, A131595.

First[RealDigits[30*Sqrt[130 - 58*Sqrt[5]], 10, 100]]

Equals 30*sqrt(130 - 58*sqrt(5)).
Equals 60*sqrt(3 - A001622)/A098317.
Equals 4*Pi/A374772.
Equals 3*A374753.
Minimal polynomial: x^4 - 234000*x^2 + 64800000. - Stefano Spezia, Sep 03 2025

A139347 Decimal expansion of negated tangent of the golden ratio. That is, the decimal expansion of -tan((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			-21.15380078249327464858628117032582559788124367464826...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139348.

RealDigits[Tan[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 07 2022 *)

-tan((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals tan(A001622).
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139348.
Equals A139345/A139346. (End)

Offset corrected by Mohammad K. Azarian, Dec 13 2008

Sign added to definition by R. J. Mathar, Feb 05 2009

A139348 Decimal expansion of negated cotangent of the golden ratio. That is, the decimal expansion of -cot((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.04727282866479448118935650960621633420056105722556...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347.

Join[{0},RealDigits[-Cot[GoldenRatio],10,120][[1]]] (* Harvey P. Dale, Sep 18 2013 *)

-cotan((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals cot(A001622).
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139347.
Equals A139346/A139345. (End)

Added sign in definition. Leading zero dropped by R. J. Mathar, Feb 05 2009

A261391 a(n) = n^5 + 5n^3 + 5n.

Original entry on oeis.org

0, 11, 82, 393, 1364, 3775, 8886, 18557, 35368, 62739, 105050, 167761, 257532, 382343, 551614, 776325, 1069136, 1444507, 1918818, 2510489, 3240100, 4130511, 5206982, 6497293, 8031864, 9843875, 11969386, 14447457, 17320268, 20633239, 24435150, 28778261, 33718432, 39315243, 45632114

Offset: 0

Raphael Ranna, Aug 17 2015

Also numbers of the form (n-th metallic mean)^5 - 1/(n-th metallic mean)^5, see link to Wikipedia.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Metallic mean.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458, A176522.

Array[#^5 + 5 #^3 + 5 # &, 34] (* Michael De Vlieger, Aug 18 2015 *)
Table[n^5 + 5*n^3 + 5*n, {n,0, 50}] (* G. C. Greubel, Aug 21 2015 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,11,82,393,1364,3775},40] (* Harvey P. Dale, May 07 2018 *)

concat(0, Vec(x*(11*x^4+16*x^3+66*x^2+16*x+11)/(x-1)^6 + O(x^100))) \\ Colin Barker, Aug 18 2015

a(n) = ( (n+sqrt(n^2+4))/2 )^5 - 1/( (n+sqrt(n^2+4))/2 )^5.
a(n) = -a(-n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Colin Barker, Aug 18 2015
G.f.: x*(11*x^4+16*x^3+66*x^2+16*x+11) / (x-1)^6. - Colin Barker, Aug 18 2015
E.g.f.: (x^5 + 15*x^4 + 70*x^3 + 120*x^2 + 71*x + 11)*e^x. - G. C. Greubel, Aug 21 2015

Offset changed from 1 to 0, initial 0 added and b-file adapted from Bruno Berselli, Aug 25 2015

A037451 a(n) = Fibonacci(n) * Fibonacci(2*n).

Original entry on oeis.org

0, 1, 3, 16, 63, 275, 1152, 4901, 20727, 87856, 372075, 1576279, 6676992, 28284569, 119814747, 507544400, 2149990983, 9107510539, 38580029568, 163427634589, 692290558575, 2932589884016, 12422650070163, 52623190204271, 222915410823168, 944284833600625, 4000054745057907, 16944503814103696, 71778070001033487

Offset: 0

Gary W. Adamson, Feb 01 2000

Let F(n) = Fibonacci(n), then abs(det([F(n), F(n+k); F(n+2k), F(n+3k)])) = a(k), independent of n. - R. M. Welukar, Aug 26 2014
From Joerg Arndt, Aug 26 2014: (Start)
This is a special case of Johnson's identity (relation 32 in the Mathworld link).
F(a)*F(b) - F(c)*F(d) = (-1)^r*(F(a-r)*F(b-r) - F(c-r)*F(d-r)), where a+b = c+d and r arbitrary.
Here a = n, b = n+3*k, c = n+k, d = n+2*k, and r = c, so that
(-1)^r*(F(a-r)*F(b-r) - F(c-r)*F(d-r)) =
(-1)^c*(F(a-c)*F(b-c) - F(c-c)*F(d-c)) =
(-1)^c*(F(a-c)*F(b-c) - 0) =
(-1)^c*(F(-k)*F(-2*k)), taking the absolute value gives a(k).
(End)
Let L(n) = A000032(n), then abs(det([L(n), L(n+k); L(n+2k), L(n+3k)])) = 5*a(k), independent of n. - M. N. Deshpande and R. M. Welukar, Aug 30 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein, Fibonacci Number (MathWorld).
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000032, A000045, A098317, A273622.

[Fibonacci(n)*Fibonacci(2*n): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

seq((fibonacci(2*n)*fibonacci(n)), n=0..25); # Zerinvary Lajos, Jun 24 2006

Table[Fibonacci[n]Fibonacci[2n],{n,0,40}] (* Harvey P. Dale, Mar 13 2011 *)

concat([0], Vec( x*(1+x^2) / ((1+x-x^2)*(1-4*x-x^2)) + O(x^66) ) ) \\ Joerg Arndt, Aug 26 2014

From Emanuele Munarini, Jul 18 2003: (Start)
G.f.: ( x + x^3 )/( 1 - 3 x - 6 x^2 + 3 x^3 + x^4 ).
a(n+4) = 3*a(n+3) + 6*a(n+2) - 3*a(n+1) - a(n).
(End)
G.f.: x*(1+x^2) / ((1+x-x^2)*(1-4*x-x^2)). - Joerg Arndt, Aug 26 2014
a(n) = (1/5)*(Lucas(3*n) - (-1)^n*Lucas(n)) = (1/5)*(Lucas(3*n) - Lucas(-n)). In general, for r = s (mod 2) the sequence Lucas(r*n) - Lucas(s*n) is a divisibility sequence. Cf. A273622. - Peter Bala, May 27 2016
Lim_{n->infinity} a(n+1)/a(n) = 2 + sqrt(5) = A098317. - Ilya Gutkovskiy, Jun 01 2016
a(n) = (-(1/2*(-1-sqrt(5)))^n+(2-sqrt(5))^n-(1/2*(-1+sqrt(5)))^n+(2+sqrt(5))^n)/5. - Colin Barker, Jun 03 2016

A139349 Decimal expansion of negated secant of the golden ratio. That is, the decimal expansion of -sec((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			21.17742400636614440872804040937130213307185355364174...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348.

RealDigits[-Sec[GoldenRatio],10,120][[1]] (* Harvey P. Dale, Dec 08 2011 *)

-1/cos((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals sec(A001622).

Equals 1/A139346. - Amiram Eldar, Feb 07 2022

Offset corrected by Mohammad K. Azarian, Dec 13 2008

Sign in definition added by R. J. Mathar, Feb 05 2009

A139350 Decimal expansion of csc((1+sqrt(5))/2), where (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			1.00111673661465225489616711351705587794461531806624...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348, A139349.

RealDigits[Csc[GoldenRatio], 10, 100][[1]] (* Vincenzo Librandi, Feb 19 2013 *)

1/sin((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals 1/A139345. - Amiram Eldar, Feb 07 2022

Edited by Bruno Berselli, Feb 19 2013

A140232 a(n) = ceiling(n*exp((1+sqrt(5))/2)).

Original entry on oeis.org

6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96, 101, 106, 111, 116, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 172, 177, 182, 187, 192, 197, 202, 207, 212, 217, 222, 227, 232, 238, 243, 248, 253, 258, 263, 268, 273, 278, 283

Offset: 1

Mohammad K. Azarian, May 13 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348, A139349, A139350, A140231, A016861.

phi:=(1+Sqrt(5))/2; [Ceiling(n*Exp(phi)): n in [1..60]]; // G. C. Greubel, Jun 30 2019

Ceiling[Exp[GoldenRatio]*Range[60]] (* G. C. Greubel, Jun 30 2019 *)

phi=(1+sqrt(5))/2; vector(60, n, ceil(n*exp(phi)) ) \\ G. C. Greubel, Jun 30 2019

[ceil(n*exp(golden_ratio)) for n in (1..60)] # G. C. Greubel, Jun 30 2019

a(n) = ceiling(n*A139341). - R. J. Mathar, Feb 06 2009

cp's OEIS Frontend

A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

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A144749 Decimal expansion of the golden ratio powered to itself.

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A374755 Decimal expansion of the surface area of a regular dodecahedron having unit inradius.

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A139347 Decimal expansion of negated tangent of the golden ratio. That is, the decimal expansion of -tan((1+sqrt(5))/2).

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A139348 Decimal expansion of negated cotangent of the golden ratio. That is, the decimal expansion of -cot((1+sqrt(5))/2).

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A261391 a(n) = n^5 + 5*n^3 + 5*n.

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A037451 a(n) = Fibonacci(n) * Fibonacci(2*n).

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A261391 a(n) = n^5 + 5n^3 + 5n.