A139339 -id:A139339 - cp's OEIS frontend

A144749 Decimal expansion of the golden ratio powered to itself.

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.

A176343 a(n) = Fibonacci(n)*a(n-1) + 1, a(0) = 0.

Original entry on oeis.org

0, 1, 2, 5, 16, 81, 649, 8438, 177199, 6024767, 331362186, 29491234555, 4246737775921, 989489901789594, 373037692974676939, 227552992714552932791, 224594803809263744664718, 358677901683394200229554647, 926823697949890613393169207849

Offset: 0

Roger L. Bagula, Apr 15 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000045, A003266, A062073, A101689, A139339, A238243, A238244.

a:= function(n)
    if n=0 then return 0;
    else return 1 + Fibonacci(n)*a(n-1);
    fi; end;
List([0..20], n-> a(n) ); # G. C. Greubel, Dec 07 2019

function a(n)
  if n eq 0 then return 0;
  else return 1 + Fibonacci(n)*a(n-1);
  end if; return a; end function;
[a(n): n in [0..20]]; // G. C. Greubel, Dec 07 2019

with(combinat);
a:= proc(n) option remember;
      if n=0 then 0
    else 1 + fibonacci(n)*a(n-1)
      fi; end:
seq( a(n), n=0..20); # G. C. Greubel, Dec 07 2019

a[n_]:= a[n]= If[n==0, 0, Fibonacci[n]*a[n-1] +1]; Table[a[n], {n,0,20}]

a(n) = if(n==0, 0, 1 + fibonacci(n)*a(n-1) ); \\ G. C. Greubel, Dec 07 2019

def a(n):
    if (n==0): return 0
    else: return 1 + fibonacci(n)*a(n-1)
[a(n) for n in (0..20)] # G. C. Greubel, Dec 07 2019

a(n) = Fibonacci(n)*a(n-1) + 1, a(0) = 0.

a(n) ~ c * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where c = A062073 * A101689 = 3.317727324507285486862890025085971028467... is product of Fibonacci factorial constant (see A062073) and Sum_{n>=1} 1/(Product_{k=1..n} A000045(k) ). - Vaclav Kotesovec, Feb 20 2014

A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 19 2011

The hyperbolas y^2-x^2=1 and xy=1 meet at (1/c,c) and (-1/c,c), where c=sqrt(golden ratio); see the Mathematica program for a graph; see A189339 for hyperbolas meeting at (c,1/c) and (-c,-1/c).
This number is the eccentricity of an ellipse inscribed in a golden rectangle. - Jean-François Alcover, Sep 03 2015
c/sqrt(-1) is the limit of Pi(a;n)/2 := a^n * sqrt(a - f(a;n))  with f(a;0) = 0, and f(a;n) = sqrt(a + f(a;n-1)) for n >= 1, if one takes a = 1. For a=2 this gives Viète's formula for Pi/2 (see A019669). - Wolfdieter Lang, Jul 06 2018

			0.786151377757423286069558585842958929523122057...

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A001622, A019669, A139339.

N[1/Sqrt[GoldenRatio], 110]
RealDigits[%]
FindRoot[x*Sqrt[1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]
Plot[{Sqrt[1 + x^2], 1/x}, {x, 0, 3}]

sqrt(2/(1+sqrt(5))) \\ Michel Marcus, Sep 03 2015

my(c=1/quadgen(5)); a_vector(len) = digits(sqrtint(floor(c*100^len))); \\ Kevin Ryde, Jul 12 2025

Equals sqrt(1/phi) = sqrt(phi-1), with phi = A001622.
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139339.
Equals tan(arcsin(1/phi)).
Equals sin(arccos(1/phi)).
Equals cos(arcsin(1/phi)).
Equals cot(arccos(1/phi)). (End)

A339856 Primitive triples for integer-sided triangles whose sides a < b < c form a geometric progression.

Original entry on oeis.org

4, 6, 9, 9, 12, 16, 16, 20, 25, 25, 30, 36, 25, 35, 49, 25, 40, 64, 36, 42, 49, 49, 56, 64, 49, 63, 81, 49, 70, 100, 49, 77, 121, 64, 72, 81, 64, 88, 121, 81, 90, 100, 81, 99, 121, 81, 117, 169, 81, 126, 196, 100, 110, 121, 100, 130, 169, 121, 132, 144, 121, 143, 169

Offset: 1

Bernard Schott, Dec 19 2020

These triangles are called "geometric triangles" in Project Euler problem 370 (see link).
The triples are displayed in increasing lexicographic order (a, b, c).
Equivalently: triples of integer-sided triangles such that b^2 = a*c with a < c and gcd(a, c) = 1.
When a < b < c are in geometric progression with b = a*q, c = b*q, q is the constant, then 1 < q < (1+sqrt(5))/2 = phi = A001622 = 1.6180... (this bound is used in Maple code).
For each triple (a, b, c), there exists (r, s), 0 < r < s such that a = r^2, b = r*s, c = s^2, q = s/r.
Angle C < 90 degrees if 1 < q < sqrt(phi) and angle C > 90 degrees if sqrt(phi) < q < phi with sqrt(phi) = A139339 = 1.2720...
For k >= 2, each triple (a, b, c) of the form (k^2, k*(k+1), (k+1)^2) is (A008133(3k+1), A008133(3k+2), A008133(3k+3)).
Three geometrical properties about these triangles:
  1) The sinus satisfy sin^2(B) = sin(A) * sin(C) with sin(A) < sin(B) < sin(C) that form a geometric progression.
  2) The heights satisfy h_b^2 = h_a * h_c with h_c < h_b < h_a that form a geometric progression.
  3) b^2 = 2 * R * h_b, with R = circumradius of the triangle ABC.

			The smallest such triangle is (4, 6, 9) with 4*9 = 6^2.
There exist four triangles with small side = 49 corresponding to triples (49, 56, 64), (49, 63, 81), (49, 70, 100) and (49, 77, 121).
The table begins:
   4,  6,  9;
   9, 12, 16;
  16, 20, 25;
  25, 30, 36;
  25, 35, 49;
  25, 40, 64;
  36, 42, 49;
  ...

David A. Corneth, Table of n, a(n) for n = 1..10002
Project Euler, Problem 370: Geometric Triangles.

Cf. A339857 (smallest side), A339858 (middle side), A339859 (largest side), A339860 (perimeter).
Cf. A336755 (similar for sides in arithmetic progression).
Cf. A335893 (similar for angles in arithmetic progression).
Cf. A001622 (phi), A139339 (sqrt(phi)), A008133.

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 * a) do
for c from b+1 to floor((1+sqrt(5))/2 * b) do
k:=a*c;
if k=b^2 and igcd(a,b,c)=1 then print(a,b,c); end if;
end do;
end do;
end do;

lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a,b,c])==1), print([a,b,c])););););} \\ Michel Marcus, Dec 25 2020

upto(n) = my(res=List(), phi = (sqrt(5)+1) / 2); for(i = 2, sqrtint(n), for(j = i+1, (i*phi)\1, if(gcd(i, j)==1, listput(res, [i^2, i*j, j^2])))); concat(Vec(res)) \\ David A. Corneth, Dec 25 2020

Data corrected by David A. Corneth, Dec 25 2020

A063827 a(n) = round(sqrt(a(n-2)^2 + a(n-1)^2)) with a(0) = 1 and a(1) = 2.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 26, 33, 42, 53, 68, 86, 110, 140, 178, 226, 288, 366, 466, 593, 754, 959, 1220, 1552, 1974, 2511, 3194, 4063, 5168, 6574, 8362, 10637, 13530, 17211, 21892, 27847, 35422, 45057, 57314, 72904, 92736, 117962, 150050

Offset: 0

Henry Bottomley, Aug 20 2001

nonn

a(n)/a(n-1) tends towards 1.27201964951... = sqrt((1+sqrt(5))/2). See A139339.

			a(7) = 8 since round(sqrt(5^2 + 6^2)) = round(sqrt(61)) = round(7.8102...) = 8.

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A000283, A139339.

nxt[{a_,b_}]:={b,Round[Sqrt[a^2+b^2]]}; NestList[nxt,{1,2},50][[;;,1]] (* Harvey P. Dale, Dec 18 2024 *)

{ default(realprecision, 50); for (n=0, 500, if (n>1, a=round(sqrt(a2^2 + a1^2)); a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b063827.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 01 2009

Missing parenthesis added to definition by Harry J. Smith, Sep 01 2009

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

A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 22 2011

Every cyclic quadrilateral all of whose angles are greater than arccos((sqrt(5)-1)/2) admits a 3 × 1 grid dissection into three cyclic quadrilaterals [Thm. 2.3 in Choi et al. p. 2]. - Michel Marcus, Aug 13 2019
The base angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
Smallest positive root of the equation sin(x) = cot(x). - Wolfe Padawer, Apr 11 2023

			arccos(1/phi) = 0.904556894302381364127316795661958721...
cos(0.904556894302381364127316795661958721...) = 1/(golden ratio) = 0.618...
sec(0.904556894302381364127316795661958721...) = (golden ratio) = 1.618...

Erica Choi, Dan Ismailescu, Jiho Lee and Joonsoo Lee, Grid dissections of tangential quadrilaterals, arXiv:1908.02251 [math.MG], 2019.
Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
Index entries for transcendental numbers

Cf. A001622, A019669, A139339, A175288, A195693, A195694, A197762.

r = 1/GoldenRatio;
N[ArcCos[r], 100]
RealDigits[%]

acos(2/(sqrt(5)+1)) \\ Charles R Greathouse IV, Nov 21 2024

From Amiram Eldar, Feb 07 2022: (Start)
Equals Pi/2 - A175288.
Equals arcsin(1/sqrt(phi)).
Equals arctan(sqrt(phi)). (End)

Terms replaced with intended terms by Rick L. Shepherd, Jan 30 2013

A238244 A recursive sequence: a(n) = Fibonacci(n)*a(n-1) + 3.

Original entry on oeis.org

1, 4, 11, 36, 183, 1467, 19074, 400557, 13618941, 749041758, 66664716465, 9599719170963, 2236734566834382, 843248931696562017, 514381848334902830373, 507694884306549093578154, 810788730237558902444311941, 2095078078933852203916102055547

Offset: 1

Vaclav Kotesovec, Feb 20 2014

nonn

Generally, sequence a(n) = Fibonacci(n)*a(n-1) + p, with a(1)=1 and fixed p, is asymptotic to c(p) * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where constant c(p) = A062073 * (p*A101689 - p + 1).

Harvey P. Dale, Table of n, a(n) for n = 1..98

Cf. A176343, A238243, A003266, A101689, A062073, A000045, A139339.

RecurrenceTable[{a[n]==Fibonacci[n]*a[n-1]+3,a[1]==1},a,{n,1,20}]
nxt[{n_,a_}]:={n+1,a*Fibonacci[n+1]+3}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Sep 04 2024 *)

a(n) ~ c * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where c = A062073 * (3*A101689-2) = 7.4996979520811499717534... is product of Fibonacci factorial constant (see A062073) and -2+3*sum_{n>=1} 1/product(A000045(k), k=1..n).

A317974 a(n) = 2*(a(n-1)+a(n-2)+a(n-3))-a(n-4) for n >= 4, with initial terms 0,0,1,1.

Original entry on oeis.org

0, 0, 1, 1, 4, 12, 33, 97, 280, 808, 2337, 6753, 19516, 56404, 163009, 471105, 1361520, 3934864, 11371969, 32865601, 94983348, 274506972, 793339873, 2292794785, 6626299912, 19150362168, 55345573857, 159951677089, 462268926316, 1335981992356, 3861059617665

Offset: 0

N. J. A. Sloane, Sep 03 2018

A.H.M. Smeets, Table of n, a(n) for n = 0..2172
H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.
Eric Weisstein's World of Mathematics, Coxeter's Loxodromic Sequence of Tangent Circles
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

nxt[{a_,b_,c_,d_}]:={b,c,d,2(b+c+d)-a}; NestList[nxt,{0,0,1,1},30][[;;,1]] (* or *) LinearRecurrence[{2,2,2,-1},{0,0,1,1},40] (* Harvey P. Dale, Dec 10 2024 *)

concat(vector(2), Vec(x^2*(1 - x) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + O(x^40))) \\ Colin Barker, Sep 04 2018

a1,a2,a3,a4,n = 1,1,0,0,3
print(0,0)
print(1,0)
print(2,1)
print(3,1)
while n < 2172:
    a1,a2,a3,a4,n = 2*(a1+a2+a3)-a4,a1,a2,a3,n+1
    print(n,a1) # A.H.M. Smeets, Sep 04 2018

Lim {n -> infinity} log(a(n))/n = 1.0612750619050... = log(phi+sqrt(phi)) = log(A001622+A139339), where phi is the golden ratio. - A.H.M. Smeets, Sep 04 2018

G.f.: x^2*(1 - x) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Sep 04 2018

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A144749 Decimal expansion of the golden ratio powered to itself.

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A176343 a(n) = Fibonacci(n)*a(n-1) + 1, a(0) = 0.

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A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio.

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A339856 Primitive triples for integer-sided triangles whose sides a < b < c form a geometric progression.

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A063827 a(n) = round(sqrt(a(n-2)^2 + a(n-1)^2)) with a(0) = 1 and a(1) = 2.

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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).