A159990 -id:A159990 - cp's OEIS frontend

A159993 Denominator of Sum_{k=0..n} A159990(k)/A159991(k), numerator=A159992.

1, 30, 3600, 36000, 4320000, 24300000, 23328000000, 93312000000, 3359232000000, 223948800000000, 604661760000000000, 36279705600000000000, 544195584000000000000, 1360488960000000000000, 326517350400000000000000

Offset: 0

Reinhard Zumkeller, May 01 2009

R. Zumkeller, Table of n, a(n) for n = 0..30

A159991, subsequence of A051037, the 5-smooth numbers.

A159992 Numerator of Sum_{k=0..n} A159990(k)/A159991(k).

Original entry on oeis.org

1, 41, 4927, 49277, 5913251, 33262037, 31931555539, 127726222157, 4598143997653, 306542933176867, 827665919577540943, 49659955174652456593, 744899327619786848909, 1862248319049467122273, 446939596571872109345521

Offset: 0

Reinhard Zumkeller, May 01 2009

a(n)/A159993(n) approximates the positive root of x^3+2*x^2+10*x=20:
A159994(n)/A159995(n) = f(a(n)/A159993(n)) --> 0, where f(x) = x^3 + 2*x^2 + 10*x - 20;
a(n)/A159993(n) = a(n-1)/A159993(n-1) + A159990(n)/A159991(n).
Limit can be found at A202300. - Jason Bard, Jul 26 2025

			a(0)/A159993(0) = 1;
a(1)/A159993(1) = 41/30;
a(2)/A159993(2) = 4927/3600;
a(3)/A159993(3) = 49277/36000;
a(4)/A159993(4) = 5913251/4320000;
a(5)/A159993(5) = 33262037/24300000;
a(6)/A159993(6) = 31931555539/23328000000;
a(7)/A159993(7) = 127726222157/93312000000;
a(8)/A159993(8) = 4598143997653/3359232000000;
and written as decimal fractions:
a(0)/A159993(0) = 1;
a(1)/A159993(1) ~= 1.3666666666666667;
a(2)/A159993(2) ~= 1.3686111111111111;
a(3)/A159993(3) ~= 1.3688055555555556;
a(4)/A159993(4) ~= 1.3688081018518519;
a(5)/A159993(5) ~= 1.3688081069958847;
a(6)/A159993(6) ~= 1.3688081078103566;
a(7)/A159993(7) ~= 1.3688081078210733;
a(8)/A159993(8) ~= 1.3688081078213710.

Reinhard Zumkeller, Table of n, a(n) for n = 0..30

Cf. A159990, A159991, A159993 (denominator), A159994, A159995, A202300.

A159991 Powers of 60: a(n) = 60^n.

Original entry on oeis.org

1, 60, 3600, 216000, 12960000, 777600000, 46656000000, 2799360000000, 167961600000000, 10077696000000000, 604661760000000000, 36279705600000000000, 2176782336000000000000, 130606940160000000000000, 7836416409600000000000000, 470184984576000000000000000

Offset: 0

Reinhard Zumkeller, May 01 2009

			G.f. = 1 + 60*x + 3600*x^2 + 216000*x^3 + 12960000*x^4 + 77600000*x^5 + ... - _Michael Somos_, Jan 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Wikipedia, Sexagesimal
Index entries for linear recurrences with constant coefficients, signature (60).

Subsequence of A051037.

Cf. A159990, A159993, A159995, A088157, A091649, A125628, A091720, A091721, A091722, A060707, A070197.

List([0..20],n->60^n); # Muniru A Asiru, Nov 21 2018

[60^n: n in [0..20]]; // Vincenzo Librandi, May 02 2011

[60^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

60^Range[0,15] (* Harvey P. Dale, Jun 02 2011 *)

A159991(n):=60^n$
makelist(A159991(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=60^n \\ Charles R Greathouse IV, May 02 2011

a(n)=60^n \\ Charles R Greathouse IV, Jun 19 2015

powers(60,8) \\ Charles R Greathouse IV, Jun 19 2015

for n in range(0,20): print(60**n, end=', ') # Stefano Spezia, Nov 21 2018

a(n) = A000400(n)*A011557(n) = A000351(n)*A001021(n) = A000302(n)*A001024(n) = A000244(n)*A009964(n). (Corrected by Robert B Fowler, Jan 25 2023)
From Muniru A Asiru, Nov 21 2018: (Start)
a(n) = 60^n.
a(n) = 60*a(n-1) for n > 0, a(0) = 1.
G.f.: 1/(1-60*x).
E.g.f: exp(60*x). (End)
a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 01 2019

A159994 Numerator of f(A159992(n)/A159993(n)) with f(x)=x^3+2x^2+10x-20, denominator=A159995.

Original entry on oeis.org

-7, -1219, -193885217, -2512095067, -10152983807749, -249880575515347, -2950249420928771944181, -5129149052857317896107, -1247467412339070464235923, -16941291362994850503969493637

Offset: 0

Reinhard Zumkeller, May 01 2009

a(n)/A159995(n) = f(A159992(n)/A159993(n));
a(n)/A159995(n) < 0; a(n)/A159995(n) <= a(n+1)/A159995(n+1);
a(n)/A159995(n) --> 0.

			a(0)/A159995(0)=-7;
a(1)/A159995(1)=-1216/27000;
a(2)/A159995(2)=-193885217/46656000000;
a(3)/A159995(3)=-2512095067/46656000000000;
a(4)/A159995(4)=-10152983807749/80621568000000000000;
a(5)/A159995(5)=-249880575515347/14348907000000000000000;
a(6)/A159995(6)=-2950249420928771944181/12694994583552000000000000000000;
a(7)/A159995(7)=-5129149052857317896107/812479653347328000000000000000000;
a(8)/A159995(8)=-1247467412339070464235923/37907050706572935168000000000000000000;
written as decimal fractions:
a(1)/A159995(1) ~= -0.045037037037037037037037;
a(2)/A159995(2) ~= -0.004155633080418381344307;
a(3)/A159995(3) ~= -0.000053842915530692729766;
a(4)/A159995(4) ~= -0.000000125933842017920068;
a(5)/A159995(5) ~= -0.000000017414606946392990;
a(6)/A159995(6) ~= -0.000000000232394697099847;
a(7)/A159995(7) ~= -0.000000000006312956923568;
a(8)/A159995(8) ~= -0.000000000000032908585318.

R. Zumkeller, Table of n, a(n) for n = 0..30

A159990.

A202300 Decimal expansion of the real root of x^3 + 2x^2 + 10x - 20.

Original entry on oeis.org

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

Offset: 1

Charles R Greathouse IV, Jan 11 2012

There is a small typo in Posamentier & Lehmann (2007): this number is given as approximately 1.3688081075 rather than 1.3688081078, a mistake that can't be justified by rounding rather than truncating nor a loss of machine precision. - Alonso del Arte, Mar 24 2012
Perhaps the reason for the mistake is that the authors got the correct answer mixed up with Fibonacci's answer, which, though wrong, was very good for the time: 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6 = 1.36880810785322... But apparently they truncated at the first 5 and left out the 8 before that 5. - Alonso del Arte, Jun 09 2014
The complex roots are -1.68440405391... +- 3.43133135... * i. - Alonso del Arte, Jun 21 2014
Fibonacci calculated this constant to six sexagesimal digits and proved that it was neither rational nor a square root of a rational. - Charles R Greathouse IV, Oct 21 2022

			x = 1.36880810782137263522741433002132553954243554148753653...

John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press (2006): 69-70.
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 63-64.
Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007) p. 21.

Ezra Brown and Jason C. Brunson, Fibonacci's forgotten number (archived link)
Stanislaw Glushkov, On approximation methods of Leonardo Fibonacci, Historia Mathematica 3 (1976), pp. 291-296.
Wolfram|Alpha, real root of x^3 + 2x^2 + 10x - 20 = 0
Index entries for algebraic numbers, degree 3

Cf. A159990, A243629, A244467.

RealDigits[x /. FindRoot[x^3 + 2x^2 + 10x - 20 == 0, {x, 1.4}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Feb 27 2013 *)

PARI
```
real(polroots(x^3+2*x^2+10*x-20)[1])
```

polrootsreal(x^3+2*x^2+10*x-20)[1] \\ Charles R Greathouse IV, Jan 05 2016

x = (2*sqrt(3930)/9 - 352/27)^(1/3) + (2*sqrt(3930)/9 + 352/27)^(1/3) - 2/3;
x = (1/3)*(-2 - 13 * 2^(2/3)/(176 + 3*sqrt(3930))^(1/3) + (2*(176 + 3*sqrt(3930)))^(1/3)).
The first formula comes from Posamentier & Lehmann (2007), the second from Wolfram|Alpha. - Alonso del Arte, Mar 24 2012

A243629 Fibonacci's solution to x^3 + 2x^2 + 10x = 20.

Original entry on oeis.org

1, 22, 7, 42, 33, 4, 40

Offset: 0

Jason Chabora, Jun 10 2014

nonn

Fibonacci attempted to solve x^3 + 2x^2 + 10x = 20 and managed to approximate an answer in base 60 with this sequence without a calculator or computer, but he gave a(6) as 40 instead of the correct answer of 38 (see A159990). Unfortunately, Fibonacci did not explain how he calculated this sequence, so it is not possible to explore the precision to a greater degree.

			x = 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6. Then x^3 + 2x^2 + 10x = 20.0000000006719323...

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007): 21.

Cf. A159990 shows the sequence corrected.

A244467 Decimal expansion of 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6, Fibonacci's solution to x^3 + 2x^2 + 10x = 20.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Jun 28 2014

The equation has no rational solutions. Despite Fibonacci's mistake with the sixth coefficient (40 rather than 38), his solution is a remarkable achievement for the time. Plugged into the equation, it gives 20.0000000006719323...

			= 1.36880810785322...

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007): 21.

Cf. A243629, A159990.

RealDigits[1596577777/1166400000, 10, 100][[1]]

1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6 = 1596577777/1166400000.

cp's OEIS Frontend

A159993 Denominator of Sum_{k=0..n} A159990(k)/A159991(k), numerator=A159992.

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A159992 Numerator of Sum_{k=0..n} A159990(k)/A159991(k).

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A159991 Powers of 60: a(n) = 60^n.

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A159994 Numerator of f(A159992(n)/A159993(n)) with f(x)=x^3+2*x^2+10*x-20, denominator=A159995.

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A202300 Decimal expansion of the real root of x^3 + 2x^2 + 10x - 20.

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A243629 Fibonacci's solution to x^3 + 2x^2 + 10x = 20.

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A244467 Decimal expansion of 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6, Fibonacci's solution to x^3 + 2x^2 + 10x = 20.

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A159994 Numerator of f(A159992(n)/A159993(n)) with f(x)=x^3+2x^2+10x-20, denominator=A159995.