A159991 -id:A159991 - cp's OEIS frontend

A001024 Powers of 15.

1, 15, 225, 3375, 50625, 759375, 11390625, 170859375, 2562890625, 38443359375, 576650390625, 8649755859375, 129746337890625, 1946195068359375, 29192926025390625, 437893890380859375, 6568408355712890625, 98526125335693359375, 1477891880035400390625, 22168378200531005859375, 332525673007965087890625

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 15), L(1, 15), P(1, 15), T(1, 15). Essentially same as Pisot sequences E(15, 225), L(15, 225), P(15, 225), T(15, 225). See A008776 for definitions of Pisot sequences.
A000005(a(n)) = A000290(n+1). - Reinhard Zumkeller, Mar 04 2007
If X_1, X_2, ..., X_n is a partition of the set {1,2,...,2*n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1,2,..., 2*n}->{1,2,3,4} such that for fixed y_1,y_2,...,y_n in {1,2,3,4} we have f(X_i)<>{y_i}, (i=1,2,...,n). - Milan Janjic, May 24 2007
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 15-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Number of ways to assign truth values to n quaternary disjunctions connected by conjunctions such that the proposition is true. For example, a(2) = 225, since for the proposition (a v b v c v d) & (e v f v g v h) there are 225 assignments that make the proposition true. - Ori Milstein, Jan 26 2023

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 279
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (15).

a(n) = A159991(n)/A000302(n). - Reinhard Zumkeller, May 02 2009

Row 6 of A329332.

[ 15^n: n in [0..20] ]; // Vincenzo Librandi, Nov 21 2010

[ n eq 1 select 1 else 15*Self(n-1): n in [1..21] ];

A001024:=-1/(-1+15*z); # Simon Plouffe in his 1992 dissertation

Table[15^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=15^n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-15x), e.g.f.: exp(15x)

a(n) = 15^n; a(n) = 15*a(n-1) with a(0)=1. - Vincenzo Librandi, Nov 21 2010

More terms from James Sellers, Sep 19 2000

A060707 Base-60 (Babylonian or sexagesimal) expansion of Pi.

Original entry on oeis.org

3, 8, 29, 44, 0, 47, 25, 53, 7, 24, 57, 36, 17, 43, 4, 29, 7, 10, 3, 41, 17, 52, 36, 12, 14, 36, 44, 51, 50, 15, 33, 7, 23, 59, 9, 13, 48, 22, 12, 21, 45, 22, 56, 47, 39, 44, 28, 37, 58, 23, 21, 11, 56, 33, 22, 40, 42, 31, 6, 6, 3, 46, 16, 52, 2, 48, 33, 24, 38, 33, 22, 1, 0, 1

Offset: 1

Robert G. Wilson v, Feb 05 2001

Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009, pp. 903-914.
Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005.
Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.

Harry J. Smith, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796, A091649, A159991.

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), this sequence (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60). - Jason Kimberley, Dec 06 2012

Mathematica
```
RealDigits[ Pi, 60, 75][[1]]
```

{ default(realprecision, 17900); x=Pi; for (n=1, 10000, d=floor(x); x=(x-d)*60; write("b060707.txt", n, " ", d)); } \\ Harry J. Smith, Jul 09 2009

A088157 Value of (n+1)-th digit in sexagesimal representation of n^n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 21, 2, 1, 59, 5, 49, 2, 19, 57, 20, 45, 35, 30, 0, 5, 28, 50, 4, 19, 50, 23, 32, 10, 23, 38, 16, 45, 29, 6

Offset: 0

Reinhard Zumkeller, Sep 20 2003

a(n) = d(n) with n^n = Sum(d(k)*60^k: 0 <= d(k) < 60, k >= 0).

			a(0) = 1, a(k) = 0 for 0 < k < 60 and a(60) = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sexagesimal

Cf. A000312, A088150, A088151, A088152, A088153, A088154, A088155, A088156, A159991.

a088157 n = mod (div (n ^ n) (60 ^ n)) 60
-- Reinhard Zumkeller, Mar 14 2014

f[n_] := IntegerDigits[n^n, 60, n + 1][[1]]; f[0] = 1; Array[f, 92, 0] (* Robert G. Wilson v, Dec 27 2012 *)

a(n)=lift(chinese(chinese(Mod(n,3^(n+1))^n,Mod(n,4^(n+1))^n), Mod(n,5^(n+1))^n))\60^n \\ Charles R Greathouse IV, Dec 27 2012

a(n) = floor(n^n / 60^n) mod 60.

A159990 Coefficients in sexagesimal expansion of the positive root of x^3 + 2x^2 + 10x = 20, first studied by Leonardo of Pisa (Fibonacci) in 1225.

Original entry on oeis.org

1, 22, 7, 42, 33, 4, 38, 30, 50, 15, 43, 13, 56, 48, 24, 41, 0, 48, 22, 40, 39, 37, 23, 53, 55, 57, 45, 40, 5, 46, 50, 57, 28, 45, 46, 34, 2, 6, 7, 15, 25, 25, 13, 10, 59, 30, 13, 14, 7, 6, 15, 46, 23, 53, 59, 32, 24, 20, 11, 48, 35, 4, 4, 18, 33, 50, 7, 40, 16, 16, 1, 32, 24, 10, 43, 59, 23, 44, 51, 58, 11, 22, 26, 17

Offset: 0

Reinhard Zumkeller, May 01 2009

Leonardo of Pisa (Fibonacci) found a(0), ..., a(5) but gave a(6) as 40.
A159992(n)/A159993(n) = Sum_{k=0..n} a(k)/60^k = (Sum_{k=0..n} a(k)*60^(n-k))/60^n; let f(x) = x^3 + 2*x^2 + 10*x - 20, then for n > 0:
a(n) = Max(k: f(A159992(n-1)/A159993(n - 1) + k/60^n)) < 0),
a(n) + 1 = Min(k: f(A159992(n - 1)/A159993(n - 1) + k/60^n)) > 0);
A159994(n)/A159995(n) = f(A159992(n)/A159993(n)).

			The root is 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 38/60^6 + 30/60^7 + 50/60^8 + ...
Leonardo's approximation 1;22.7.42.33.4.40 is to be read as 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6 = A159992(5)/A159993(5) + 40/60^6 = 1596577777 / 1166400000 ~= 1.3688081078532235 and f(1596577777/1166400000) ~= +6.7193226361369/10^10; compare this to A159992(6)/A159993(6) = A159992(5)/A159993(5) + 38/60^6 = 31931555539 / 23328000000 ~= 1.3688081078103566 and f(31931555539/23328000000) ~= -2.3239469709985/10^10.
Assuming that Leonardo did similar calculations, the question may arise: why he didn't find a(6) = 38 instead of 40? Supposedly he just avoided the effort to calculate f(A159992(5)/A159993(5) + k/60^6) for k = 37, 38, or 39: 37/60^6 = 37/46656000000, 38/60^6 = 19/23328000000, or 39/60^6 = 13/15552000000; finally, he did calculate only f(A159992(5)/A159993(5) + k/60^6) for k = 36 and k = 40, the less complex cases concerning sexagesimal fractional arithmetic with 36/60^6 = 1/1296000000 and 40/60^6 = 1/1166400000: f(A159992(5)/A159993(5) + 36/60^6) ~= -1.9999999988632783, f(A159992(5)/A159993(5) + 40/60^6) ~= +0.0000000006719323.
The latter result looks precise enough and could explain and justify Leonardo's 'rounding'.

Cox, David A., Galois theory. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2004. xx+559 pp. ISBN: 0-471-43419-1 MR2119052 (2006a:12001). See page 9.
A. F. Horadam, Eight hundred years young, The Australian Mathematics Teacher 31 (1975) 123-134.
Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007): 21.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Ezra Brown and Jason C. Brunson, Fibonacci's forgotten number
Stanislaw Glushkov, On approximation methods of Leonardo Fibonacci, Historia Mathematica 3 (1976), pp. 291-296.
J. J. O'Connor and E. F. Robertson, Fibonacci
Clark Kimberling, Fibonacci [containing the Horadam article]
Trond Steihaug, Fibonacci and digit-by-digit computation; An example of reverse engineering in computational mathematics, arXiv:2211.00504 [math.HO], 2022.
Wikipedia, Sexagesimal
Index entries for algebraic numbers, degree 3

Cf. A159991, A202300.

RealDigits[ Solve[x^3 + 2 x^2 + 10 x - 20 == 0, x][[3, 1, 2]], 60, 111][[1]] (* Robert G. Wilson v, May 11 2012 *)
RealDigits[Root[x^3+2x^2+10x-20,1],60,90][[1]] (* Harvey P. Dale, Jun 16 2025 *)

polrootsreal(x^3+2*x^2+10*x-20)[1] \\ Charles R Greathouse IV, Apr 14 2014

A009974 Powers of 30.

Original entry on oeis.org

1, 30, 900, 27000, 810000, 24300000, 729000000, 21870000000, 656100000000, 19683000000000, 590490000000000, 17714700000000000, 531441000000000000, 15943230000000000000, 478296900000000000000, 14348907000000000000000, 430467210000000000000000, 12914016300000000000000000

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 30), L(1, 30), P(1, 30), T(1, 30). Essentially same as Pisot sequences E(30, 900), L(30, 900), P(30, 900), T(30, 900). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 30-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (30).

Row 7 of A329332.

Cf. A000079, A000244, A008776, A011557, A159991.

[30^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

NestList[30#&,1,20] (* Harvey P. Dale, Jul 21 2023 *)

a(n)=30^n \\ Charles R Greathouse IV, Jun 19 2015

powers(30,12) \\ Charles R Greathouse IV, Jun 19 2015

[lucas_number1(n,30,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-30*x). - Philippe Deléham, Nov 24 2008
a(n) = 30^n; a(n) = 30*a(n-1), n > 0; a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(30*x).
a(n) = A000244(n)*A011557(n) = A159991(n)/A000079(n). (End)

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

Original entry on oeis.org

1, 27000, 46656000000, 46656000000000, 80621568000000000000, 14348907000000000000000, 12694994583552000000000000000000, 812479653347328000000000000000000

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.

A269025 a(n) = Sum_{k = 0..n} 60^k.

Original entry on oeis.org

1, 61, 3661, 219661, 13179661, 790779661, 47446779661, 2846806779661, 170808406779661, 10248504406779661, 614910264406779661, 36894615864406779661, 2213676951864406779661, 132820617111864406779661, 7969237026711864406779661, 478154221602711864406779661

Offset: 0

Ilya Gutkovskiy, Feb 18 2016

Partial sums of powers of 60 (A159991).
Converges in a 10-adic sense to ...762711864406779661.
More generally, the ordinary generating function for the Sum_{k = 0..n} m^k is 1/((1 - m*x)*(1 - x)). Also, Sum_{k = 0..n} m^k = (m^(n + 1) - 1)/(m - 1).

Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (61,-60).

Cf. A159991.

Cf. similar sequences of the form (k^n-1)/(k-1): A000225 (k=2), A003462 (k=3), A002450 (k=4), A003463 (k=5), A003464 (k=6), A023000 (k=7), A023001 (k=8), A002452 (k=9), A002275 (k=10), A016123 (k=11), A016125 (k=12), A091030 (k=13), A135519 (k=14), A135518 (k=15), A131865 (k=16), A091045 (k=17), A218721 (k=18), A218722 (k=19), A064108 (k=20), A218724-A218734 (k=21..31), A132469 (k=32), A218736-A218753 (k=33..50), this sequence (k=60), A133853 (k=64), A094028 (k=100), A218723 (k=256), A261544 (k=1000).

Table[Sum[60^k, {k, 0, n}], {n, 0, 15}]
Table[(60^(n + 1) - 1)/59, {n, 0, 15}]
LinearRecurrence[{61, -60}, {1, 61}, 15]

a(n)=60^n + 60^n\59 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: 1/((1 - 60*x)*(1 - x)).
a(n) = (60^(n + 1) - 1)/59 = 60^n + floor(60^n/59).
a(n+1) = 60*a(n) + 1, a(0)=1.
a(n) = Sum_{k = 0..n} A159991(k).
Sum_{n>=0} 1/a(n) = 1.016671221665660580331...
E.g.f.: exp(x)*(60*exp(59*x) - 1)/59. - Stefano Spezia, Mar 23 2023

A091722 Babylonian sexagesimal (base 60) expansion of 1/13.

Original entry on oeis.org

4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55

Offset: 0

Jeppe Stig Nielsen, Feb 01 2004

Index entries for linear recurrences with constant coefficients, signature (1, -1, 1).

Cf. A021017, A091721, A091722, A055643.

Cf. A159991. - Reinhard Zumkeller, May 02 2009

RealDigits[ 1/13, 60, 75] [[1]] (* Robert G. Wilson v, Feb 02 2004 *)

A091649 Base-60 (Babylonian or sexagesimal) expansion of 2 Pi.

Original entry on oeis.org

6, 16, 59, 28, 1, 34, 51, 46, 14, 49, 55, 12, 35, 26, 8, 58, 14, 20, 7, 22, 35, 45, 12, 24, 29, 13, 29, 43, 40, 31, 6, 14, 47, 58, 18, 27, 36, 44, 24, 43, 30, 45, 53, 35, 19, 28, 57, 15, 56, 46, 42, 23, 53, 6, 45, 21, 25, 2, 12, 12, 7, 32, 33, 44, 5, 37, 6, 49, 17, 6, 44, 2, 0, 3, 20

Offset: 1

Eric W. Weisstein, Jan 25 2004

			2 pi = 6 + 16/60 + 59/60^2 + 28/60^3 + ....

Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009 , pp. 903-914.
Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005.
Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.

Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796, A060707.

A159991. - Reinhard Zumkeller, May 02 2009

RealDigits[ 2Pi, 60, 75][[1]] (* Robert G. Wilson v, Jul 08 2004 *)

A091721 Babylonian sexagesimal (base 60) expansion of 1/11.

Original entry on oeis.org

5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, 5, 27, 16, 21

Offset: 0

Jeppe Stig Nielsen, Feb 01 2004

Period 5: repeat [5, 27, 16, 21, 49]. - Wesley Ivan Hurt, May 25 2024

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 1).

Cf. A092291, A091720, A091722, A055643.

Cf. A159991. - Reinhard Zumkeller, May 02 2009

RealDigits[ 1/11, 60, 75] [[1]] (* Robert G. Wilson v, Feb 02 2004 *)
CoefficientList[Series[(5 + 27 x + 16 x^2 + 21 x^3 + 49 x^4)/(1 - x^5), {x, 0, 40}], x] (* Wesley Ivan Hurt, May 25 2024 *)

From Wesley Ivan Hurt, May 25 2024: (Start)
a(n+5) = a(n).
G.f.: (5+27*x+16*x^2+21*x^3+49*x^4)/(1-x^5). (End)

cp's OEIS Frontend

A001024 Powers of 15.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

Formula

Extensions

A060707 Base-60 (Babylonian or sexagesimal) expansion of Pi.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

A088157 Value of (n+1)-th digit in sexagesimal representation of n^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A159990 Coefficients in sexagesimal expansion of the positive root of x^3 + 2*x^2 + 10*x = 20, first studied by Leonardo of Pisa (Fibonacci) in 1225.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A009974 Powers of 30.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

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

Views

Author

Keywords

Links

Crossrefs

A269025 a(n) = Sum_{k = 0..n} 60^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A159990 Coefficients in sexagesimal expansion of the positive root of x^3 + 2x^2 + 10x = 20, first studied by Leonardo of Pisa (Fibonacci) in 1225.

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