A020729 -id:A020729 - cp's OEIS frontend

A020699 Expansion of (1-3x)/(1-5x).

1, 2, 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250, 19531250, 97656250, 488281250, 2441406250, 12207031250, 61035156250, 305175781250, 1525878906250, 7629394531250, 38146972656250, 190734863281250, 953674316406250

Offset: 0

David W. Wilson

Partial sums are A034478.
Except for the first two terms 1 and 2, these are the integers that satisfy phi(n) = 2*n/5. - Michel Marcus, Jul 14 2015
For n>=1, period of powers of 4 mod 10^n. See A000302. - Martin Renner, Jun 12 2020

Nathaniel Johnston, Table of n, a(n) for n = 0..250
D Bevan, D Levin, P Nugent, J Pantone, L Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1037
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1037 (archived version of page)
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A034478, A001045, A020729, A000302.

seq(`if`(n=0,1,2*5^(n-1)), n=0..22); # Nathaniel Johnston, Jun 26 2011

CoefficientList[Series[(1 - 3 x)/(1 - 5 x), {x, 0, 22}], x] (* Michael De Vlieger, Jul 14 2015 *)

Vec((1-3*x)/(1-5*x) + O(x^30)) \\ Michel Marcus, Jul 14 2015

a(n) = 2*5^(n-1) for n>0.
E.g.f.: (2*exp(5*x)+3)/5; a(n)=(2*5^n+3*0^n)/5. - Paul Barry, Sep 03 2003
a(n) = sum{k=0..n, C(n-1, k)*(Jac(2n-2k)+Jac(2n-2k-1))}+0^n/2, where Jac(n)=A001045(n). - Paul Barry, Jun 07 2005
a(0)=1, a(1)=2, a(n) = 5*a(n-1) for n>=2. [Vincenzo Librandi, Jan 01 2011]
a(n) = A020729(n-1), n>0. - R. J. Mathar, Sep 16 2016

A154692 Triangle read by rows: T(n, k) = (2^(n-k)3^k + 2^k3^(n-k))*binomial(n, k).

Original entry on oeis.org

2, 5, 5, 13, 24, 13, 35, 90, 90, 35, 97, 312, 432, 312, 97, 275, 1050, 1800, 1800, 1050, 275, 793, 3492, 7020, 8640, 7020, 3492, 793, 2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315, 6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

			Triangle begins
     2;
     5,     5;
    13,    24,    13;
    35,    90,    90,     35;
    97,   312,   432,    312,     97;
   275,  1050,  1800,   1800,   1050,    275;
   793,  3492,  7020,   8640,   7020,   3492,   793;
  2315, 11550, 26460,  37800,  37800,  26460, 11550,  2315;
  6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000225, A007482, A013620, A088137, A119309.
Sums include: A010673 (alternating sign row), A020699 (row), A020729 (row).
Related sequences: A007318, A154690,

A154692:= func< n,k | (2^(n-k)*3^k + 2^k*3^(n-k))*Binomial(n,k) >;
[A154692(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154692 := proc(n,m)
        (2^(n-m)*3^m+2^m*3^(n-m))*binomial(n,m) ;
end proc:
seq(seq(A154692(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Oct 24 2011

p=2; q=3;
T[n_, m_]= (p^(n-m)*q^m + p^m*q^(n-m))*Binomial[n,m];
Table[T[n,m], {n,0,10}, {m,0,n}]//Flatten

from sage.all import *
def A154692(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*binomial(n,k)
print(flatten([[A154692(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Sum_{k=0..n} T(n, k) = A020729(n) = A020699(n+1).
T(n,m) = A013620(n,m) + A013620(m,n). - R. J. Mathar, Oct 24 2011
From G. C. Greubel, Jan 18 2025: (Start)
T(2*n, n) = A119309(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1) + A007482(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A088137(n+1) + A000225(n+1). (End)

A132021 Decimal expansion of Product_{k>=0} 1-1/(2*5^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.438796837203638531...

Cf. A048651, A098844, A067080, A132019, A132026, A132029, A100222, A000217, A020729.

digits = 103; NProduct[1-1/(2*5^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/5], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_5(n))} floor(n/5^k)*5^k/n.
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^(1/2*(1+floor(log_5(n)))*floor(log_5(n))).
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^A000217(floor(log_5(n))).
Equals (1/2)*exp(-Sum_{n>0} 5^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132029(n)/A132029(n+1).
Equals Product_{n>=0} (1 - 1/A020729(n)). - Amiram Eldar, May 08 2023

A276508 a(n) = (25^n + 3(-1)^(floor((n-1)/3)) + (-1)^n)/6.

Original entry on oeis.org

0, 2, 9, 42, 208, 1041, 5208, 26042, 130209, 651042, 3255208, 16276041, 81380208, 406901042, 2034505209, 10172526042, 50862630208, 254313151041, 1271565755208, 6357828776042, 31789143880209, 158945719401042, 794728597005208, 3973642985026041, 19868214925130208, 99341074625651042

Offset: 0

Ilya Gutkovskiy, Sep 06 2016

Number of 1’s in substitution system {1 -> 12321, 2 -> 23132, 3 -> 31213} at step n from initial string "3" (see example). Number of 2’s: A000351(n) - A010892(n+1) - 2*a(n). Number of 3’s: A010892(n+1) + a(n).

Excluding zero, convolution of A000351 and A174737.

			Evolution from initial string "3": 3 -> 31213 -> 3121312321231321232131213 -> ...
Therefore, number of 1’s at step n:
a(0) = 0;
a(1) = 2;
a(2) = 9, etc.

Ilya Gutkovskiy, Illustration (substitution system {1 -> 12321, 2 -> 23132, 3 -> 31213})
Eric Weisstein's World of Mathematics, Substitution System
Index entries for linear recurrences with constant coefficients, signature (6,-6,5)

Cf. A000351, A010892, A020729, A057079, A174737.

A276508:=n->(2*5^n + 3*(-1)^(floor((n-1)/3)) + (-1)^n)/6: seq(A276508(n), n=0..30); # Wesley Ivan Hurt, Sep 07 2016

Table[(2 5^n + 3 (-1)^Floor[(n - 1)/3] + (-1)^n)/6, {n, 0, 25}]
LinearRecurrence[{6, -6, 5}, {0, 2, 9}, 26]

concat(0, Vec(x*(2-3*x)/((1-5*x)*(1-x+x^2)) + O(x^99))) \\ Altug Alkan, Sep 06 2016

O.g.f.: x*(2 - 3*x)/((1 - 5 x)*(1 - x + x^2)).
E.g.f.: (exp(9*x/2) - 2*sin(Pi/6-sqrt(3)*x/2))*exp(x/2)/3.
a(n) = 6*a(n-1) - 6*a(n-2) + 5*a(n-3).
a(n) = (5^n + sqrt(3)*sin(Pi*n/3) - cos(Pi*n/3))/3.
a(n) = (A020729(n) + A057079(n-1))/3.

A095687 Numbers n such that n-th Pisano number = 6*n.

Original entry on oeis.org

0, 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250, 19531250, 97656250, 488281250, 2441406250, 12207031250, 61035156250, 305175781250, 1525878906250, 7629394531250, 38146972656250, 190734863281250

Offset: 1

Lekraj Beedassy, Jul 05 2004

nonn

Period of Fibonacci sequence(mod m)<=6*m.

Effectively the same as A020699 and A020729 apart from the initial terms. [From Carl R. White, Sep 22 2009]

K. S. Brown, Periods of Fibonacci Sequences Mod m

Cf. A001175.

a(0) = 0, a(1) = 10, a(n) = 5*a(n-1); a(n) = sgn(n)*2*5^n [From Carl R. White, Sep 22 2009]

More terms from Carl R. White, Sep 22 2009

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A020699 Expansion of (1-3*x)/(1-5*x).

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A154692 Triangle read by rows: T(n, k) = (2^(n-k)*3^k + 2^k*3^(n-k))*binomial(n, k).

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A132021 Decimal expansion of Product_{k>=0} 1-1/(2*5^k).

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A276508 a(n) = (2*5^n + 3*(-1)^(floor((n-1)/3)) + (-1)^n)/6.

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A095687 Numbers n such that n-th Pisano number = 6*n.

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A020699 Expansion of (1-3x)/(1-5x).

A154692 Triangle read by rows: T(n, k) = (2^(n-k)3^k + 2^k3^(n-k))*binomial(n, k).

A276508 a(n) = (25^n + 3(-1)^(floor((n-1)/3)) + (-1)^n)/6.