A002266 -id:A002266 - cp's OEIS frontend

A133875 n modulo 5 repeated 5 times.

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 5^2 = 25.

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

Cf. A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133885, A133880, A133890, A133900, A133910.

[(1 + Floor(n/5)) mod 5 : n in [0..50]]; // Wesley Ivan Hurt, Jun 06 2014

A133875:=n->((1+floor(n/5)) mod 5); seq(A133875(n), n=0..100); # Wesley Ivan Hurt, Jun 06 2014

Table[Mod[1 + Floor[n/5], 5], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 06 2014 *)
LinearRecurrence[{1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1},{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,0},120] (* Harvey P. Dale, Dec 14 2017 *)

a(n) = (1 + floor(n/5)) mod 5.
a(n) = A010874(A002266(n+5)).
a(n) = 1 + floor(n/5) - 5*floor((n+5)/25).
a(n) = (((n+5) mod 25) - (n mod 5)) / 5.
a(n) = ((n + 5 - (n mod 5)) / 5) mod 5.
a(n) = A010874((n + 5 - A010874(n))/5).
a(n) = binomial(n+5, n) mod 5 = binomial(n+5, 5) mod 5.
a(n) = +a(n-1) -a(n-5) +a(n-6) -a(n-10) +a(n-11) -a(n-15) +a(n-16) -a(n-20) +a(n-21). - R. J. Mathar, Sep 03 2011
G.f.: ( 1+2*x^5+3*x^10+4*x^15 ) / ( (1-x)*(x^20+x^15+x^10+x^5+1) ). - R. J. Mathar, Sep 03 2011

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 18 2015

			The 5 partitions of n=4 are 1+1+1+1 (lcm=1), 1+1+2 (lcm=2), 2+2 (lcm=2), 1+3 (lcm=3) and 4 (lcm=4). So k=1, 3 and 4 appear once, k=2 appears twice.
The triangle starts:
  1 ;
  1 ;
  1  1;
  1  1  1;
  1  2  1  1;
  1  2  1  1  1  1;
  1  3  2  2  1  2;
  1  3  2  2  1  3  1  0  0  1  0  1;
  ...

Alois P. Heinz, Rows n = 0..30, flattened

Cf. A000041 (row sums), A000793 (row lengths), A213952, A074761 (diagonal), A074752 (6th column), A008642 (4th column), A002266 (5th column), A002264 (3rd column), A132270 (7th column), A008643 (8th column), A008649 (9th column), A258470 (10th column).

Cf. A009490 (number of nonzero terms of rows), A074064 (last elements of rows), A168532 (the same for gcd), A181844 (Sum k*T(n,k)).

A256067 := proc(n,k)
        local a,p ;
        a := 0 ;
        for p in combinat[partition](n) do
                ilcm(op(p)) ;
                if % = k then
                        a := a+1 ;
                end if;
        end do:
        a;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 27 2015

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i-1] + Function[{p}, Sum[ Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

T(0,1)=1 prepended by Alois P. Heinz, Mar 27 2015

A010883 Simple periodic sequence: repeat 1,2,3,4.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1

Offset: 0

N. J. A. Sloane

Partial sums are given by A130482(n) + n + 1. - Hieronymus Fischer, Jun 08 2007

1234/9999 = 0.123412341234... - Eric Desbiaux, Nov 03 2008

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

Cf. A010872, A010873, A010874, A010875, A010876, A004526, A002264, A002265, A002266.

Cf. A177037 (decimal expansion of (9+2*sqrt(39))/15). - Klaus Brockhaus, May 01 2010

PadRight[{},120,{1,2,3,4}] (* Harvey P. Dale, Aug 02 2016 *)

a(n)=(n-1)%4+1 \\ Charles R Greathouse IV, Jun 11 2015

def A010883(n): return 1 + (n & 3) # Chai Wah Wu, May 25 2022

a(n) = 1 + (n mod 4). - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = A010873(n) + 1.
Also a(n) = (1/2)*(5 - (-1)^n - 2*(-1)^((2*n - 1 + (-1)^n)/4)).
G.f.: g(x) = (4*x^3 + 3*x^2 + 2*x + 1)/(1 - x^4) = (4*x^5 - 5*x^4 + 1)/((1 - x^4)*(1-x)^2). (End)
a(n) = 5/2 - cos(Pi*n/2) - sin(Pi*n/2) - (-1)^n/2. - R. J. Mathar, Oct 08 2011

A132292 Integers repeated 8 times: a(n) = floor((n-1)/8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Mohammad K. Azarian, Nov 06 2007

Also floor((n^8-1)/(8*n^7)).

Cf. A004526, A002264, A002265, A002266, A054895.

[(n - 1) div 8 : n in [1..90]]; // Vincenzo Librandi, Oct 05 2018

A132292:=n->floor((n-1)/8); seq(A132292(n), n=1..100); # Wesley Ivan Hurt, Feb 27 2014

Table[Floor[(n-1)/8], {n, 100}] (* Wesley Ivan Hurt, Feb 27 2014 *)
Table[PadRight[{},8,n],{n,0,10}]//Flatten (* Harvey P. Dale, Apr 13 2020 *)

a(n)=(n-1)\8 \\ Charles R Greathouse IV, Jun 18 2013

def A132292(n): return n-1>>3 # Chai Wah Wu, Jul 27 2022

Also, a(n) = floor((n^8-n^7)/(8n^7-7n^6)). - Mohammad K. Azarian, Nov 18 2007
a(n) = A180969(3,n).
a(n) = (r - 8 + 4*sin(r*Pi/8))/16 where r = 2*n - 1 - 2*cos(n*Pi/2) - cos(n*Pi) + 2*sin(n*Pi/2). - Wesley Ivan Hurt, Oct 04 2018

Offset corrected by Mohammad K. Azarian, Nov 20 2008

New name from Wesley Ivan Hurt, Jun 17 2013

A090223 Nonnegative integers with doubled multiples of 4.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 56, 56, 57, 58

Offset: 0

Wolfdieter Lang, Dec 01 2003

Degrees of row-polynomials of array A090222.

a(n) is the number of full orbits completed by body A for n full orbits completed by body B in a celestial system with two orbiting bodies A and B with orbital resonance A:B equal to 4:5. This resonance is exhibited by the planets Kepler-90b and Kepler-90c in the planetary system of the star Kepler-90. - Felix Fröhlich, May 03 2021

Felix Fröhlich, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A057353 and other floors of ratios references there.

Cf. A090222.

[4*n div 5: n in [0..80]]; // Bruno Berselli, Dec 07 2016

A090223:=n->floor(4*n/5); seq(A090223(n), n=0..100); # Wesley Ivan Hurt, Nov 15 2013

Table[Floor[4n/5], {n,0,100}] (* Wesley Ivan Hurt, Nov 15 2013 *)

a(n)=4*n\5 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = floor(4*n/5).
G.f.:  x^2 *(1+x^2)*(1+x)/((1-x^5)*(1-x)) = (x^2)*(1-x^4)/((1-x^5)*(1-x)^2).
a(n) = n - 1 - A002266(n - 1). - Wesley Ivan Hurt, Nov 15 2013
a(n) = A057354(2*n). - R. J. Mathar, Jul 21 2020
5*a(n) = 4*n-2+A117444(n+2) . - R. J. Mathar, Jul 21 2020
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-1)*Pi/8. - Amiram Eldar, Sep 30 2022

A110532 a(n) = floor(n/2) + floor(n/5).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 4, 5, 5, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 14, 14, 15, 15, 16, 17, 18, 18, 19, 19, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 28, 28, 29, 29, 30, 31, 32, 32, 33, 33, 35, 35, 36, 36, 37, 38, 39, 39, 40, 40, 42, 42, 43, 43, 44, 45, 46, 46, 47, 47, 49, 49, 50, 50

Offset: 0

Reinhard Zumkeller, Jul 25 2005

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,1,0,-1).

Cf. A010761, A110533.

Table[Floor[n/2]+Floor[n/5],{n,0,80}] (* or *) LinearRecurrence[ {0,1,0,0,1,0,-1},{0,0,1,1,2,3,4},80] (* Harvey P. Dale, Dec 26 2015 *)

a(n)=n\2 + n\5 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A004526(n) + A002266(n).
G.f.: x^2*(1+x+x^2+2*x^3+2*x^4) / ( (1+x)*(x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Feb 20 2011
a(n) = 7n/10 + O(1). - Charles R Greathouse IV, Jun 11 2015

A110533 a(n) = floor(n/2) * floor(n/5).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 3, 4, 4, 10, 10, 12, 12, 14, 21, 24, 24, 27, 27, 40, 40, 44, 44, 48, 60, 65, 65, 70, 70, 90, 90, 96, 96, 102, 119, 126, 126, 133, 133, 160, 160, 168, 168, 176, 198, 207, 207, 216, 216, 250, 250, 260, 260, 270, 297, 308, 308, 319, 319, 360, 360, 372

Offset: 0

Reinhard Zumkeller, Jul 25 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A001622, A002266, A004526, A010762, A110532.

Table[Floor[n/2]*Floor[n/5], {n, 0, 50}] (* G. C. Greubel, Aug 30 2017 *)

for(n=0,50, print1(floor(n/2)*floor(n/5), ", ")) \\ G. C. Greubel, Aug 30 2017

a(n) = A004526(n)*A002266(n).
From R. J. Mathar, Feb 20 2011: (Start)
a(n) = +a(n-2) +a(n-5) -a(n-7) +a(n-10) -a(n-12) -a(n-15) +a(n-17).
G.f.: -x^5*(2+3*x+x^2+x^3+x^4+4*x^5+3*x^6+x^7+x^8+x^9+x^10+x^11) / ( (x^4-x^3+x^2-x+1) *(1+x)^2 *(x^4+x^3+x^2+x+1)^2 *(x-1)^3 ). (End)
Sum_{n>=5} (-1)^(n+1)/a(n) = sqrt(5*(5-2*sqrt(5)))*Pi/8 - (5/8)*(1 + sqrt(5)*log(phi)) + (25/16)*log(5) - 2*log(2), where phi is the golden ratio (A001622). - Amiram Eldar, Mar 30 2023

A256554 Number T(n,k) of cycle types of degree-n permutations having the k-th smallest possible order; triangle T(n,k), n>=0, 1<=k<=A009490(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 1, 1, 1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1, 1, 5, 3, 6, 2, 12, 1, 2, 1, 4, 1, 6, 2, 2, 1, 2, 1, 1, 1, 2

Offset: 0

Alois P. Heinz, Apr 01 2015

Sum_{k>=0} A256553(n,k)*T(n,k) = A181844(n).

			Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 2, 1, 1, 1, 1;
  1, 3, 2, 2, 1, 2;
  1, 3, 2, 2, 1, 3, 1, 1, 1;
  1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1;
  1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1;
  1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1;

Alois P. Heinz, Rows n = 0..60, flattened

Row sums give A000041.
Row lengths give A009490.
Columns k=1-9 give: A000012, A004526, A002264, A008642(n-4), A002266, A074752, A132270, A008643(n-8) for n>7, A008649(n-9) for n>8.
Last elements of rows give A074064.
Main diagonal gives A074761.
Cf. A181844, A256067, A256553.

b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n->(p->seq((h->`if`(h=0, [][], h))(coeff(p, x, i))
     , i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; T[n_] := Function[p, Table[Function[h, If[h == 0, {{}, {}}, h]][Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 23 2017, translated from Maple *)

A356858 a(n) is the product of the first n numbers not divisible by 5.

Original entry on oeis.org

1, 1, 2, 6, 24, 144, 1008, 8064, 72576, 798336, 9580032, 124540416, 1743565824, 27897053184, 474249904128, 8536498274304, 162193467211776, 3406062811447296, 74933381851840512, 1723467782592331776, 41363226782215962624, 1075443896337615028224, 29036985201115605762048

Offset: 0

Stefano Spezia, Sep 01 2022

nonn

Unlike the factorial number n!, a(n) does not have trailing zeros.

Harvey P. Dale, Table of n, a(n) for n = 0..434

Cf. A000142, A000351, A002266, A047201.

Cf. A356859 (number of zero digits), A356860 (number of digits), A356861 (number of nonzero digits).

Table[Product[Floor[(5k-1)/4], {k,n}], {n,0,22}] (* or *)
Join[{1}, Table[Floor[(5n-1)/4]!/(Floor[Floor[(5n-1)/4]/5]!*5^Floor[Floor[(5n-1)/4]/5]), {n,22}]]
Join[{1},FoldList[Times,Table[If[Mod[n,5]==0,Nothing,n],{n,30}]]] (* Harvey P. Dale, Nov 03 2024 *)

from math import prod
def a(n): return prod((5*k-1)//4 for k in range(1, n+1))
print([a(n) for n in range(23)]) # Michael S. Branicky, Sep 01 2022

a(n) = Product_{k=1..n} A047201(k).

a(n) = A047201(n)!/(floor(A047201(n)/5)!*5^floor(A047201(n)/5)) for n > 0.

A010887 Simple periodic sequence: repeat 1,2,3,4,5,6,7,8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130486(n)+n+1. - Hieronymus Fischer, Jun 08 2007

1371742/11111111 = 0.123456781234567812345678... - Eric Desbiaux, Nov 03 2008

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

Cf. A010872, A010873, A010874, A010875, A010876, A010878, A004526, A002264, A002265, A002266.

Cf. A177034 (decimal expansion of (9280+3*sqrt(13493990))/14165). - Klaus Brockhaus, May 01 2010

a010887 = (+ 1) . flip mod 8
a010887_list = cycle [1..8]
-- Reinhard Zumkeller, Nov 09 2014, Mar 04 2014

PadRight[{},90,Range[8]] (* Harvey P. Dale, May 10 2022 *)

def A010887(n): return 1 + (n & 7) # Chai Wah Wu, May 25 2022

a(n) = 1 + (n mod 8) - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = (1/2)*(9 - (-1)^n - 2*(-1)^(b/4) - 4*(-1)^((b - 2 + 2*(-1)^(b/4))/8)) where b = 2n - 1 + (-1)^n.
Also a(n) = A010877(n) + 1.
G.f.: g(x) = (1/(1-x^8))*Sum_{k=0..7} (k+1)*x^k.
Also: g(x) = (8x^9 - 9x^8 + 1)/((1-x^8)*(1-x)^2). (End)

cp's OEIS Frontend

A133875 n modulo 5 repeated 5 times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A010883 Simple periodic sequence: repeat 1,2,3,4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A132292 Integers repeated 8 times: a(n) = floor((n-1)/8).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A090223 Nonnegative integers with doubled multiples of 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A110532 a(n) = floor(n/2) + floor(n/5).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A110533 a(n) = floor(n/2) * floor(n/5).

Views

Author

Keywords

Links

Crossrefs

Programs