A133886 -id:A133886 - cp's OEIS frontend

A133900 a(n) = period of the sequence {b(m), m>=0}, defined by b(m):=binomial(m+n,n) mod n.

1, 4, 9, 16, 25, 72, 49, 64, 81, 400, 121, 864, 169, 784, 675, 256, 289, 2592, 361, 1600, 1323, 3872, 529, 3456, 625, 5408, 729, 3136, 841, 324000, 961, 1024, 9801, 18496, 6125, 31104, 1369, 23104, 13689, 32000, 1681, 254016, 1849, 15488, 30375, 33856

Offset: 1

Hieronymus Fischer, Oct 15 2007, Oct 20 2007

nonn

This is the analog of the sequence of Pisano periods (A001175) for binomial factors.
n^2 always divides a(n).
A prime p is a factor of a(n) if and only if it is a factor of n (i.e., a(n) and n have the same prime factors).

			a(3)=9 since binomial(m+3,3) mod 3, m>=0, is periodic with period length 3^2=9 (see A133883).
a(6)=72 since binomial(m+6,6) mod 6, m>=0, is periodic with period length 4*6^2=72 (see A133886).

Hieronymus Fischer, Table of n, a(n) for n = 1..111

Cf. A000040, A001175, A133872-A133880, A133882-A133890, A133910.
Cf. A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133905.

a(n)=n^2 if n is a prime or a power of a prime.

A133636 Nonprime numbers k such that binomial(k+p,k) mod k = 1, where p=6.

Original entry on oeis.org

9, 27, 49, 63, 77, 81, 91, 99, 117, 119, 121, 133, 143, 153, 161, 169, 171, 187, 189, 203, 207, 209, 217, 221, 243, 247, 253, 259, 261, 279, 287, 289, 297, 299, 301, 319, 323, 329, 333, 341, 343, 351, 361, 369, 371, 377, 387, 391, 403, 407, 413, 423, 427, 437

Offset: 1

Hieronymus Fischer, Sep 30 2007

nonn

Also composite n such that binomial(7*n,7)== n (mod n^2). - Gary Detlefs, Sep 24 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133876, A133886, A133880, A133890, A133900, A133910.

Select[Range[500],CompositeQ[#]&&Mod[Binomial[#+6,#],#]==1&] (* Harvey P. Dale, Jan 30 2025 *)

isok(n) = ! isprime(n) && ((binomial(n+6, n) % n) == 1); \\ Michel Marcus, Sep 25 2013

isok(n) = ! isprime(n) && ((binomial(7*n, 7) % n^2) == n); \\ Michel Marcus, Sep 25 2013

A133876 n modulo 6 repeated 6 times.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 0, 0

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 6^2=36.

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

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133886, A133880, A133890, A133900, A133910.

Table[PadRight[{},6,Mod[n,6]],{n,20}]//Flatten (* Harvey P. Dale, Nov 15 2023 *)

a(n)=(1+floor(n/6)) mod 6.
a(n)=1+floor(n/6)-6*floor((n+6)/36).
a(n)=(((n+6) mod 36)-(n mod 6))/6.
a(n)=((n+6-(n mod 6))/6) mod 6.
G.f. g(x)=(1-x^6)(1+2x^5+3x^12+4x^18+5x^24)/((1-x)(1-x^36)).
G.f. g(x)=(5x^36-6x^30+1)/((1-x)(1-x^6)(1-x^36)).

A133891 a(n) = binomial(n+p,n) mod p, where p=12.

Original entry on oeis.org

1, 1, 7, 11, 8, 8, 0, 0, 6, 2, 2, 2, 4, 4, 4, 0, 3, 3, 9, 9, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 8, 8, 5, 9, 3, 3, 8, 8, 8, 4, 10, 10, 6, 6, 0, 0, 0, 0, 3, 3, 9, 9, 0, 0, 4, 4, 4, 8, 8, 8, 0, 0, 0, 8, 5, 5, 7, 7, 4, 0, 0, 0, 6, 6, 6, 6, 0, 0, 0, 0, 3, 7, 1, 1, 8, 8, 8, 0, 0, 0, 8, 8, 8, 4, 4, 4, 9, 9, 3, 3, 0, 0, 0, 0

Offset: 0

Hieronymus Fischer, Oct 16 2007

nonn

Periodic with length 6*12^2 = 864 = A133900(12).

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636, A133910.
See A133872, A133873, A133875, A133877, A133884, A133886, A133888, A133889, A133890 for sequences with different values of p.
See A133900 for the respective periods regarding other values of p.

Table[Mod[Binomial[n+12,n],12],{n,0,110}] (* Harvey P. Dale, Oct 13 2017 *)

a(n) = binomial(n+12,12) mod 12.

A133896 Numbers m such that binomial(m+6,m) mod 6 = 0.

Original entry on oeis.org

3, 4, 5, 6, 7, 12, 13, 14, 15, 21, 22, 23, 26, 30, 31, 34, 35, 39, 42, 43, 44, 50, 51, 52, 53, 58, 59, 60, 61, 62, 66, 67, 68, 69, 70, 71, 75, 76, 77, 78, 79, 84, 85, 86, 87, 93, 94, 95, 98, 102, 103, 106, 107, 111, 114, 115, 116, 122, 123, 124, 125, 130, 131, 132, 133, 134

Offset: 0

Hieronymus Fischer, Oct 20 2007

nonn

Partial sums of the sequence 3,1,1,1,1,5,1,1,1,6,1,1,3,4,1,3,1,4,3,1,1,6,1,1,1,5,1,1,1,1,4,1,1,1,1,1,4, ... which has period 36.

Cf. A000040, A133620, A133621, A133623, A133630, A133635.

Cf. A133876, A133886, A133890, A133900, A133910.

Select[Range[140], Mod[Binomial[# + 6, #], 6] == 0&] (* Jean-François Alcover, Nov 12 2017 *)

isok(n) = !(binomial(n+6, n) % 6); \\ Michel Marcus, Nov 12 2017

G.f.: g(x)=3/(1-x)+ x/(1-x)^2+(4x^5+5x^9+2x^12+3x^13+2x^15+3x^17+2x^18+5x^21+3x^26+3x^32) /((1-x^36)(1-x)).

G.f.: g(x)=(3-2x+4x^5+5x^9+2x^12+3x^13+2x^15+3x^17+2x^18+5x^21+3x^26+3x^32-x^37) /((1-x^36)(1-x)^2).

cp's OEIS Frontend

A133900 a(n) = period of the sequence {b(m), m>=0}, defined by b(m):=binomial(m+n,n) mod n.

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A133636 Nonprime numbers k such that binomial(k+p,k) mod k = 1, where p=6.

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A133876 n modulo 6 repeated 6 times.

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A133891 a(n) = binomial(n+p,n) mod p, where p=12.

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A133896 Numbers m such that binomial(m+6,m) mod 6 = 0.

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