A133625 -id:A133625 - 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.

A133620 Binomial(n+p,n) mod n where p=10.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 2, 6, 2, 6, 1, 2, 1, 10, 5, 7, 1, 12, 1, 15, 18, 12, 1, 12, 21, 14, 4, 12, 1, 28, 1, 29, 1, 18, 6, 5, 1, 20, 14, 10, 1, 14, 1, 34, 15, 24, 1, 3, 8, 16, 18, 27, 1, 34, 23, 16, 1, 30, 1, 16, 1, 32, 17, 57, 40, 56, 1, 1, 47, 60, 1, 54, 1, 38, 36, 58, 12, 66, 1, 63, 10, 42, 1

Offset: 1

Hieronymus Fischer, Sep 30 2007

Let d(m)...d(2)d(1)d(0) be the base-n representation of n+p. The relation a(n)=d(1) holds, if n is a prime index. For this reason there are infinitely many terms which are equal to 1.

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 58060800.

Cf. A000040, A133621-A133625, A133630, A038509, A133634, A133635, A133636.

Cf. A133880, A133890, A133900, A133910, A362686, A362687, A362688, A362689.

Table[Mod[Binomial[n + 10, n], n], {n, 90}] (* Harvey P. Dale, Apr 04 2015 *)

a(n) = binomial(n+10, n) % n \\ Michel Marcus, Jul 15 2013

a(n) = binomial(n+p,p) mod n.
a(n) = 1 if n is a prime > p, since binomial(n+p,n)==(1+floor(p/n))(mod n), provided n is a prime.
a(n) = A001287(n+10) mod n. - Michel Marcus, Jul 15 2013; corrected by Michel Marcus, Jan 27 2020
For n > 58060802, a(n) = 2*a(n-29030400) - a(n-58060800). - Ray Chandler, Apr 29 2023

A133872 Period 4: repeat [1, 1, 0, 0].

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1

Offset: 0

Hieronymus Fischer, Oct 10 2007

Partial sums of A056594.
Let i=sqrt(-1) and S(n) = Sum_{k=0..n-1} exp(2*Pi*i*k^2/n) for n>=1 the famous Gauss sum. Then S(n) = (a(n)+a(n+1)*i)*sqrt(n). - Franz Vrabec, Nov 08 2007
a(A042948(n)) = 1; a(A042964(n)) = 0. - Reinhard Zumkeller, Oct 03 2008
a(n) is also the real part of partial sum of powers of the complex unit i. - Enrique Pérez Herrero, Aug 16 2009
Periodic sequences having a period of 2k and composed of k ones followed by k zeros have a closed formula of floor(((n+k) mod 2k)/k). Listed sequences of this form are: k=1..A000035(n+1), k=2..A133872(n), k=3..A088911, k=4..A131078(n), k=5..A112713(n-1). - Gary Detlefs, May 17 2011
0.repeat(0,0,1,1) is 1/5 in base 2, due to 1/5 = (3/16)/(1-1/16). For the general case see 1/A062158(n) in base n >= 2. Here n = 2. - Wolfdieter Lang, Jun 20 2014
a(n) (for n>=1) is the determinant of the n X n Toeplitz matrix M satisfying: M(i,j)=1 if -1<=j-i<=2 and 0 otherwise. - Dmitry Efimov, Jun 23 2015
a(n) (for n>=1) is the difference between numbers of even and odd permutations p of 1,2,...,n such that -1 <= p(i)-i <= 2 for i=1,2,...,n. - Dmitry Efimov, Jan 08 2016
The binomial transform is 1, 2, 3, 4, 6, 12,... (see A038504). - R. J. Mathar, Feb 25 2023

			G.f. = 1 + x + x^4 + x^5 + x^8 + x^9 + x^12 + x^13 + x^16 + x^17 + x^20 + ...

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Psychedelic Geometry Blogspot, Curious Series-001 [_Enrique Pérez Herrero_, Aug 16 2009]
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A056594, A133620-A133625, A133630, A038509, A133634-A133636, A021913, A000217, A133882, A133880, A133890, A133900, A133910, A000035, A088911, A131078, A112713.

[ (1 + (-1)^Floor(n/2))/2 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

&cat[[1,1,0,0]^^25]; // Vincenzo Librandi, Jan 09 2016

A133872:=n->(1+(-1)^((2n-1+(-1)^n)/4))/2;
seq(A133872(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Table[(1 + (-1)^((2 n - 1 + (-1)^n)/4))/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 06 2013 *)
PadRight[{},120,{1,1,0,0}] (* Harvey P. Dale, Jan 26 2014 *)

a(n)=n%4<2 \\ Jaume Oliver Lafont, Mar 17 2009

Vec((1+x)/(1-x^4) + O(x^100)) \\ Altug Alkan, Jan 08 2015

def A133872(n): return int(not n&2) # Chai Wah Wu, Jan 31 2023

maxn <- 63 # by choice
a <- c(1,0,0)
for(n in 4:maxn) a[n] <- a[n-1] - a[n-2] + a[n-3]
(a <- a(1,a))
# Yosu Yurramendi, Oct 25 2020

a(n) = (1 + floor(n/2)) mod 2.
a(n) = A004526(A000035(n+2)).
a(n) = 1 + floor(n/2) - 2*floor((n+2)/4).
a(n) = (((n+2) mod 4) - (n mod 2))/2.
a(n) = ((n + 2 - (n mod 2))/2) mod 2.
a(n) = ((2*n + 3 + (-1)^n)/4) mod 2.
a(n) = (1 + (-1)^((2*n - 1 + (-1)^n)/4))/2.
a(n) = binomial(n+2, n) mod 2 = binomial(n+2, 2) mod 2.
a(n) = A000217(n+1) mod 2.
G.f.: (1+x)/(1-x^4) = 1/((1-x)(1+x^2)).
a(n) = 1/2 + (1/2)*cos(Pi*n/2) + (1/2)*sin(Pi*n/2). a(n) = A021913(n+2). - R. J. Mathar, Nov 15 2007
From Jaume Oliver Lafont, Dec 05 2008: (Start)
a(n) = 1/2 + sin((2n+1)Pi/4)/sqrt(2).
a(n) = 1/2 + cos((2n-1)Pi/4)/sqrt(2). (End)
a(n) = Re(Sum_{k=0..n} i^k), where i=sqrt(-1) and Re is the real part of a complex number. a(n) = (1/2)*((Sum_{k=0..n} i^k) + Sum_{k=0..n} i^-k) = Re((1/2)*(1 + i)*(1 - i^(n+1))). - Enrique Pérez Herrero, Aug 16 2009
a(n) = (1 + i^(n*(n-1)))/2, where i=sqrt(-1). - Bruno Berselli, May 18 2011
a(n) = (Sum_{k=1..n} k^j) mod 2, for any j. - Gary Detlefs, Dec 28 2011
a(n) = a(n-1) - a(n-2) + a(n-3) for n>2. - Jean-Christophe Hervé, May 01 2013
a(n) = 1 - floor(n/2) + 2*floor(n/4) = 1 - A004526(n) + A122461(n). - Wesley Ivan Hurt, Dec 06 2013
a(n) = (1 + (-1)^floor(n/2))/2. - Wesley Ivan Hurt, Apr 17 2014
a(n) = A054925(n+2) - A011848(n+2). - Wesley Ivan Hurt, Jun 09 2014
Euler transform of length 4 sequence [1, -1, 0, 1]. - Michael Somos, Sep 26 2014
a(n) = a(1-n) for all n in Z. - Michael Somos, Sep 26 2014
From Ilya Gutkovskiy, Jul 09 2016: (Start)
Inverse binomial transform of A038504(n+1).
E.g.f.: (exp(x) + sin(x) + cos(x))/2. (End)
a(n) = (1 + (-1)^(n*(n-1)/2))/2. - Guenther Schrack, Apr 04 2019

Definition rewritten by N. J. A. Sloane, Apr 30 2009

A133890 Binomial(n+10,n) mod 10.

Original entry on oeis.org

1, 1, 6, 6, 1, 3, 8, 8, 8, 8, 6, 6, 6, 6, 6, 0, 5, 5, 0, 0, 5, 5, 0, 0, 0, 6, 6, 6, 6, 6, 8, 8, 3, 3, 8, 6, 1, 1, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 6, 6, 1, 1, 6, 8, 8, 8, 8, 8, 6, 6, 6, 6, 1, 5, 0, 0, 5, 5, 0, 0, 0, 0, 0, 6, 6, 6, 6, 6, 3, 3, 8, 8, 3, 1, 6, 6, 6, 6, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 1, 1, 6, 6, 6

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length A133900(10)=4*10^2=400.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 231.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636, A133880, A133900, A133910.

Table[Mod[Binomial[10+n,n],10],{n,0,120}] (* Harvey P. Dale, Sep 19 2018 *)

a(n)=binomial(n+10,10)%10 \\ Charles R Greathouse IV, Nov 22 2011

a(n) = binomial(n+10,10) mod 10.

A133630 Nonprime numbers k such that binomial(k+p,k) mod k = 1, where p=10.

Original entry on oeis.org

4, 33, 57, 68, 85, 87, 111, 121, 141, 143, 164, 169, 185, 187, 209, 219, 221, 235, 247, 249, 253, 260, 289, 292, 299, 303, 319, 323, 327, 335, 341, 356, 361, 377, 381, 388, 391, 403, 407, 411, 435, 437, 451, 452, 473, 481, 484, 485, 489, 493, 516, 517, 519

Offset: 1

Hieronymus Fischer, Sep 30 2007

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636, A133880, A133890, A133900, A133910.

is(n)=binomial(n+10,10)%n==1 && !isprime(n) \\ Charles R Greathouse IV, Nov 04 2015

A133880 n modulo p repeated p times (where p=10).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length p^2=100.

a(n) = A179051(n) for n < 90. - Reinhard Zumkeller, Jun 27 2010

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

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

Cf. A133890, A133900, A133910.

Flatten[Table[PadRight[{},10,Mod[n,10]],{n,11}]] (* Harvey P. Dale, May 10 2012 *)

a(n)=(n\10+1)%10 \\ Charles R Greathouse IV, Nov 15 2022

The following formulas are given for a general parameter p (p=10 for this sequence).
a(n)=(1+floor(n/p)) mod p.
a(n)=1+floor(n/p)-p*floor((n+p)/p^2).
a(n)=(((n+p) mod p^2)-(n mod p))/p.
a(n)=((n+p-(n mod p))/p) mod p.
G.f. g(x)=((p-1)x^(p^2)-px^(p(p-1))+1)/((1-x)(1-x^p)(1-x^(p^2))).
G.f. g(x)=(1-x^p)*sum{0<=k<(p-1), (k+1)*x^(k*p)}/((1-x)(1-x^(p^2))).

A038509 Composite numbers congruent to +-1 mod 6.

Original entry on oeis.org

25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169, 175, 185, 187, 203, 205, 209, 215, 217, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, 355, 361, 365, 371, 377, 385

Offset: 1

Jeff Burch

Or, composite numbers with smallest prime factor >= 5.
Or, nonprime numbers n such that binomial(n+3, 3) mod n == 1. - Hieronymus Fischer, Sep 30 2007
Note that the primes > 3 are congruent to +-1 mod 6.
This sequence differs from A067793 (composite n such that phi(n) > 2n/3) starting at 385. Numbers in this sequence but not in A067793 are 385, 455, 595, 665, 805, 1015, 1085, 1925, 2275, 2695, etc. See A069043. - R. J. Mathar, Jun 08 2008 and Zak Seidov, Nov 02 2011
Intersection of A002808 and A007310. - Reinhard Zumkeller, Jun 30 2012
The product (24/25) * (36/35) * (48/49) * (54/55) * (66/65) * (78/77) * (84/85) * (90/91) * ... * ((6*k)/a(n)) * ... = Pi^2/(6*sqrt(3)), where 6*k is the nearest number to a(n), with k in A067611 but not in A002822. (See A258414.) - Dimitris Valianatos, Mar 27 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A171993 (nonprimes of the form 3*k+-1).
Cf. A000040, A133620-A133625, A133630, A133634-A133636.
Cf. A133873, A133883, A133880, A133890, A133900, A133910.
Cf. A069043, A067793 (composite n such that phi(n) > 2n/3).

a038509 n = a038509_list !! (n-1)
a038509_list = [x | x <- a002808_list, gcd x 6 == 1]
-- Reinhard Zumkeller, Aug 05 2014, Jun 30 2012

A038509 := proc(n)
    option remember;
    if n = 1 then
        25;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and modp(a,6) in {1,5} then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A038509(n),n=1..30) ; # R. J. Mathar, Feb 28 2020

Select[Range[1000], FactorInteger[#][[1,1]] >= 5 && ! PrimeQ[#] &] (* Robert G. Wilson v, Dec 19 2009 *)
With[{nn=400},Select[Rest[Complement[Range[nn],Prime[Range[ PrimePi[ nn]]]]], MemberQ[ {1,5},Mod[#,6]]&]] (* Harvey P. Dale, Feb 21 2013 *)
Select[Range[400],CompositeQ[#]&&MemberQ[{1,5},Mod[#,6]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2019 *)

is(n)=gcd(n,6)==1 && !isprime(n) && n>7 \\ Charles R Greathouse IV, Nov 20 2012

a(n) ~ 3n. - Charles R Greathouse IV, Nov 20 2012

More terms from Robert G. Wilson v, Dec 19 2009

Entry revised by N. J. A. Sloane, Dec 31 2011, at the suggestion of Gary Detlefs

A133634 Nonprime numbers k such that binomial(k+p,k) mod k = 1, where p=4.

Original entry on oeis.org

10, 25, 26, 34, 35, 49, 50, 55, 58, 65, 74, 77, 82, 85, 91, 95, 98, 106, 115, 119, 121, 122, 125, 130, 133, 143, 145, 146, 154, 155, 161, 169, 170, 175, 178, 185, 187, 194, 202, 203, 205, 209, 215, 217, 218, 221, 226, 235, 242, 245, 247, 250, 253, 259, 265, 266

Offset: 1

Hieronymus Fischer, Sep 30 2007

nonn

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

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

Cf. A133874, A133884, A133880, A133890, A133900, A133910.

Select[Range[300],!PrimeQ[#]&&Mod[Binomial[#+4,#],#]==1&] (* Harvey P. Dale, Oct 09 2011 *)

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

A133621 Numbers k such that binomial(k+p,k) mod k = 1, where p=10.

Original entry on oeis.org

3, 4, 11, 13, 17, 19, 23, 29, 31, 33, 37, 41, 43, 47, 53, 57, 59, 61, 67, 68, 71, 73, 79, 83, 85, 87, 89, 97, 101, 103, 107, 109, 111, 113, 121, 127, 131, 137, 139, 141, 143, 149, 151, 157, 163, 164, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 209, 211

Offset: 1

Hieronymus Fischer, Sep 30 2007

nonn

All primes q > p are included, in that binomial(q+p,q) == (1+floor(p/q)) == 1 (mod q) holds for those primes.

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

Cf. A133880, A133890, A133900, A133910.

isok(n) = ((binomial(n+10, n) % n) == 1) \\ Michel Marcus, Jul 15 2013

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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A133620 Binomial(n+p,n) mod n where p=10.

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A133872 Period 4: repeat [1, 1, 0, 0].

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A133890 Binomial(n+10,n) mod 10.

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A133630 Nonprime numbers k such that binomial(k+p,k) mod k = 1, where p=10.

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A133880 n modulo p repeated p times (where p=10).

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A038509 Composite numbers congruent to +-1 mod 6.

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