A133624 Binomial(n+p, n) mod n, where p=4.

0, 1, 2, 2, 1, 0, 1, 7, 4, 1, 1, 8, 1, 8, 6, 13, 1, 7, 1, 6, 8, 12, 1, 3, 1, 1, 10, 8, 1, 26, 1, 25, 12, 1, 1, 22, 1, 20, 14, 31, 1, 15, 1, 12, 16, 24, 1, 5, 1, 1, 18, 14, 1, 46, 1, 43, 20, 1, 1, 36, 1, 32, 22, 49, 1, 23, 1, 18, 24, 36, 1, 7, 1, 1, 26, 20, 1, 66, 1, 61, 28, 1, 1, 50, 1, 44, 30

Offset

1,3

Comments

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Formula

a(n) = binomial(n+4,4) mod n.

a(n)=1 if n is a prime > 4, since binomial(n+4,n) == (1+floor(4/n))(mod n), provided n is a prime.

From Chai Wah Wu, May 26 2016: (Start)

a(n) = (n^4 + 10*n^3 + 11*n^2 + 2*n + 24)/24 mod n.

For n > 6:

if n mod 24 == 0, then a(n) = n/12 + 1.

if n mod 24 is in {1, 2, 5, 7, 10, 11, 13, 17, 19, 23}, then a(n) = 1.

if n mod 24 is in {3, 9, 15, 18, 21}, then a(n) = n/3 + 1.

if n mod 24 is in {4, 20}, then a(n) = n/4 + 1.

if n mod 24 == 6, then a(n) = 5*n/6 + 1.

if n mod 24 is in {8, 16}, then a(n) = 3*n/4 + 1.

if n mod 24 == 12, then a(n) = 7*n/12 + 1.

if n mod 24 is in {14, 22}, then a(n) = n/2 + 1.

(End)

For n > 54, a(n) = 2*a(n-24) - a(n-48). - Ray Chandler, Apr 23 2023

Mathematica

Table[Mod[Binomial[n+4, n], n], {n, 90}] (* Harvey P. Dale, Apr 26 2014 *)

Prog

(Magma) [Binomial(n+4, 4) mod n: n in [1..100]]; // Vincenzo Librandi, Apr 27 2014

Crossrefs

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636, A133874, A133884, A133880, A133890, A133900, A133910, A362686, A362687, A362688, A362689.

Sequence in context: A339160 A355755 A141335 * A030110 A083570 A287407

Adjacent sequences: A133621 A133622 A133623 * A133625 A133626 A133627

Keyword

nonn

Author

Hieronymus Fischer, Sep 30 2007

Status

approved