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

%I A133624 #29 Apr 30 2023 13:13:36
%S A133624 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,
%T A133624 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,
%U A133624 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
%N A133624 Binomial(n+p, n) mod n, where p=4.
%C A133624 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.
%H A133624 Vincenzo Librandi, <a href="/A133624/b133624.txt">Table of n, a(n) for n = 1..1000</a>
%H A133624 <a href="/index/Rec#order_48">Index entries for linear recurrences with constant coefficients</a>, 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).
%F A133624 a(n) = binomial(n+4,4) mod n.
%F A133624 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.
%F A133624 From _Chai Wah Wu_, May 26 2016: (Start)
%F A133624 a(n) = (n^4 + 10*n^3 + 11*n^2 + 2*n + 24)/24 mod n.
%F A133624 For n > 6:
%F A133624 if n mod 24 == 0, then a(n) = n/12 + 1.
%F A133624 if n mod 24 is in {1, 2, 5, 7, 10, 11, 13, 17, 19, 23}, then a(n) = 1.
%F A133624 if n mod 24 is in {3, 9, 15, 18, 21}, then a(n) = n/3 + 1.
%F A133624 if n mod 24 is in {4, 20}, then a(n) = n/4 + 1.
%F A133624 if n mod 24 == 6, then a(n) = 5*n/6 + 1.
%F A133624 if n mod 24 is in {8, 16}, then a(n) = 3*n/4 + 1.
%F A133624 if n mod 24 == 12, then a(n) = 7*n/12 + 1.
%F A133624 if n mod 24 is in {14, 22}, then a(n) = n/2 + 1.
%F A133624 (End)
%F A133624 For n > 54, a(n) = 2*a(n-24) - a(n-48). - _Ray Chandler_, Apr 23 2023
%t A133624 Table[Mod[Binomial[n+4,n],n],{n,90}] (* _Harvey P. Dale_, Apr 26 2014 *)
%o A133624 (Magma) [Binomial(n+4,4) mod n: n in [1..100]]; // _Vincenzo Librandi_, Apr 27 2014
%Y A133624 Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636, A133874, A133884, A133880, A133890, A133900, A133910, A362686, A362687, A362688, A362689.
%K A133624 nonn
%O A133624 1,3
%A A133624 _Hieronymus Fischer_, Sep 30 2007