This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A133623 #28 Feb 21 2025 22:39:35
%S A133623 0,0,2,3,1,0,1,5,4,6,1,11,1,8,6,9,1,16,1,11,8,12,1,21,1,14,10,15,1,26,
%T A133623 1,17,12,18,1,31,1,20,14,21,1,36,1,23,16,24,1,41,1,26,18,27,1,46,1,29,
%U A133623 20,30,1,51,1,32,22,33,1,56,1,35,24,36,1,61,1,38,26,39,1,66,1,41,28,42,1
%N A133623 Binomial(n+p, n) mod n where p=3.
%C A133623 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.
%C A133623 Appears to satisfy the recurrence: a(n) = -2*a(n-1) - a(n-2) + 2*a(n-3) + 4*a(n-4) + 2*a(n-5) - a(n-6) - 2*a(n-7) - a(n-8) for n > 14. - _Chai Wah Wu_, May 25 2016
%H A133623 G. C. Greubel, <a href="/A133623/b133623.txt">Table of n, a(n) for n = 1..1000</a>
%H A133623 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-1,2,4,2,-1,-2,-1).
%F A133623 a(n) = binomial(n+3,3) mod n.
%F A133623 a(n)=1 if n is a prime > 3, since binomial(n+3,n)==(1+floor(3/n))(mod n), provided n is a prime.
%F A133623 From _Chai Wah Wu_, May 26 2016: (Start)
%F A133623 a(n) = (n^3 + 5*n + 6)/6 mod n.
%F A133623 For n > 6:
%F A133623 if n mod 6 == 0, then a(n) = 5*n/6 + 1.
%F A133623 if n mod 6 is in {1, 5}, then a(n) = 1.
%F A133623 if n mod 6 is in {2, 4}, then a(n) = n/2 + 1.
%F A133623 if n mod 6 == 3, then a(n) = n/3 + 1.
%F A133623 (End)
%t A133623 Table[Mod[Binomial[n+3,n],n],{n,90}] (* _Harvey P. Dale_, Nov 22 2011 *)
%Y A133623 Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
%Y A133623 Cf. A133873, A133883, A133880, A133890, A133900, A133910.
%K A133623 nonn,easy
%O A133623 1,3
%A A133623 _Hieronymus Fischer_, Sep 30 2007