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 A289199 #33 Jul 04 2017 21:21:57
%S A289199 0,0,2,2,3,5,5,6,7,7,9,10,10,12,12,14,14,15,17,17,18,19,19,21,22,22,
%T A289199 24,24,26,26,27,29,29,30,31,31,33,34,34,36,36,38,38,39,41,41,42,43,43,
%U A289199 45,46,46,48,48,50,50,51,53,53,54,55,55,57,58,58,60,60,62,62,63,65
%N A289199 a(n) is the number of odd integers divisible by 13 in the open interval (12*(n-1)^2, 12*n^2).
%C A289199 This sequence has the form (0+12k, 0+12k, 2+12k, 2+12k, 3+12k, 5+12k, 5+12k, 6+12k, 7+12k, 7+12k, 9+12k, 10+12k, 10+12k) for k >= 0.
%C A289199 Theorems: A) Generally for an interval (2*m*(n-1)^2,2*m*n^2) and a divisor d with 2*m < d there is a unique d-length form (e_i+2*m*k)_{i=0..d-1, k>=0} with e_i in [0,2*m]; here m = 6, d = 13.
%C A289199 B) Sum_{i=0..d-1}e_i = m*(d-2); here 66 = 6*(13-2).
%C A289199 Proof:
%C A289199 A) In d consecutive intervals
%C A289199 (2*m*(n-1)^2,2*m*(n+2)^2) there are m*d*(2*k+d) consecutive odd numbers and therefore m*(2*k+d) multiples of d where k=floor((n-1)/d).
%C A289199 B) With initial value a(0)=0 we have a(d)=2*m and thus Sum_{i=0..d-1} e_i = Sum_{i=1..d}a(i)-a(d) = m(2*0+d)-2*m = m*(d-2). Q.E.D.
%H A289199 Colin Barker, <a href="/A289199/b289199.txt">Table of n, a(n) for n = 0..1000</a>
%H A289199 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,1,-1).
%F A289199 a(n + 13*k) = a(n) + 12*k.
%F A289199 a(n) = 12n/13 + O(1). - _Charles R Greathouse IV_, Jun 29 2017
%F A289199 From _Colin Barker_, Jul 03 2017: (Start)
%F A289199 G.f.: x^2*(1 + x)*(1 - x + x^2)*(2 + x^2 + x^6 + 2*x^8) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)).
%F A289199 a(n) = a(n-1) + a(n-13) - a(n-14) for n>13.
%F A289199 (End)
%t A289199 Table[Count[Mod[Table[2(6(n-1)^2 +k)-1,{k,12 n-6}],13],0],{n,0,70}]
%o A289199 (PARI) a(n)=(12*n^2+12)\26 - (12*n^2-24*n+25)\26 \\ _Charles R Greathouse IV_, Jun 29 2017
%o A289199 (PARI) concat(vector(2), Vec(x^2*(1 + x)*(1 - x + x^2)*(2 + x^2 + x^6 + 2*x^8) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)) + O(x^100))) \\ _Colin Barker_, Jul 03 2017
%Y A289199 Cf. A004523, A288156, A289120, A289122, A289133, A289139, A289195.
%K A289199 nonn,easy
%O A289199 0,3
%A A289199 _Ralf Steiner_, Jun 28 2017