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 A130909 #21 Sep 22 2024 02:35:41
%S A130909 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,
%T A130909 13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,
%U A130909 10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9
%N A130909 Simple periodic sequence (n mod 16).
%C A130909 The value of the rightmost digit in the base-16 representation of n. Also, the equivalent value of the two rightmost digits in the base-4 representation of n. Also, the equivalent value of the four rightmost digits in the base-2 representation of n.
%H A130909 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).
%F A130909 a(n) = n mod 16 = n-16*floor(n/16).
%F A130909 G.f.: g(x) = (Sum_{k=1..15} k*x^k)/(1-x^16).
%F A130909 G.f.: g(x) = x(15x^16-16x^15+1)/((1-x^16)(1-x)^2).
%F A130909 a(n) = A000035(n) + 2*A010877(A004526(n)).
%F A130909 a(n) = A010873(n) + 4*A010873(A002265(n)).
%F A130909 a(n) = A010877(n) + 8*A000035(floor(n/8)).
%F A130909 a(n) = (1/2)*(15 - ( - 1)^n - 2*( - 1)^(b/4) - 4*( - 1)^((b - 2 + 2*( - 1)^(b/4))/8) - 8*( - 1)^((b - 6 + ( - 1)^n + 2*( - 1)^(b/4) + 4*( - 1)^((b - 2 + 2*( - 1)^(b/4))/8))/16)) where b = 2n - 1 + ( - 1)^n.
%F A130909 a(n) = n mod 2+2*(floor(n/2)mod 2)+4*(floor(n/4)mod 2)+8*(floor(n/8)mod 2).
%F A130909 a(n) = (1/2)*(15-(-1)^n-2*(-1)^floor(n/2)-4*(-1)^floor(n/4)-8*(-1)^floor(n/= 8)).
%F A130909 Complex representation: a(n) = (1/16)*(1-r^n)*sum{1<=k<16, k*product{1<=m<16,m<>k, (1-r^(n-m))}} where r=exp(Pi/8*i)=(sqrt(2+sqrt(2))+i*sqrt(2-sqrt(2)))/2 and i=sqrt(-1).
%F A130909 Trigonometric representation: a(n) = 2^22*(sin(n*Pi/16))^2*sum{1<=k<16, k*product{1<=m<16,m<>k, (sin((n-m)*Pi/16))^2}}.
%F A130909 a(n) = (1/2)*(15-(-1)^p(0,n)-2*(-1)^p(1,n)-4*(-1)^p(2,n)-8*(-1)^p(3,n)) where p(k,n) is defined recursively by p(0,n)=n, p(k,n)=1/4*(2*p(k-1,n)-1+(-1)^p(k-1,n)).
%o A130909 (PARI) a(n)=n%16 \\ _Charles R Greathouse IV_, Jul 13 2016
%o A130909 (Python)
%o A130909 def A130909(n): return n&15 # _Chai Wah Wu_, Jan 18 2023
%Y A130909 Cf. partial sums A130910. Other related sequences A010872, A010873, A130481, A130482, A130483, A130486.
%Y A130909 See A010877 for a general formula in terms of powers of -1 (for period 2^k).
%K A130909 nonn,easy
%O A130909 0,3
%A A130909 _Hieronymus Fischer_, Jun 11 2007