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 A265207 #37 Dec 07 2015 10:56:28
%S A265207 1,8,20,36,60,92,140,204,300,428,620,876,1260,1772,2540,3564,5100,
%T A265207 7148,10220,14316,20460,28652,40940,57324,81900,114668,163820,229356,
%U A265207 327660,458732,655340,917484,1310700,1834988,2621420,3669996,5242860,7340012,10485740,14680044,20971500,29360108,41943020,58720236
%N A265207 Draw a square and follow these steps: Take a square and place at its edges isosceles right triangles with the edge as hypotenuse. Draw a square at every new edge of the triangles. Repeat for all the new squares of the same size. New figures are only placed on empty space. The structure is symmetric about the first square. The sequence gives the numbers of squares of equal size in successive rings around the center.
%H A265207 Marian Kraus, <a href="/A265207/a265207_1.pdf">Illustration for a(4)</a>
%F A265207 Conjectured recurrence:
%F A265207 a(0)=1,
%F A265207 a(1)=8,
%F A265207 a(2)=20, and thereafter
%F A265207 a(n)=2*a(n-2)+20.
%F A265207 Conjectured formula: ("[]" is the floor function)
%F A265207 a(n)=4*sum_{k=1}^{[(n+1)/2]}(2^k)+6*sum_{k=1}^{[n/2]}(2^k).
%F A265207 Conjectures from _Colin Barker_, Dec 07 2015: (Start)
%F A265207 a(n) = (-20+2^(1/2*(-1+n))*(10-10*(-1)^n+7*sqrt(2)+7*(-1)^n*sqrt(2))) for n>1.
%F A265207 a(n) = 5*2^(n/2+1/2)-5*(-1)^n*2^(n/2+1/2)+7*2^(n/2)+7*(-1)^n*2^(n/2)-20 for n>1.
%F A265207 a(n) = a(n-1)+2*a(n-2)-2*a(n-3) for n>4.
%F A265207 G.f.: x*(1+7*x+10*x^2+2*x^3) / ((1-x)*(1-2*x^2)).
%F A265207 (End)
%e A265207 By recursion:
%e A265207 a(3)=2*a(1)+20=2*8+20=36
%e A265207 a(4)=2*a(2)+20=2*20+20=60
%e A265207 By function:
%e A265207 a(3)=4*sum_{k=1}^{[(3+1)/2]}(2^k)+6*sum_{k=1}^{[3/2]}(2^k)
%e A265207 =4*sum_{k=1}^{[2]}(2^k)+6*sum_{k=1}^{[1.5]}(2^k)
%e A265207 =4*sum_{k=1}^{2}(2^k)+6*sum_{k=1}^{1}(2^k)
%e A265207 =4*(2^1+2^2)+6*(2^1)
%e A265207 =4*(2+4)+6*(2)=24+12=36
%e A265207 a(4)=4*sum_{k=1}^{[(4+1)/2]}(2^k)+6*sum_{k=1}^{[4/2]}(2^k)
%e A265207 =4*sum_{k=1}^{[2.5]}(2^k)+6*sum_{k=1}^{[2]}(2^k)
%e A265207 =4*sum_{k=1}^{2}(2^k)+6*sum_{k=1}^{2}(2^k)
%e A265207 =4*(2^1+2^2)+6*(2^1+2^2)
%e A265207 =4*(2+4)+6*(2+4)=24+36=60
%o A265207 (R)
%o A265207 rm(a)
%o A265207 a <- vector() powerof2 <- vector()
%o A265207 x <- 300
%o A265207 n <- x/2
%o A265207 for (i in 1:x){
%o A265207 powerof2[i] <- 2^(i-1)}
%o A265207 powerof2 for (i in 1:n){
%o A265207 a[2*i] <- 8*(sum(powerof2[1:i]))+12*(sum(powerof2[1:i]))}
%o A265207 for (i in 1:(n+1)){
%o A265207 a[2*i+1] <- 8*(sum(powerof2[1:(i+1)]))+12*(sum(powerof2[1:i]))}
%o A265207 a[1]<-8
%o A265207 a
%Y A265207 For the differences (a(n)-a(n-1))/4, n>2, see A163978.
%Y A265207 Cf. A029744, A063759, A164090.
%K A265207 nonn,easy
%O A265207 1,2
%A A265207 _Marian Kraus_, Dec 04 2015