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 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. 1
 1, 8, 20, 36, 60, 92, 140, 204, 300, 428, 620, 876, 1260, 1772, 2540, 3564, 5100, 7148, 10220, 14316, 20460, 28652, 40940, 57324, 81900, 114668, 163820, 229356, 327660, 458732, 655340, 917484, 1310700, 1834988, 2621420, 3669996, 5242860, 7340012, 10485740, 14680044, 20971500, 29360108, 41943020, 58720236 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Marian Kraus, Illustration for a(4) FORMULA Conjectured recurrence: a(0)=1, a(1)=8, a(2)=20, and thereafter a(n)=2*a(n-2)+20. Conjectured formula: ("[]" is the floor function) a(n)=4*sum_{k=1}^{[(n+1)/2]}(2^k)+6*sum_{k=1}^{[n/2]}(2^k). Conjectures from Colin Barker, Dec 07 2015: (Start) a(n) = (-20+2^(1/2*(-1+n))*(10-10*(-1)^n+7*sqrt(2)+7*(-1)^n*sqrt(2))) for n>1. 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. a(n) = a(n-1)+2*a(n-2)-2*a(n-3) for n>4. G.f.: x*(1+7*x+10*x^2+2*x^3) / ((1-x)*(1-2*x^2)). (End) EXAMPLE By recursion: a(3)=2*a(1)+20=2*8+20=36 a(4)=2*a(2)+20=2*20+20=60 By function: a(3)=4*sum_{k=1}^{[(3+1)/2]}(2^k)+6*sum_{k=1}^{[3/2]}(2^k) =4*sum_{k=1}^{[2]}(2^k)+6*sum_{k=1}^{[1.5]}(2^k) =4*sum_{k=1}^{2}(2^k)+6*sum_{k=1}^{1}(2^k) =4*(2^1+2^2)+6*(2^1) =4*(2+4)+6*(2)=24+12=36 a(4)=4*sum_{k=1}^{[(4+1)/2]}(2^k)+6*sum_{k=1}^{[4/2]}(2^k) =4*sum_{k=1}^{[2.5]}(2^k)+6*sum_{k=1}^{[2]}(2^k) =4*sum_{k=1}^{2}(2^k)+6*sum_{k=1}^{2}(2^k) =4*(2^1+2^2)+6*(2^1+2^2) =4*(2+4)+6*(2+4)=24+36=60 PROG (R) rm(a) a <- vector() powerof2 <- vector() x <- 300 n <- x/2 for (i in 1:x){ powerof2[i] <- 2^(i-1)} powerof2 for (i in 1:n){ a[2*i] <- 8*(sum(powerof2[1:i]))+12*(sum(powerof2[1:i]))} for (i in 1:(n+1)){ a[2*i+1] <- 8*(sum(powerof2[1:(i+1)]))+12*(sum(powerof2[1:i]))} a[1]<-8 a CROSSREFS For the differences (a(n)-a(n-1))/4, n>2, see A163978. Cf. A029744, A063759, A164090. Sequence in context: A186293 A158865 A139570 * A004118 A082231 A318339 Adjacent sequences: A265204 A265205 A265206 * A265208 A265209 A265210 KEYWORD nonn,easy AUTHOR Marian Kraus, Dec 04 2015 STATUS approved

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Last modified December 8 12:03 EST 2022. Contains 358693 sequences. (Running on oeis4.)