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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
OFFSET
1,2
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.
Sequence in context: A186293 A158865 A139570 * A004118 A082231 A318339
KEYWORD
nonn,easy
AUTHOR
Marian Kraus, Dec 04 2015
STATUS
approved