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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

Table of n, a(n) for n=1..44.

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 June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)