OFFSET
0,2
COMMENTS
Refer to A247618, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e."vertex to side" expansion version. Let us assign the label "1" to the square at the origin; at n-th generation add a square at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping squares will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The squares count is A001844. See illustration ("vertex to side" is equal to "side to vertex"). For n >= 1, (a(n) - a(n-1))/4 is A027383.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms (vertex to side)
Kival Ngaokrajang, Illustration of initial terms (side to vertex)
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).
FORMULA
a(0) = 1, for n >= 1, a(n) = 4*A027383(n) + a(n-1).
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +2*a(n-4). - Colin Barker, Sep 26 2014
G.f.: (1+3*x+2*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - Colin Barker, Sep 26 2014
A(n) = 2^(n/2+1)*((1+sqrt(2))^3 + (-1)^n*(1-sqrt(2))^3) - (8*n + 27). - G. C. Greubel, Feb 18 2022
MATHEMATICA
LinearRecurrence[{2, 1, -4, 2}, {1, 5, 13, 29}, 51] (* G. C. Greubel, Feb 18 2022 *)
PROG
(PARI)
{
b=0; a=1; print1(1, ", ");
for (n=0, 50,
b=b+2^floor(n/2);
a=a+4*b;
print1(a, ", ")
)
}
(PARI)
Vec(-(2*x^3+2*x^2+3*x+1) / ((x-1)^2*(2*x^2-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014
(Magma) [2^(n/2+1)*((7+5*Sqrt(2)) + (-1)^n*(7-5*Sqrt(2))) -(8*n+27): n in [0..50]]; // G. C. Greubel, Feb 18 2022
(Sage) [2*2^(n/2)*((7+5*sqrt(2)) +(-1)^n*(7-5*sqrt(2))) -(8*n+27) for n in (0..50)] # G. C. Greubel, Feb 18 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Sep 26 2014
EXTENSIONS
More terms from Colin Barker, Sep 26 2014
STATUS
approved