

A247903


Start with a single square; at nth generation add a square at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at nth generation. (See comment for construction rules.)


7



1, 5, 13, 29, 53, 93, 149, 237, 357, 541, 789, 1165, 1669, 2429, 3445, 4973, 7013, 10077, 14165, 20301, 28485, 40765, 57141, 81709, 114469, 163613, 229141, 327437, 458501, 655101, 917237, 1310445, 1834725, 2621149, 3669717, 5242573, 7339717, 10485437
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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 nth 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 nonoverlapping 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 nth generation. The squares count is A001844. See illustration ("vertex to side" is equal to "side to vertex"). For n >= 1, (a(n)  a(n1))/4 is A027383.


LINKS



FORMULA

a(0) = 1, for n >= 1, a(n) = 4*A027383(n) + a(n1).
a(n) = 2*a(n1) +a(n2) 4*a(n3) +2*a(n4).  Colin Barker, Sep 26 2014
G.f.: (1+3*x+2*x^2+2*x^3)/((1x)^2*(12*x^2)).  Colin Barker, Sep 26 2014
A(n) = 2^(n/2+1)*((1+sqrt(2))^3 + (1)^n*(1sqrt(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) / ((x1)^2*(2*x^21)) + O(x^100)) \\ Colin Barker, Sep 26 2014
(Magma) [2^(n/2+1)*((7+5*Sqrt(2)) + (1)^n*(75*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*(75*sqrt(2))) (8*n+27) for n in (0..50)] # G. C. Greubel, Feb 18 2022


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



