A247903 Start with a single square; at n-th 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 n-th generation. (See comment for construction rules.)

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

Offset: 0

Author

Kival Ngaokrajang, Sep 26 2014

Keywords

• nonn
• easy

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).

Crossrefs

Vertex to vertex version: A061777, A247618, A247619, A247620.

Vertex to side version: A101946, A247904, A247905.

Cf. A001844, A027383.

Programs

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

Mathematica

LinearRecurrence[{2,1,-4,2}, {1,5,13,29}, 51] (* G. C. Greubel, Feb 18 2022 *)

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

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

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

Extensions

More terms from Colin Barker, Sep 26 2014