A247905 Start with a single hexagon; at n-th generation add a hexagon 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, 7, 19, 43, 79, 139, 223, 355, 535, 811, 1183, 1747, 2503, 3643, 5167, 7459, 10519, 15115, 21247, 30451, 42727, 61147, 85711, 122563, 171703, 245419, 343711, 491155, 687751, 982651, 1375855, 1965667, 2752087, 3931723, 5504575, 7863859, 11009575, 15728155, 22019599, 31456771, 44039671, 62914027, 88079839, 125828563, 176160199, 251657659

Offset: 0

Author

Kival Ngaokrajang, Sep 26 2014

Keywords

• nonn
• easy

Comments

Refer to A247620, 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" the hexagon at the origin; at n-th generation add a hexagon 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 hexagons 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 hexagons count is A003215. See illustration. For n >= 1, (a(n) - a(n-1))/6 is A027383.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Crossrefs

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

Cf. Vertex to side version: A101946, A247903, A247904.

Cf. A003215, A027383.

Programs

Magma

[3*2^(n/2)*((7+5*Sqrt(2)) + (-1)^n*(7-5*Sqrt(2))) -(12*n+41): n in [0..50]]; // G. C. Greubel, Feb 17 2022

Mathematica

LinearRecurrence[{2,1,-4,2}, {1,7,19,43}, 50] (* G. C. Greubel, Feb 17 2022 *)

PARI

{
b=0; a=1; print1(1, ", ");
for (n=0, 50,
     b=b+2^floor(n/2);
     a=a+6*b;
     print1(a, ", ")
    )
}

PARI

Vec(-(2*x^3+4*x^2+5*x+1)/((x-1)^2*(2*x^2-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

Sage

[3*2^(n/2)*((7+5*sqrt(2)) + (-1)^n*(7-5*sqrt(2))) -(12*n+41) for n in (0..50)] # G. C. Greubel, Feb 17 2022

Formula

a(0) = 1, for n >= 1, a(n) = 6*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+5*x+4*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - Colin Barker, Sep 26 2014

a(n) = 3*2^(n/2)*((1+sqrt(2))^3 + (-1)^n*(1-sqrt(2))^3) -12*n - 41. - G. C. Greubel, Feb 18 2022