A249852 a(n) is the total number of pentagons on the left or the right of the vertical symmetry axis of a pentagon expansion (vertex to vertex) after n iterations.

0, 2, 7, 14, 23, 35, 50, 67, 86, 108, 133, 160, 189, 221, 256, 293, 332, 374, 419, 466, 515, 567, 622, 679, 738, 800, 865, 932, 1001, 1073, 1148, 1225, 1304, 1386, 1471, 1558, 1647, 1739, 1834, 1931, 2030, 2132, 2237, 2344, 2453, 2565, 2680, 2797, 2916, 3038, 3163, 3290, 3419

Offset: 0

Author

Kival Ngaokrajang, Mar 07 2015

Keywords

• nonn

Comments

a(n) is also total number of pentagrams on the left or the right of the vertical symmetry axis of a pentagram expansion (vertex to vertex) after n iterations.

The total pentagons (or pentagrams) after n iterations is A005891. See illustration in the links.

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A005891, A247619.

Programs

PARI

{a=2; s=2; d=2; print1(0,", ",s,", "); for(n=2, 100, if(Mod(n,4)==3, d=2, if(Mod(n,4)==4, d=2, d=3)); a=a+d; s=s+a; print1(s, ", ");)}

Formula

Conjectures from Colin Barker, Mar 07 2015: (Start)

a(n) = 3*a(n-1)-4*a(n-2)+4*a(n-3)-3*a(n-4)+a(n-5).

G.f.: -x*(x^3+x^2+x+2) / ((x-1)^3*(x^2+1)). (End)

Conjectured e.g.f.: (exp(x)*(1 + 8*x + 5*x^2) - cos(x) - sin(x))/4. - Stefano Spezia, Feb 23 2025