A398028 Number of 7-convex polyominoes with semiperimeter n.

0, 0, 1, 2, 7, 28, 120, 528, 2344, 10416, 46160, 203676, 894228, 3906036, 16976952, 73439268, 316282360, 1356560728, 5796404302, 24681075816, 104755646898, 443313571324, 1870972607370, 7876620943924, 33083632385428, 138664016164056, 580044174368274, 2421965007900888, 10095807143173876

Offset

0,4

Links

Table of n, a(n) for n=0..28.

Andrew R. Conway and Anthony J. Guttmann, k-Convex Polyominoes by Semi-perimeter, arXiv:2606.13845 [cs.DM], Jul 17 2026.

Formula

G.f.: x^2*(24*x^10 - 198*x^9 + 867*x^8 - 2476*x^7 + 4072*x^6 - 3896*x^5 + 2251*x^4 - 798*x^3 + 170*x^2 - 20*x + 1) / ((1-2*x) * (1-4*x)^2 * (2*x^2 - 4*x + 1) * (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)) - x^4*(7*x^3 - 14*x^2 + 7*x - 1)^2 / ((1-4*x)^(3/2) * (1-2*x) * (2*x^2 - 4*x + 1) * (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)) [Conway and Guttmann, 2026].

For asymptotics see Conway and Guttmann (2026).

Mathematica

CoefficientList[Series[(-2*x^4 + x^2*(1 - 6*x + 7*x^2 + 4*x^3) * ChebyshevU[k, (1-2*x)/(2*x)] + x^3*(1 - 4*x + 8*x^2) * ChebyshevU[k-1, (1-2*x)/(2*x)]) / ((1-4*x)^2 * ChebyshevU[k, (1-2*x)/(2*x)]) - 4*x^4 * (ChebyshevT[k, 1/(2*Sqrt[x])]^2/((1 - 4*x)^(3/2) * ChebyshevU[k, (1-2*x)/(2*x)])) /. k->7, {x, 0, 28}], x]

Crossrefs

Cf. A128611, A398024, A398025, A398026, A398027.

Sequence in context: A398025 A398026 A398027 * A005436 A150656 A150657

Adjacent sequences: A398025 A398026 A398027 * A398029 A398031 A398036

Keyword

nonn,new

Author

Vaclav Kotesovec, Jul 17 2026

Status

approved