A387489 Number of packing 1X1X2 bricks into 2X2Xn boxes considering packings obtained by rigid motions equivalent.

1, 1, 2, 7, 26, 71, 258, 857, 3148, 11300, 41841, 154140, 573201, 2129726, 7935779, 29569762, 110281431, 411333271, 1534676318, 5726191937, 21367848168, 79738762725, 297573920356, 1110521036955, 4144432037026, 15467004104026, 57723125759179, 215424338586742, 803971544759711, 3000455162798396, 11197833423648453, 41790839930063492, 155965434740272813, 582070675232252525

Offset: 0

Author

R. J. Mathar, Aug 31 2025

Keywords

• nonn
• easy
• new

Comments

There seem to be several typos in Jepsen's equations. The enumeration here is derived from the expression of p(n) as 1/8ths of Psi(e)+2*Psi(rho)+Psi(rho^2)+2*Psi(sigma)+2*Psi(rho*sigma) if n>=3.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Charles H. Jepsen, Packing a box with bricks, Math. Mag. 64 (2) (1991) 92-97, Table 1.
• Index entries for linear recurrences with constant coefficients, signature (6,-4,-26,33,8,-8,24,-31,-14,12,2,-1).

Crossrefs

Cf. A109437 (is Jepsen's b(n)/4), A006253 (rigid motion symmetry ignored, Jepsen's a(n)).

Programs

Magma

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 +x +2*x^2 -x^3*(-7 +16*x +57*x^2 -118*x^3 -38*x^4 +30*x^5 -53*x^6 +127*x^7 +42*x^8 -49*x^9 -7*x^10 +4*x^11) / ( (x-1)*(1+x) *(x^2+2*x-1) *(x^2+1) *(x^2-4*x+1) *(x^4-4*x^2+1)) )); // Vincenzo Librandi, Sep 02 2025

Mathematica

CoefficientList[Series[1+x+2*x^2-x^3*(-7+16*x+57*x^2-118*x^3-38*x^4+30*x^5-53*x^6+127*x^7+42*x^8-49*x^9-7*x^10+4*x^11)/((x-1)*(1+x)*(x^2+2*x-1)*(x^2+1)*(x^2-4*x+1)*(x^4-4*x^2+1)),{x,0,33}],x] (* Vincenzo Librandi, Sep 02 2025 *)

Formula

G.f.: 1 +x +2*x^2 -x^3*(-7 +16*x +57*x^2 -118*x^3 -38*x^4 +30*x^5 -53*x^6 +127*x^7 +42*x^8 -49*x^9 -7*x^10 +4*x^11) / ( (x-1)*(1+x) *(x^2+2*x-1) *(x^2+1) *(x^2-4*x+1) *(x^4-4*x^2+1) ).