A160175 Expansion of 1/(1 - 2*x - 2*x^2 - 2*x^3 - 2*x^4).

1, 2, 6, 18, 54, 160, 476, 1416, 4212, 12528, 37264, 110840, 329688, 980640, 2916864, 8676064, 25806512, 76760160, 228319200, 679123872, 2020019488, 6008445440, 17871816000, 53158809600, 158118181056, 470314504192, 1398926621696, 4161036233088

Offset: 0

Author

Geoffrey Critzer, May 03 2009, May 06 2009

Keywords

• nonn

Comments

a(n) is the number of ways two opposing baseball teams could score a combined total of n runs (tallying the score just prior to each "batter up!") considering the order of the scoring as important. Equivalently, a(n) is the number of 2-colored tilings of an n-board with tiles of length at most 4.

a(n) is the number of compositions (ordered partitions) of n into parts <= 4 with 2 sorts of each part. - Joerg Arndt, Aug 06 2019

References

• Arthur Benjamin and Jennifer Quinn, Proofs that Really Count, Mathematical Association of America, 2003, p. 36.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,2,2).

Crossrefs

Cf. A077835, A002605, A000079.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-2*x-2*x^2-2*x^3-2*x^4))); // G. C. Greubel, Sep 24 2018

Mathematica

RecurrenceTable[{a[n] == 2(a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4]), a[0] == 1, a[1] == 2, a[2] == 6, a[3] == 18}, a, {n, 0, 20}]
LinearRecurrence[{2,2,2,2},{1,2,6,18},30] (* Harvey P. Dale, Oct 27 2013 *)
CoefficientList[Series[1/(1-2*x-2*x^2-2*x^3-2*x^4), {x, 0, 50}], x] (* G. C. Greubel, Sep 24 2018 *)

PARI

x='x+O('x^30); Vec(1/(1-2*x-2*x^2-2*x^3-2*x^4)) \\ G. C. Greubel, Sep 24 2018

Formula

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

Extensions

More terms from Harvey P. Dale, Oct 27 2013