A031940 Length of longest legal domino snake using full set of dominoes up to [n:n].

1, 3, 6, 9, 15, 19, 28, 33, 45, 51, 66, 73, 91, 99, 120, 129, 153, 163, 190, 201, 231, 243, 276, 289, 325, 339, 378, 393, 435, 451, 496, 513, 561, 579, 630, 649, 703, 723, 780, 801, 861, 883, 946, 969, 1035, 1059, 1128, 1153, 1225, 1251, 1326, 1353, 1431, 1459

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Index entries for sequences related to dominoes

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

C(n, 2) + n if n odd, C(n, 2) + n/2 + 1 if n even. - T. D. Noe, Nov 09 2006

a(n) = A204556(n+1) / (n+1). - Reinhard Zumkeller, Jan 18 2012

G.f.: -x*(1+2*x+x^2-x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 13 2012

a(n) = ((-1)^n*(2 - n) + (2 + n + 2*n^2))/4. - G. C. Greubel, Jun 15 2018

Example

E.g., for n=4 [ 1:1 ][ 1:2 ][ 2:2 ][ 2:3 ][ 3:3 ][ 3:1 ][ 1:4 ][ 4:4 ][ 4:2 ].

Mathematica

Rest[CoefficientList[Series[x*(1 + 2*x + x^2 - x^3 + x^4)/((1 + x)^2*(1 - x)^3), {x, 0, 50}], x]] (* or *) Table[((-1)^n*(2-n) + (2+n+2*n^2))/4, {n, 1, 50}] (* G. C. Greubel, Jun 15 2018 *)

Prog

(PARI) for(n=1, 60, print1(((-1)^n*(2 - n) + (2 + n + 2*n^2))/4, ", ")) \\ G. C. Greubel, Jun 15 2018
(PARI) Vec(-x*(1+2*x+x^2-x^3+x^4) / ( (1+x)^2*(x-1)^3 ) + O(x^60)) \\ Felix Fröhlich, Jun 18 2018
(Magma) [((-1)^n*(2 - n) + (2 + n + 2*n^2))/4: n in [1..60]]; // G. C. Greubel, Jun 15 2018

Crossrefs

Cf. A031878, A204556.

Sequence in context: A049991 A368611 A143981 * A007187 A337502 A082004

Adjacent sequences: A031937 A031938 A031939 * A031941 A031942 A031943

Keyword

nonn

Author

Colin Mallows

Extensions

Corrected by T. D. Noe, Nov 09 2006

More terms from Felix Fröhlich, Jun 18 2018

Status

approved