A068620 Number of strings over Z_4 of length n with trace 0 and subtrace 0.

1, 2, 4, 8, 56, 272, 1184, 4736, 17536, 65792, 254464, 1015808, 4130816, 16781312, 67641344, 270565376, 1077968896, 4295032832, 17146445824, 68585259008, 274609995776, 1099512676352, 4400196091904, 17600784367616, 70385932435456, 281474993487872

Offset

1,2

Comments

a(n; 0,0) where a(n; t,s) is the number of length n 4-ary strings whose digits sum to t mod 4 and whose sum of products of all pairs of digits sum to s mod 4.

Links

Table of n, a(n) for n=1..26.

Max Alekseyev, PARI/GP scripts for miscellaneous math problems

F. Ruskey, Strings over Z_4 with given trace and subtrace

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+3, s+3t+1) + a(n-1; t+2, s+2t) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.

Empirical g.f.: -x*(704*x^7-704*x^6+288*x^5-56*x^4+32*x^3-24*x^2+8*x-1) / ((2*x-1)*(4*x-1)*(8*x^2-4*x+1)*(16*x^4+1)). - Colin Barker, Dec 06 2014

Example

a(3;0,0)=4 since the four 4-ary strings of trace 0, subtrace 0 and length 3 are { 000, 022, 202, 220 }.

Crossrefs

Cf. A068711, A068774, A068777, A068786, A068778, A068787, A068788, A068789, A068790.

Sequence in context: A289324 A338302 A359877 * A073953 A206850 A094333

Adjacent sequences: A068617 A068618 A068619 * A068621 A068622 A068623

Keyword

easy,nonn

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 19 2007

Terms a(11) onward from Max Alekseyev, Apr 14 2013

Status

approved