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A250167 T(n,k)=Number of length n+1 0..k arrays with the sum of adjacent differences multiplied by some arrangement of +-1 equal to zero.

%I A250167 #8 Jul 23 2025 12:14:50
%S A250167 2,3,4,4,11,8,5,20,37,16,6,33,96,119,32,7,48,211,436,373,64,8,67,380,
%T A250167 1269,1880,1151,128,9,88,639,2860,7109,7836,3517,256,10,113,976,5831,
%U A250167 19896,37881,32032,10679,512,11,140,1437,10460,49037,129648,195927
%N A250167 T(n,k)=Number of length n+1 0..k arrays with the sum of adjacent differences multiplied by some arrangement of +-1 equal to zero.
%C A250167 Table starts
%C A250167 ....2.....3.......4........5.........6..........7..........8...........9
%C A250167 ....4....11......20.......33........48.........67.........88.........113
%C A250167 ....8....37......96......211.......380........639........976........1437
%C A250167 ...16...119.....436.....1269......2860.......5831......10460.......17765
%C A250167 ...32...373....1880.....7109.....19896......49037.....103556......203615
%C A250167 ...64..1151....7836....37881....129648.....380939.....938128.....2121089
%C A250167 ..128..3517...32032...195927....810964....2810751....7989940....20567199
%C A250167 ..256.10679..129572...996933...4962056...20169871...65768448...191480917
%C A250167 ..512.32293..521256..5029417..30034672..142786013..532548628..1748028901
%C A250167 .1024.97391.2091052.25262121.180893724.1004527983.4281269376.15822382297
%H A250167 R. H. Hardin, <a href="/A250167/b250167.txt">Table of n, a(n) for n = 1..263</a>
%F A250167 Empirical for column k:
%F A250167 k=1: a(n) = 2*a(n-1)
%F A250167 k=2: a(n) = 5*a(n-1) -6*a(n-2)
%F A250167 k=3: a(n) = 8*a(n-1) -21*a(n-2) +22*a(n-3) -8*a(n-4)
%F A250167 k=4: [order 8]
%F A250167 Empirical for row n:
%F A250167 n=1: a(n) = n + 1
%F A250167 n=2: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2
%F A250167 n=3: a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6); also a cubic polynomial plus a linear quasipolynomial with period 2
%F A250167 n=4: [order 12; also a quartic polynomial plus a quadratic quasipolynomial with period 12]
%F A250167 n=5: [order 24; also a polynomial of degree 5 plus a cubic quasipolynomialwith period 60]
%e A250167 Some solutions for n=5 k=4
%e A250167 ..3....0....3....4....0....3....4....4....2....4....4....2....0....4....3....1
%e A250167 ..2....0....4....2....0....4....1....4....4....1....3....1....1....2....1....1
%e A250167 ..4....4....0....4....4....2....2....2....4....3....4....3....1....2....3....3
%e A250167 ..0....2....0....1....2....1....3....2....1....0....3....2....3....1....0....4
%e A250167 ..1....2....4....1....3....1....3....3....3....0....0....2....0....4....3....3
%e A250167 ..1....0....3....2....0....1....2....4....0....4....0....2....0....2....3....1
%Y A250167 Column 1 is A000079
%Y A250167 Column 2 is A084171
%Y A250167 Row 2 is A212959
%K A250167 nonn,tabl
%O A250167 1,1
%A A250167 _R. H. Hardin_, Nov 13 2014