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A200838 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any two consecutive increases or two consecutive decreases.

%I A200838 #7 Jul 22 2025 15:58:14
%S A200838 8,25,16,56,69,32,105,194,191,64,176,435,676,529,128,273,846,1817,
%T A200838 2356,1465,256,400,1491,4108,7587,8210,4057,512,561,2444,8239,19930,
%U A200838 31677,28610,11235,1024,760,3789,15128,45465,96690,132263,99700,31113,2048
%N A200838 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any two consecutive increases or two consecutive decreases.
%C A200838 Table starts
%C A200838 ....8.....25......56......105.......176........273........400.........561
%C A200838 ...16.....69.....194......435.......846.......1491.......2444........3789
%C A200838 ...32....191.....676.....1817......4108.......8239......15128.......25953
%C A200838 ...64....529....2356.....7587.....19930......45465......93472......177381
%C A200838 ..128...1465....8210....31677.....96690.....250913.....577660.....1212729
%C A200838 ..256...4057...28610...132263....469116....1384813....3570086.....8291391
%C A200838 ..512..11235...99700...552247...2276028....7642875...22063924....56687801
%C A200838 .1024..31113..347434..2305835..11042700...42181611..136360286...387572529
%C A200838 .2048..86161.1210736..9627715..53576350..232803603..842739040..2649819955
%C A200838 .4096.238605.4219166.40199277.259938722.1284861277.5208328180.18116728573
%H A200838 R. H. Hardin, <a href="/A200838/b200838.txt">Table of n, a(n) for n = 1..9999</a>
%F A200838 Empirical for columns:
%F A200838 k=1: a(n) = 2*a(n-1)
%F A200838 k=2: a(n) = 3*a(n-1) -a(n-2) +a(n-3)
%F A200838 k=3: a(n) = 4*a(n-1) -2*a(n-2) +a(n-3) -a(n-4)
%F A200838 k=4: a(n) = 5*a(n-1) -4*a(n-2) +3*a(n-3) -3*a(n-4) +a(n-5) -a(n-6)
%F A200838 k=5: a(n) = 6*a(n-1) -6*a(n-2) +3*a(n-3) -5*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7)
%F A200838 k=6: a(n) = 7*a(n-1) -9*a(n-2) +6*a(n-3) -9*a(n-4) +7*a(n-5) -7*a(n-6) +5*a(n-7) -2*a(n-8) +a(n-9)
%F A200838 k=7: a(n) = 8*a(n-1) -12*a(n-2) +6*a(n-3) -10*a(n-4) +12*a(n-5) -11*a(n-6) +11*a(n-7) -6*a(n-8) +3*a(n-9) -a(n-10)
%F A200838 Empirical for rows:
%F A200838 n=1: a(k) = (2/3)*k^3 + 3*k^2 + (10/3)*k + 1
%F A200838 n=2: a(k) = (5/12)*k^4 + (19/6)*k^3 + (79/12)*k^2 + (29/6)*k + 1
%F A200838 n=3: a(k) = (4/15)*k^5 + (17/6)*k^4 + (28/3)*k^3 + (73/6)*k^2 + (32/5)*k + 1
%F A200838 n=4: a(k) = (61/360)*k^6 + (93/40)*k^5 + (779/72)*k^4 + (521/24)*k^3 + (1801/90)*k^2 + (239/30)*k + 1
%F A200838 n=5: a(k) = (34/315)*k^7 + (163/90)*k^6 + (1981/180)*k^5 + (557/18)*k^4 + (7807/180)*k^3 + (1361/45)*k^2 + (333/35)*k + 1
%F A200838 n=6: a(k) = (277/4032)*k^8 + (1375/1008)*k^7 + (4933/480)*k^6 + (2723/72)*k^5 + (14161/192)*k^4 + (11197/144)*k^3 + (216211/5040)*k^2 + (929/84)*k + 1
%F A200838 n=7: a(k) = (124/2835)*k^9 + (1123/1120)*k^8 + (244/27)*k^7 + (1991/48)*k^6 + (57133/540)*k^5 + (74183/480)*k^4 + (291427/2268)*k^3 + (9739/168)*k^2 + (568/45)*k + 1
%e A200838 Some solutions for n=4 k=3
%e A200838 ..1....2....3....0....1....1....2....1....3....3....3....1....2....0....1....1
%e A200838 ..0....0....0....2....1....0....3....3....1....3....0....3....2....3....1....0
%e A200838 ..0....0....2....2....0....3....0....0....1....2....1....3....2....0....1....1
%e A200838 ..3....0....1....3....3....3....3....2....1....2....0....1....2....0....0....1
%e A200838 ..3....3....3....0....3....0....1....2....1....1....3....3....3....2....2....3
%e A200838 ..1....3....2....0....1....3....3....2....2....1....0....1....2....1....1....0
%Y A200838 Column 1 is A000079(n+2)
%Y A200838 Column 2 is A098182(n+3)
%Y A200838 Row 1 is A131423(n+1)
%K A200838 nonn,tabl
%O A200838 1,1
%A A200838 _R. H. Hardin_ Nov 23 2011