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 A200871 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors 13
 6, 17, 10, 36, 37, 16, 65, 94, 77, 26, 106, 195, 236, 163, 42, 161, 356, 567, 602, 343, 68, 232, 595, 1168, 1673, 1528, 723, 110, 321, 932, 2163, 3886, 4917, 3882, 1523, 178, 430, 1389, 3704, 7973, 12890, 14455, 9858, 3209, 288, 561, 1990, 5973, 14932, 29325 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Table starts ...6....17.....36......65.....106......161......232.......321.......430 ..10....37.....94.....195.....356......595......932......1389......1990 ..16....77....236.....567....1168.....2163.....3704......5973......9184 ..26...163....602....1673....3886.....7973....14932.....26073.....43066 ..42...343...1528....4917...12890....29325....60112....113745....201994 ..68...723...3882...14455...42744...107777...241718....495495....945790 .110..1523...9858...42479..141688...395929...971416...2156867...4424298 .178..3209..25038..124851..469726..1454643..3904290...9389377..20696974 .288..6761..63592..366959.1557320..5344795.15693816..40880321..96838448 .466.14245.161514.1078565.5163158.19638715.63085186.177996275.453123270 LINKS R. H. Hardin, Table of n, a(n) for n = 1..9999 FORMULA Empirical for columns: k=1: a(n) = a(n-1) +a(n-2) k=2: a(n) = 2*a(n-1) +a(n-4) k=3: a(n) = 2*a(n-1) +a(n-2) +2*a(n-4) +a(n-5) k=4: a(n) = 3*a(n-1) -a(n-2) +a(n-3) +4*a(n-4) +a(n-6) +a(n-7) k=5: a(n) = 3*a(n-1) +a(n-3) +7*a(n-4) +3*a(n-5) +2*a(n-6) +3*a(n-7) +a(n-8) k=6: a(n) = 4*a(n-1) -3*a(n-2) +4*a(n-3) +9*a(n-4) +7*a(n-6) +6*a(n-7) +a(n-8) +2*a(n-9) +a(n-10) k=7: a(n) = 4*a(n-1) -2*a(n-2) +4*a(n-3) +15*a(n-4) +6*a(n-5) +12*a(n-6) +16*a(n-7) +7*a(n-8) +5*a(n-9) +4*a(n-10) +a(n-11) Empirical for rows: n=1: a(k) = (1/3)*k^3 + 2*k^2 + (8/3)*k + 1 n=2: a(k) = (1/12)*k^4 + (3/2)*k^3 + (47/12)*k^2 + (7/2)*k + 1 n=3: a(k) = (1/60)*k^5 + (3/4)*k^4 + (15/4)*k^3 + (25/4)*k^2 + (127/30)*k + 1 n=4: a(k) = (1/360)*k^6 + (7/24)*k^5 + (197/72)*k^4 + (185/24)*k^3 + (1667/180)*k^2 + 5*k + 1 n=5: a(k) = (1/2520)*k^7 + (17/180)*k^6 + (281/180)*k^5 + (64/9)*k^4 + (4927/360)*k^3 + (2303/180)*k^2 + (604/105)*k + 1 n=6: a(k) = (1/20160)*k^8 + (19/720)*k^7 + (211/288)*k^6 + (1889/360)*k^5 + (44167/2880)*k^4 + (15991/720)*k^3 + (5689/336)*k^2 + (391/60)*k + 1 n=7: a(k) = (1/181440)*k^9 + (131/20160)*k^8 + (8893/30240)*k^7 + (4621/1440)*k^6 + (118933/8640)*k^5 + (83957/2880)*k^4 + (763489/22680)*k^3 + (36343/1680)*k^2 + (9169/1260)*k + 1 EXAMPLE Some solutions for n=4 k=3 ..3....2....0....0....2....0....1....0....0....2....3....3....1....1....1....3 ..2....2....0....2....0....2....2....2....0....3....1....3....2....1....2....3 ..2....1....3....3....0....2....2....2....0....3....0....3....2....2....2....3 ..2....0....3....3....3....0....1....0....2....3....0....2....2....2....0....2 ..2....0....0....1....3....0....1....0....2....3....0....2....2....2....0....2 ..3....2....0....1....1....0....2....3....3....2....2....1....3....2....0....0 MATHEMATICA t[0, k_, x_, y_] := 1; t[n_, k_, x_, y_] := t[n, k, x, y] = Sum[If[z <= x <= y || y <= x <= z, t[n-1, k, z, x], 0], {z, k+1}]; t[n_, k_] := Sum[t[n, k, x, y], {x, k+1}, {y, k+1}]; TableForm@ Table[t[n, k], {n, 8}, {k, 8}] (* Giovanni Resta, Mar 05 2014 *) CROSSREFS Column 1 is A006355(n+4) Row 1 is A084990(n+1) Sequence in context: A323516 A120930 A070395 * A112366 A095421 A063584 Adjacent sequences: A200868 A200869 A200870 * A200872 A200873 A200874 KEYWORD nonn,tabl AUTHOR R. H. Hardin Nov 23 2011 STATUS approved

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Last modified June 8 14:02 EDT 2023. Contains 363165 sequences. (Running on oeis4.)