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A250351 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 2 times.

%I A250351 #10 Jul 23 2025 12:20:10
%S A250351 2,3,4,4,9,8,5,16,26,16,6,25,62,75,32,7,36,122,235,216,64,8,49,212,
%T A250351 581,888,622,128,9,64,338,1221,2724,3349,1791,256,10,81,506,2287,6900,
%U A250351 12734,12620,5157,512,11,100,722,3935,15186,38543,59406,47545,14849,1024,12
%N A250351 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 2 times.
%C A250351 Table starts
%C A250351 ....2.....3......4.......5........6.........7.........8..........9.........10
%C A250351 ....4.....9.....16......25.......36........49........64.........81........100
%C A250351 ....8....26.....62.....122......212.......338.......506........722........992
%C A250351 ...16....75....235.....581.....1221......2287......3935.......6345.......9721
%C A250351 ...32...216....888....2724.....6900.....15186.....30072......54888......93924
%C A250351 ...64...622...3349...12734....38543.....99344....226247.....467642.....894599
%C A250351 ..128..1791..12620...59406...214716....644040...1681860....3932472....8409252
%C A250351 ..256..5157..47545..276816..1193739...4164930..12411486...32743710...78177402
%C A250351 ..512.14849.179104.1289208..6628042..26882466..91384716..270990642..720784488
%C A250351 .1024.42756.674666.6002949.36773706.173276640.671639928.2238089580.6611373660
%H A250351 R. H. Hardin, <a href="/A250351/b250351.txt">Table of n, a(n) for n = 1..418</a>
%F A250351 Empirical for column k:
%F A250351 k=1: a(n) = 2*a(n-1)
%F A250351 k=2: a(n) = 3*a(n-1) -a(n-3)
%F A250351 k=3: a(n) = 4*a(n-1) -2*a(n-3) -5*a(n-4) +a(n-6) = A250346(n)
%F A250351 k=4: [order 12] = A250347(n)
%F A250351 k=5: [order 24] = A250348(n)
%F A250351 k=6: [order 48] = A250349(n).
%F A250351 k=7: [order 96] = A250350(n).
%F A250351 Empirical for row n:
%F A250351 n=1: a(n) = n + 1
%F A250351 n=2: a(n) = n^2 + 2*n + 1
%F A250351 n=3: a(n) = n^3 + 3*n^2 + 2*n + 2 = A250352(n).
%F A250351 n=4: a(n) = n^4 + 4*n^3 + 2*n^2 + 9*n + 1 for n>1 = A250353(n).
%F A250351 n=5: a(n) = n^5 + 5*n^4 + 25*n^2 + 5*n for n>2 = A250354(n).
%F A250351 n=6: a(n) = n^6 + 6*n^5 - 5*n^4 + 55*n^3 + 25*n^2 - 61*n + 98 for n>3 = A250355(n).
%F A250351 n=7: a(n) = n^7 + 7*n^6 - 14*n^5 + 105*n^4 + 105*n^3 - 532*n^2 + 1252*n - 744 for n>4 = A250356(n).
%e A250351 Some solutions for n=6 k=4
%e A250351 ..2....3....2....0....4....2....4....3....4....0....3....1....0....4....4....4
%e A250351 ..3....1....2....1....3....2....1....4....1....4....5....4....5....4....5....1
%e A250351 ..3....4....4....4....2....3....4....5....5....2....4....5....6....6....3....2
%e A250351 ..5....7....7....4....5....6....5....3....4....5....4....6....5....6....6....5
%e A250351 ..7....6....7....7....7....6....5....5....7....6....8....5....7....7....7....7
%e A250351 ..9....6....9....9....6....5....9....8....7....6....5....6....8....5....7....9
%Y A250351 Column 1 is A000079
%Y A250351 Column 2 is A076264(n)
%Y A250351 Row 1 is A000027(n+1)
%Y A250351 Row 2 is A000290(n+1)
%Y A250351 Cf. A248944
%K A250351 nonn,tabl
%O A250351 1,1
%A A250351 _R. H. Hardin_, Nov 19 2014