A269656 - OEIS

%I #15 Mar 01 2020 12:16:47

%S 2,3,4,4,9,8,5,16,27,15,6,25,64,79,26,7,36,125,253,225,42,8,49,216,

%T 621,988,626,64,9,64,343,1291,3065,3816,1710,93,10,81,512,2395,7686,

%U 15036,14596,4605,130,11,100,729,4089,16681,45590,73348,55344,12259,176,12,121

%N T(n,k) = number of length-n 0..k arrays with no adjacent pair x,x+1 repeated: infinite square array read by falling antidiagonals.

%C The table could be extended to T(0,k) = T(n,0) = 1, since there is exactly one length-0 array {()} and exactly one length-n array with coefficients in 0..0, {(0,...,0)}, each of which satisfies the requirement. The "empirical" formulas for n = 1, ..., 5 are easily proved, cf., e.g., A269657. - _M. F. Hasler_, Feb 29 2020

%H R. H. Hardin, <a href="/A269656/b269656.txt">Table of n, a(n) for n = 1..759</a>

%F Empirical for column k, apparently a recurrence of order (k+1)^2:

%F k=1: a(n) = (1/6)*n^3 + (5/6)*n + 1

%F k=2: [linear recurrence of order 9]

%F k=3: [order 16]

%F k=4: [order 25]

%F k=5: [order 36]

%F k=6: [order 49]

%F k=7: [order 64]

%F Empirical for row n:

%F n=1: a(n) = n + 1 = #{ v = (m); 0 <= m <= n }.

%F n=2: a(n) = n^2 + 2*n + 1 = (n+1)^2 = #{ v in {0..n}^2 }.

%F n=3: a(n) = n^3 + 3*n^2 + 3*n + 1 = (n+1)^3 = #{ v in {0..n}^3 }.

%F n=4: a(n) = n^4 + 4*n^3 + 6*n^2 + 3*n + 1 = (n+1)^4 - n, cf. A269657.

%F n=5: a(n) = n^5 + 5*n^4 + 10*n^3 + 7*n^2 + 2*n + 1 = (n+1)^5 - 3*n*(n+1).

%F n=6: a(n) = n^6 + 6*n^5 + 15*n^4 + 14*n^3 + 3*n^2 + 3*n.

%F n=7: a(n) = n^7 + 7*n^6 + 21*n^5 + 25*n^4 + 5*n^3 + 2*n^2 + 11*n - 8.

%e Table starts

%e     2     3      4       5        6         7          8          9         10

%e     4     9     16      25       36        49         64         81        100

%e     8    27     64     125      216       343        512        729       1000

%e    15    79    253     621     1291      2395       4089       6553       9991

%e    26   225    988    3065     7686     16681      32600      58833      99730

%e    42   626   3816   15036    45590    115902     259476     527576     994626

%e    64  1710  14596   73348   269472    803434    2061940    4725456    9911008

%e    93  4605  55344  355921  1587450   5556909   16359580   42277329   98674806

%e   130 12259 208196 1718569  9321628  38350583  129599404  377821501  981592964

%e   176 32320 777582 8259567 54569340 264117327 1025145474 3372803487 9756620832

%e Some solutions for n=6, k=4:

%e   1  2  2  3  1  0  3  3  3  3  1  2  3  4  3  3

%e   3  2  4  3  3  1  4  3  4  0  4  2  2  3  0  3

%e   2  3  1  3  2  2  4  2  0  1  0  3  2  3  0  2

%e   0  2  4  4  1  4  2  3  0  4  0  3  2  3  0  3

%e   4  2  0  1  4  4  2  1  1  2  4  0  0  3  2  1

%e   0  1  1  0  0  4  4  2  4  3  3  3  1  0  3  0

%Y Column 1 is A000125.

%Y Row 1 is A000027(n+1).

%Y Row 2 is A000290(n+1).

%Y Row 3 is A000578(n+1).

%Y Rows 4, ..., 7: A269657, A269658, A269659 and A269660 (see there for formulas).

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Mar 02 2016

%E Edited by _M. F. Hasler_, Feb 29 2020