This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A269656 #15 Mar 01 2020 12:16:47
%S A269656 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 A269656 621,988,626,64,9,64,343,1291,3065,3816,1710,93,10,81,512,2395,7686,
%U A269656 15036,14596,4605,130,11,100,729,4089,16681,45590,73348,55344,12259,176,12,121
%N A269656 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 A269656 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 A269656 R. H. Hardin, <a href="/A269656/b269656.txt">Table of n, a(n) for n = 1..759</a>
%F A269656 Empirical for column k, apparently a recurrence of order (k+1)^2:
%F A269656 k=1: a(n) = (1/6)*n^3 + (5/6)*n + 1
%F A269656 k=2: [linear recurrence of order 9]
%F A269656 k=3: [order 16]
%F A269656 k=4: [order 25]
%F A269656 k=5: [order 36]
%F A269656 k=6: [order 49]
%F A269656 k=7: [order 64]
%F A269656 Empirical for row n:
%F A269656 n=1: a(n) = n + 1 = #{ v = (m); 0 <= m <= n }.
%F A269656 n=2: a(n) = n^2 + 2*n + 1 = (n+1)^2 = #{ v in {0..n}^2 }.
%F A269656 n=3: a(n) = n^3 + 3*n^2 + 3*n + 1 = (n+1)^3 = #{ v in {0..n}^3 }.
%F A269656 n=4: a(n) = n^4 + 4*n^3 + 6*n^2 + 3*n + 1 = (n+1)^4 - n, cf. A269657.
%F A269656 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 A269656 n=6: a(n) = n^6 + 6*n^5 + 15*n^4 + 14*n^3 + 3*n^2 + 3*n.
%F A269656 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 A269656 Table starts
%e A269656 2 3 4 5 6 7 8 9 10
%e A269656 4 9 16 25 36 49 64 81 100
%e A269656 8 27 64 125 216 343 512 729 1000
%e A269656 15 79 253 621 1291 2395 4089 6553 9991
%e A269656 26 225 988 3065 7686 16681 32600 58833 99730
%e A269656 42 626 3816 15036 45590 115902 259476 527576 994626
%e A269656 64 1710 14596 73348 269472 803434 2061940 4725456 9911008
%e A269656 93 4605 55344 355921 1587450 5556909 16359580 42277329 98674806
%e A269656 130 12259 208196 1718569 9321628 38350583 129599404 377821501 981592964
%e A269656 176 32320 777582 8259567 54569340 264117327 1025145474 3372803487 9756620832
%e A269656 Some solutions for n=6, k=4:
%e A269656 1 2 2 3 1 0 3 3 3 3 1 2 3 4 3 3
%e A269656 3 2 4 3 3 1 4 3 4 0 4 2 2 3 0 3
%e A269656 2 3 1 3 2 2 4 2 0 1 0 3 2 3 0 2
%e A269656 0 2 4 4 1 4 2 3 0 4 0 3 2 3 0 3
%e A269656 4 2 0 1 4 4 2 1 1 2 4 0 0 3 2 1
%e A269656 0 1 1 0 0 4 4 2 4 3 3 3 1 0 3 0
%Y A269656 Column 1 is A000125.
%Y A269656 Row 1 is A000027(n+1).
%Y A269656 Row 2 is A000290(n+1).
%Y A269656 Row 3 is A000578(n+1).
%Y A269656 Rows 4, ..., 7: A269657, A269658, A269659 and A269660 (see there for formulas).
%K A269656 nonn,tabl
%O A269656 1,1
%A A269656 _R. H. Hardin_, Mar 02 2016
%E A269656 Edited by _M. F. Hasler_, Feb 29 2020