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 A376276 #12 Oct 17 2024 08:21:50
%S A376276 1,3,1,4,4,1,2,3,4,1,8,5,5,5,1,7,2,3,4,5,1,9,10,6,6,6,6,1,6,11,2,3,4,
%T A376276 5,6,1,10,9,13,7,7,7,7,7,1,5,12,12,2,3,4,5,6,7,1,16,8,14,15,8,8,8,8,8,
%U A376276 8,1,15,13,11,16,2,3,4,5,6,7,8,1,17,7,15,14,18,9,9,9,9,9,9,9,1,14,14,10,17,17,2,3,4,5,6,7,8,9,1,18,6,16,13,19,20,10,10,10
%N A376276 Table T(n, k) n > 0, k > 2 read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. The length of the row number n in column k is equal to the n-th k-gonal number A086270.
%C A376276 A209278 presents an algorithm for generating permutations.
%C A376276 The sequence is an intra-block permutation of integer positive numbers.
%D A376276 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 45.
%H A376276 Boris Putievskiy, <a href="/A376276/b376276.txt">Table of n, a(n) for n = 1..9870</a>
%H A376276 Boris Putievskiy, <a href="https://arxiv.org/abs/2310.18466">Integer Sequences: Irregular Arrays and Intra-Block Permutations</a>, arXiv:2310.18466 [math.CO], 2023.
%H A376276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>.
%H A376276 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%H A376276 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A376276 T(n,k) = P(n,k) + ((L(n,k)-1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6, where L(n,k) = ceiling(x(n,k)), x(n,k) is largest real root of the equation x^3*(k - 2) + 3*x^2 - x*(k - 5) - 6*n = 0. R(n,k) = n - ((L(n,k) - 1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6. P(n,k) = ((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 2 - R(n,k)) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = (R(n,k) + (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) + (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) - (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even.
%F A376276 T(1,n) = A000012(n). T(2,n) = A004526(n+7). T(3,n) = A028242(n+6). T(4,n) = A084964(n+5). T(n-2,n) = A000027(n) for n > 3. L(n,3) = A360010(n). L(n,4) = A074279(n).
%e A376276 Table begins:
%e A376276 k = 3 4 5 6 7 8
%e A376276 --------------------------------------
%e A376276 n = 1: 1, 1, 1, 1, 1, 1, ...
%e A376276 n = 2: 3, 4, 4, 5, 5, 6, ...
%e A376276 n = 3: 4, 3, 5, 4, 6, 5, ...
%e A376276 n = 4: 2, 5, 3, 6, 4, 7, ...
%e A376276 n = 5: 8, 2, 6, 3, 7, 4, ...
%e A376276 n = 6: 7, 10, 2, 7, 3, 8, ...
%e A376276 n = 7: 9, 11, 13, 2, 8, 3, ...
%e A376276 n = 8: 6, 9, 12, 15, 2, 9, ...
%e A376276 n = 9: 10, 12, 14, 16, 18, 2, ...
%e A376276 n =10: 5, 8, 11, 14, 17, 20, ...
%e A376276 n =11: 16, 13, 15, 17, 19, 21, ...
%e A376276 n =12: 15, 7, 10, 13, 16, 19, ...
%e A376276 n =13: 17, 14, 16, 18, 20, 22, ...
%e A376276 n =14: 14, 6, 9, 12, 15, 18, ...
%e A376276 n =15: 18, 23, 17, 19, 21, 23, ...
%e A376276 n =16: 13, 22, 8, 11, 14, 17, ...
%e A376276 n =17: 19, 24, 18, 20, 22, 24, ...
%e A376276 n =18: 12, 21, 7, 10, 13, 16, ...
%e A376276 n =19: 20, 25, 30, 21, 23, 25, ...
%e A376276 n =20: 11, 20, 29, 9, 12, 15, ...
%e A376276 ... .
%e A376276 For k = 3 the first 4 blocks have lengths 1,3,6 and 10.
%e A376276 For k = 4 the first 3 blocks have lengths 1,4, and 9.
%e A376276 For k = 5 the first 3 blocks have lengths 1,5, and 12.
%e A376276 Each block is a permutation of the numbers of its constituents.
%e A376276 The first 6 antidiagonals are:
%e A376276 1;
%e A376276 3, 1;
%e A376276 4, 4, 1;
%e A376276 2, 3, 4, 1;
%e A376276 8, 5, 5, 5, 1;
%e A376276 7, 2, 3, 4, 5, 1;
%t A376276 T[n_,k_]:=Module[{L,R,P,Res,result},L=Ceiling[Max[x/.NSolve[x^3*(k-2)+3*x^2-x*(k-5)-6*n==0,x,Reals]]];
%t A376276 R=n-(((L-1)^3)*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;P=Which[OddQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],((k*L*(L-1)/2-L^2+2*L+1-R)+1)/2,OddQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],(R+k*L*(L-1)/2-L^2+2*L+1)/2,EvenQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]+R/2,EvenQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]-R/2];
%t A376276 Res=P+((L-1)^3*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;result=Res;result]
%t A376276 Nmax=6;Table[T[n,k],{n,1,Nmax},{k,3,Nmax+2}]
%Y A376276 Cf. A000012, A004526, A028242, A074279, A084964, A086270, A209278, A360010, A375725, A375797
%K A376276 nonn,tabl
%O A376276 1,2
%A A376276 _Boris Putievskiy_, Sep 18 2024