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 A376353 #13 Feb 16 2025 08:34:07
%S A376353 1,4,1,3,4,1,5,5,5,1,2,3,4,5,1,11,6,6,6,6,1,10,2,3,4,5,6,1,12,14,7,7,
%T A376353 7,7,7,1,9,13,2,3,4,5,6,7,1,13,15,17,8,8,8,8,8,8,1,8,12,16,2,3,4,5,6,
%U A376353 7,8,1,14,16,18,20,9,9,9,9,9,9,9,1,7,11,15,19,2,3,4,5,6,7,8,9,1,15,17,19,21,23,10,10,10,10,10,10,10,10,1,6,10,14,18,22,2,3
%N A376353 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-pyramidal number A261720.
%C A376353 A209278 presents an algorithm for generating permutations.
%C A376353 The sequence is an intra-block permutation of integer positive numbers.
%D A376353 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%H A376353 Boris Putievskiy, <a href="/A376353/b376353.txt">Table of n, a(n) for n = 1..9870</a>
%H A376353 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 A376353 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>.
%H A376353 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%H A376353 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%F A376353 T(n,k) = P(n,k) + ((k-2)*(L(n,k)-1)^4+2*k*(L(n,k)-1)^3+(14-k)*(L(n,k)-1)^2+(12-2*k)*(L(n,k)-1))/24, where L(n,k) = ceiling(x(n,k)), x(n,k) is largest real root of the equation (k-2)*x^4+2*k*x^3+(14-k)*x^2+(12-2*k)*x-24*n = 0. R(n,k) = n - ((k-2)*(L(n,k)-1)^4+2*k*(L(n,k)-1)^3+(14-k)*(L(n,k)-1)^2+(12-2*k)*(L(n,k)-1))/24. P(n,k) = ((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+2-R(n,k))/2 if R(n,k) is odd and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is odd, P(n,k) = ((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)+R(n,k))/2 if R(n,k) is odd and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is even, P = ceiling(((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)/2)+R(n,k)/2) if R(n,k) is even and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is odd, P = ceiling(((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)/2)-R(n,k)/2) if R(n,k) is even and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is even.
%F A376353 T(1,n) = A000012(n). T(2,n) = A004526(n+8). T(3,n) = A028242(n+7). T(4,n) = A084964(n+6). T(5,n) = A168230(n+5). T(n-2,n) = 4*A000012(n) for n > 3. T(n-1,n) = A000027(n) for n > 2.
%e A376353 Table begins:
%e A376353 k = 3 4 5 6 7 8
%e A376353 --------------------------------------
%e A376353 n = 1: 1, 1, 1, 1, 1, 1, ...
%e A376353 n = 2: 4, 4, 5, 5, 6, 6, ...
%e A376353 n = 3: 3, 5, 4, 6, 5, 7, ...
%e A376353 n = 4: 5, 3, 6, 4, 7, 5, ...
%e A376353 n = 5: 2, 6, 3, 7, 4, 8, ...
%e A376353 n = 6: 11, 2, 7, 3, 8, 4, ...
%e A376353 n = 7: 10, 14, 2, 8, 3, 9, ...
%e A376353 n = 8: 12, 13, 17, 2, 9, 3, ...
%e A376353 n = 9: 9, 15, 16, 20, 2, 10, ...
%e A376353 n = 10: 13, 12, 18, 19, 23, 2, ...
%e A376353 n = 11: 8, 16, 15, 21, 22, 26, ...
%e A376353 n = 12: 14, 11, 19, 18, 24, 25, ...
%e A376353 n = 12: 7, 17, 14, 22, 21, 27, ...
%e A376353 n = 14: 15, 10, 20, 17, 25, 24, ...
%e A376353 n = 15: 6, 18, 13, 23, 20, 28, ...
%e A376353 ... .
%e A376353 For k = 3 the first 3 blocks have lengths 1,4 and 10.
%e A376353 For k = 4 the first 2 blocks have lengths 1 and 5.
%e A376353 For k = 5 the first 2 blocks have lengths 1 and 6.
%e A376353 Each block is a permutation of the numbers of its constituents.
%e A376353 The first 6 antidiagonals are:
%e A376353 1;
%e A376353 4, 1;
%e A376353 3, 4, 1;
%e A376353 5, 5, 5, 1;
%e A376353 2, 3, 4, 5, 1;
%e A376353 11, 6, 6, 6, 6, 1;
%t A376353 T[n_,k_]:=Module[{L,R,result},L=Ceiling[Max[x/.NSolve[(k-2)*x^4+2*k*x^3+(14-k)*x^2+(12-2*k)*x-24*n==0,x,Reals]]]; R=n-((k-2)*(L-1)^4+2*k*(L-1)^3+(14-k)*(L-1)^2+(12-2*k)*(L-1))/24; P=Which[OddQ[R]&&OddQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],((L^3*(k-2)+3*L^2-L*(k-5))/6+2-R)/2,OddQ[R]&&EvenQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],(R+(L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2,EvenQ[R]&&OddQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],Ceiling[((L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2]+R/2,EvenQ[R]&&EvenQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],Ceiling[((L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2]-R/2]; Res= P +((k-2)*(L-1)^4+2*k*(L-1)^3+(14-k)*(L-1)^2+(12-2*k)*(L-1))/24; result=Res] Nmax=6; Table[T[n,k],{n,1,Nmax},{k,3,Nmax+2}]
%Y A376353 Cf. A000012, A000027, A004526, A028242, A084964, A168230, A209278, A261720, A375725, A375797.
%K A376353 nonn,tabl
%O A376353 1,2
%A A376353 _Boris Putievskiy_, Sep 21 2024