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 A211394 #28 Feb 16 2025 08:33:17
%S A211394 1,5,6,2,3,4,12,13,14,15,7,8,9,10,11,23,24,25,26,27,28,16,17,18,19,20,
%T A211394 21,22,38,39,40,41,42,43,44,45,29,30,31,32,33,34,35,36,37,57,58,59,60,
%U A211394 61,62,63,64,65,66,46,47,48,49,50,51,52,53,54,55,56,80
%N A211394 T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.
%C A211394 Permutation of the natural numbers.
%C A211394 a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
%C A211394 Enumeration table T(n,k). The order of the list:
%C A211394 T(1,1)=1;
%C A211394 T(1,3), T(2,2), T(3,1);
%C A211394 T(1,2), T(2,1);
%C A211394 . . .
%C A211394 T(1,n), T(2,n-1), T(3,n-2), ... T(n,1);
%C A211394 T(1,n-1), T(2,n-3), T(3,n-4),...T(n-1,1);
%C A211394 . . .
%C A211394 First row matches with the elements antidiagonal {T(1,n), ... T(n,1)},
%C A211394 second row matches with the elements antidiagonal {T(1,n-1), ... T(n-1,1)}.
%C A211394 Table contains:
%C A211394 row 1 is alternation of elements A130883 and A096376,
%C A211394 row 2 accommodates elements A033816 in even places,
%C A211394 row 3 accommodates elements A100037 in odd places,
%C A211394 row 5 accommodates elements A100038 in odd places;
%C A211394 column 1 is alternation of elements A084849 and A000384,
%C A211394 column 2 is alternation of elements A014106 and A014105,
%C A211394 column 3 is alternation of elements A014107 and A091823,
%C A211394 column 4 is alternation of elements A071355 and |A168244|,
%C A211394 column 5 accommodates elements A033537 in even places,
%C A211394 column 7 is alternation of elements A100040 and A130861,
%C A211394 column 9 accommodates elements A100041 in even places;
%C A211394 the main diagonal is A058331,
%C A211394 diagonal 1, located above the main diagonal is A001844,
%C A211394 diagonal 2, located above the main diagonal is A001105,
%C A211394 diagonal 3, located above the main diagonal is A046092,
%C A211394 diagonal 4, located above the main diagonal is A056220,
%C A211394 diagonal 5, located above the main diagonal is A142463,
%C A211394 diagonal 6, located above the main diagonal is A054000,
%C A211394 diagonal 7, located above the main diagonal is A090288,
%C A211394 diagonal 9, located above the main diagonal is A059993,
%C A211394 diagonal 10, located above the main diagonal is |A147973|,
%C A211394 diagonal 11, located above the main diagonal is A139570;
%C A211394 diagonal 1, located under the main diagonal is A051890,
%C A211394 diagonal 2, located under the main diagonal is A005893,
%C A211394 diagonal 3, located under the main diagonal is A097080,
%C A211394 diagonal 4, located under the main diagonal is A093328,
%C A211394 diagonal 5, located under the main diagonal is A137882.
%H A211394 Boris Putievskiy, <a href="/A211394/b211394.txt">Rows n = 1..140 of triangle, flattened</a>
%H A211394 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H A211394 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H A211394 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A211394 T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2.
%F A211394 As linear sequence
%F A211394 a(n) = A003057(n)*A002024(n)/2- A002024(n)*(-1)^A003056(n)-A004736(n)+2.
%F A211394 a(n) = (t+2)*(t+1)/2 - (t+1)*(-1)^t-j+2, where j=(t*t+3*t+4)/2-n and t=int((math.sqrt(8*n-7) - 1)/ 2).
%e A211394 The start of the sequence as table:
%e A211394 1....5...2..12...7..23..16...
%e A211394 6....3..13...8..24..17..39...
%e A211394 4...14...9..25..18..40..31...
%e A211394 15..10..26..19..41..32..60...
%e A211394 11..27..20..42..33..61..50...
%e A211394 28..21..43..34..62..51..85...
%e A211394 22..44..35..63..52..86..73...
%e A211394 . . .
%e A211394 The start of the sequence as triangle array read by rows:
%e A211394 1;
%e A211394 5,6;
%e A211394 2,3,4;
%e A211394 12,13,14,15;
%e A211394 7,8,9,10,11;
%e A211394 23,24,25,26,27,28;
%e A211394 16,17,18,19,20,21,22;
%e A211394 . . .
%e A211394 Row number r matches with r numbers segment {(r+1)*r/2-r*(-1)^(r+1)-r+2,... (r+1)*r/2-r*(-1)^(r+1)+1}.
%t A211394 T[n_, k_] := (n+k)(n+k-1)/2 - (-1)^(n+k)(n+k-1) - k + 2;
%t A211394 Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Dec 06 2018 *)
%o A211394 (Python)
%o A211394 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A211394 j=(t*t+3*t+4)/2-n
%o A211394 result=(t+2)*(t+1)/2-(t+1)*(-1)**t-j+2
%Y A211394 Cf. A130883, A096376, A033816, A100037, A100038, A084849, A000384, A014106, A014105, A014107, A091823, A071355, A168244, A033537, A100040, A130861, A100041, A058331, A001844, A001105, A046092, A056220, A142463, A054000, A090288, A059993, A147973, A139570, A051890, A005893, A097080, A093328, A137882.
%K A211394 nonn,tabl
%O A211394 1,2
%A A211394 _Boris Putievskiy_, Feb 08 2013