cp's OEIS Frontend

A221217 T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

%I A221217 #24 Feb 16 2025 08:33:19
%S A221217 1,6,5,4,3,2,15,14,13,12,11,10,9,8,7,28,27,26,25,24,23,22,21,20,19,18,
%T A221217 17,16,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,66,65,64,63,
%U A221217 62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,91
%N A221217 T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.
%C A221217 Permutation of the natural numbers.
%C A221217 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 A221217 Enumeration table T(n,k). Let m be natural number. The order of the list:
%C A221217   T(1,1)=1;
%C A221217   T(3,1), T(2,2), T(1,3);
%C A221217   T(2,1), T(1,2);
%C A221217   . . .
%C A221217   T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(1,2*m+1);
%C A221217   T(2*m,1), T(2*m-1,2), T(2*m-2,3),...T(1,2*m);
%C A221217   . . .
%C A221217 First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
%C A221217 Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read upwards.
%H A221217 Boris Putievskiy, <a href="/A221217/b221217.txt">Rows n = 1..140 of triangle, flattened</a>
%H A221217 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 A221217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H A221217 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A221217 As table
%F A221217 T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2.
%F A221217 As linear sequence
%F A221217 a(n) = (A003057(n)^2-2*A002260(n)+3-A002024(n)*(1+2*(-1)^A003056(n)))/2;
%F A221217 a(n) = ((t+2)^2-2*i+3-(t+1)*(1+2*(-1)**t))/2, where i=n-t*(t+1)/2,
%F A221217   j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
%e A221217 The start of the sequence as table:
%e A221217   1....6...4..15..11..28..22...
%e A221217   5....3..14..10..27..21..44...
%e A221217   2...13...9..26..20..43..35...
%e A221217   12...8..25..19..42..34..63...
%e A221217   7...24..18..41..33..62..52...
%e A221217   23..17..40..32..61..51..86...
%e A221217   16..39..31..60..50..85..73...
%e A221217   . . .
%e A221217 The start of the sequence as triangle array read by rows:
%e A221217   1;
%e A221217   6,5;
%e A221217   4,3,2;
%e A221217   15,14,13,12;
%e A221217   11,10,9,8,7;
%e A221217   28,27,26,25,24,23;
%e A221217   22,21,20,19,18,17,16;
%e A221217   . . .
%e A221217 Row number r consecutive contains r numbers in decreasing order.
%o A221217 (Python)
%o A221217 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A221217 i=n-t*(t+1)/2
%o A221217 j=(t*t+3*t+4)/2-n
%o A221217 result=((t+2)**2-2*i+3-(t+1)*(1+2*(-1)**t))/2
%Y A221217 Cf. A211394, A221215, A002260, A004736, A003057, A002024;
%Y A221217 table T(n,k) contains: in rows A084849, A000384, A014106, A014105, A014107, A091823, A071355, A091823, A071355,  A100040, A130861, A100041;
%Y A221217 in columns A130883, A096376, A033816, A100037, A100038, A100039;
%Y A221217 main diagonal and parallel diagonals are A058331, A051890, A005893, A097080, A093328, A137882, A001844, A001105, A056220, A142463, A054000, A090288, A059993.
%K A221217 nonn,tabl
%O A221217 1,2
%A A221217 _Boris Putievskiy_, Feb 22 2013