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A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.

%I A377802 #9 Nov 15 2024 23:33:42
%S A377802 1,2,1,4,3,2,6,5,4,3,9,8,7,6,5,12,11,10,9,8,7,16,15,14,13,12,11,10,20,
%T A377802 19,18,17,16,15,14,13,25,24,23,22,21,20,19,18,17,30,29,28,27,26,25,24,
%U A377802 23,22,21,36,35,34,33,32,31,30,29,28,27,26,42,41,40,39,38,37,36,35,34,33,32,31
%N A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.
%C A377802 The natural numbers, based on quarter-squares (A002620 and A033638); every natural number occurs exactly twice.
%F A377802 T(n, k) = A002620(n+1) + 1 - k.
%F A377802 T(2*n-1, n) = n^2 - n + 1 = A002061(n); T(2*n-2, n) = (n-1)^2 = A000290(n-1) for n > 1; T(2*n-3, n) = (n-1) * (n-2) = A002378(n-2) for n > 2; T(2*n-4, n) = (n-1) * (n-3) = A005563(n-3) for n > 3.
%F A377802 Row sums are (2 * n^3 + 5 * n - n * (-1)^n) / 8 = (A006003(n) + A026741(n)) / 2.
%F A377802 G.f.: x*y*(1 - x*y + x^2*y + x^4*y^2 - x^5*y^3 + x^6*y^3 - x^3*y*(1 + 2*y - y^2))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - _Stefano Spezia_, Nov 08 2024
%e A377802 Triangle T(n, k) for 1 <= k <= n starts:
%e A377802 n\ k :   1   2   3   4   5   6   7   8   9  10  11  12  13
%e A377802 ==========================================================
%e A377802    1 :   1
%e A377802    2 :   2   1
%e A377802    3 :   4   3   2
%e A377802    4 :   6   5   4   3
%e A377802    5 :   9   8   7   6   5
%e A377802    6 :  12  11  10   9   8   7
%e A377802    7 :  16  15  14  13  12  11  10
%e A377802    8 :  20  19  18  17  16  15  14  13
%e A377802    9 :  25  24  23  22  21  20  19  18  17
%e A377802   10 :  30  29  28  27  26  25  24  23  22  21
%e A377802   11 :  36  35  34  33  32  31  30  29  28  27  26
%e A377802   12 :  42  41  40  39  38  37  36  35  34  33  32  31
%e A377802   13 :  49  48  47  46  45  44  43  42  41  40  39  38  37
%e A377802   etc.
%o A377802 (PARI) T(n,k)=(2*(n+1)^2+7-(-1)^n)/8-k
%Y A377802 A002620 (column 1), A024206 (column 2), A014616 (column 3), A004116 (column 4), A033638 (main diagonal), A290743 (1st subdiagonal).
%Y A377802 Cf. A006003, A026741, A002061, A000290, A002378, A005563, A246694.
%K A377802 nonn,easy,tabl
%O A377802 1,2
%A A377802 _Werner Schulte_, Nov 07 2024