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A154685 Triangle read by rows: T(n, k) = 2*n*k + n + k + 4.

%I A154685 #24 Jan 23 2025 00:17:43
%S A154685 8,11,16,14,21,28,17,26,35,44,20,31,42,53,64,23,36,49,62,75,88,26,41,
%T A154685 56,71,86,101,116,29,46,63,80,97,114,131,148,32,51,70,89,108,127,146,
%U A154685 165,184,35,56,77,98,119,140,161,182,203,224,38,61,84,107,130,153,176,199,222,245,268
%N A154685 Triangle read by rows: T(n, k) = 2*n*k + n + k + 4.
%C A154685 The terms form a subset of A153039 because 2*T(n, k) - 7 = (2*n+1)*(2*k+1) are not prime.
%H A154685 Vincenzo Librandi, <a href="/A154685/b154685.txt">Rows n = 1..100, flattened</a>
%F A154685 Sum_{k=1..n} T(n, k) = A151675(n). - _N. J. A. Sloane_, May 31 2009
%F A154685 T(n, k) = A155724(n,k) + 8. - _L. Edson Jeffery_, Oct 12 2012
%F A154685 From _G. C. Greubel_, Jan 21 2025: (Start)
%F A154685 T(2*n-1, n) = 4*n^2 + n + 3.
%F A154685 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*(9*(1-(-1)^n) + 2*(2-3*(-1)^n)*n - 4*(-1)^n*n^2).
%F A154685 G.f.: x*y*(8 - 5*(x+y) + 4*x*y)/((1-x)*(1-y))^2.
%F A154685 E.g.f.: 4 - (4+x)*exp(x) - (4+y)*exp(y) + (4+x+y+2*x*y)*exp(x+y).
%F A154685  (End)
%e A154685 Triangle begins:
%e A154685    8;
%e A154685   11, 16;
%e A154685   14, 21, 28;
%e A154685   17, 26, 35, 44;
%e A154685   20, 31, 42, 53,  64;
%e A154685   23, 36, 49, 62,  75,  88;
%e A154685   26, 41, 56, 71,  86, 101, 116;
%e A154685   29, 46, 63, 80,  97, 114, 131, 148;
%e A154685   32, 51, 70, 89, 108, 127, 146, 165, 184;
%e A154685   35, 56, 77, 98, 119, 140, 161, 182, 203, 224;
%t A154685 Flatten@Table[2*n*m+m+n+4,{n,20},{m,n}] (* _Vincenzo Librandi_, Jan 29 2012 *)
%o A154685 (PARI) for(m=1,9,for(n=1,m,print1(2*m*n+m+n+4", "))) \\ _Charles R Greathouse IV_, Dec 27 2011
%o A154685 (Magma)
%o A154685 A154685:= func< n,k | 2*n*k+n+k+4 >;
%o A154685 [A154685(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jan 21 2025
%o A154685 (Python)
%o A154685 def A154685(n,k): return 2*n*k+n+k+4
%o A154685 print(flatten([[A154685(n,k) for k in range(1,n+1)] for n in range(1,16)])) # _G. C. Greubel_, Jan 21 2025
%Y A154685 Cf. A089192, A153039.
%Y A154685 Cf. A151675 (row sums).
%Y A154685 Similar triangle: A155724.
%Y A154685 Columns k: A016789 (k=1), A016861 (k=2).
%Y A154685 Main diagonal: A137882, A271649.
%K A154685 nonn,tabl,easy
%O A154685 1,1
%A A154685 _Vincenzo Librandi_, Jan 18 2009
%E A154685 Clarified comment. - _R. J. Mathar_, Jan 24 2009