A154685 Triangle read by rows: T(n, k) = 2*n*k + n + k + 4.
8, 11, 16, 14, 21, 28, 17, 26, 35, 44, 20, 31, 42, 53, 64, 23, 36, 49, 62, 75, 88, 26, 41, 56, 71, 86, 101, 116, 29, 46, 63, 80, 97, 114, 131, 148, 32, 51, 70, 89, 108, 127, 146, 165, 184, 35, 56, 77, 98, 119, 140, 161, 182, 203, 224, 38, 61, 84, 107, 130, 153, 176, 199, 222, 245, 268
Offset: 1
Examples
Triangle begins: 8; 11, 16; 14, 21, 28; 17, 26, 35, 44; 20, 31, 42, 53, 64; 23, 36, 49, 62, 75, 88; 26, 41, 56, 71, 86, 101, 116; 29, 46, 63, 80, 97, 114, 131, 148; 32, 51, 70, 89, 108, 127, 146, 165, 184; 35, 56, 77, 98, 119, 140, 161, 182, 203, 224;
Links
- Vincenzo Librandi, Rows n = 1..100, flattened
Crossrefs
Programs
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Magma
A154685:= func< n,k | 2*n*k+n+k+4 >; [A154685(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jan 21 2025
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Mathematica
Flatten@Table[2*n*m+m+n+4,{n,20},{m,n}] (* Vincenzo Librandi, Jan 29 2012 *) -
PARI
for(m=1,9,for(n=1,m,print1(2*m*n+m+n+4", "))) \\ Charles R Greathouse IV, Dec 27 2011
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Python
def A154685(n,k): return 2*n*k+n+k+4 print(flatten([[A154685(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025
Formula
Sum_{k=1..n} T(n, k) = A151675(n). - N. J. A. Sloane, May 31 2009
T(n, k) = A155724(n,k) + 8. - L. Edson Jeffery, Oct 12 2012
From G. C. Greubel, Jan 21 2025: (Start)
T(2*n-1, n) = 4*n^2 + n + 3.
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).
G.f.: x*y*(8 - 5*(x+y) + 4*x*y)/((1-x)*(1-y))^2.
E.g.f.: 4 - (4+x)*exp(x) - (4+y)*exp(y) + (4+x+y+2*x*y)*exp(x+y).
(End)
Extensions
Clarified comment. - R. J. Mathar, Jan 24 2009
Comments