login
Irregular triangle read by rows: T(n,k) = n*(n-2) - (k-n)*(k-n-2), with 0 <= k <= 2*n.
2

%I #13 Sep 08 2022 08:45:39

%S 0,-4,-1,0,-8,-3,0,1,0,-12,-5,0,3,4,3,0,-16,-7,0,5,8,9,8,5,0,-20,-9,0,

%T 7,12,15,16,15,12,7,0,-24,-11,0,9,16,21,24,25,24,21,16,9,0,-28,-13,0,

%U 11,20,27,32,35,36,35,32,27,20,11,0,-32,-15,0,13,24,33,40,45,48,49,48,45,40,33,24,13,0

%N Irregular triangle read by rows: T(n,k) = n*(n-2) - (k-n)*(k-n-2), with 0 <= k <= 2*n.

%C The row sums are: {0, -5, -10, -7, 12, 55, 130, 245, 408, 627, 910, ...}.

%H G. C. Greubel, <a href="/A152420/b152420.txt">Rows n = 0..50 of triangle, flattened</a>

%F T(n, k) = 4*( n*(n-1) - k*(k-1) ) = 4*( (n-k)*(n+k-1) ) with n and k ranging over half-integer steps.

%F T(n, k) = n*(n-2) - (k-n)*(k-n-2), with 0 <= k <= 2*n, n >= 0. - _G. C. Greubel_, Dec 04 2019

%e Irregular triangle begins as:

%e 0;

%e -4, -1, 0;

%e -8, -3, 0, 1, 0;

%e -12, -5, 0, 3, 4, 3, 0;

%e -16, -7, 0, 5, 8, 9, 8, 5, 0;

%e -20, -9, 0, 7, 12, 15, 16, 15, 12, 7, 0;

%e -24, -11, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0;

%e -28, -13, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0;

%e -32, -15, 0, 13, 24, 33, 40, 45, 48, 49, 48, 45, 40, 33, 24, 13, 0;

%p seq(seq( n*(n-2) - (k-n)*(k-n-2), k=0..2*n), n=0..10); # _G. C. Greubel_, Dec 04 2019

%t Table[4*(n*(n-1) - k*(k-1)), {n,0,5,1/2}, {k,-n,n,1/2}]//FlattenTable[n*(n-2) - (k-n)*(k-n-2), {n,0,5}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Dec 04 2019 *)

%o (PARI) T(n,k) = n*(n-2) - (k-n)*(k-n-2); \\ _G. C. Greubel_, Dec 04 2019

%o (Magma) [n*(n-2) - (k-n)*(k-n-2): k in [0..2*n], n in [0..10]]; // _G. C. Greubel_, Dec 04 2019

%o (Sage) [[n*(n-2) - (k-n)*(k-n-2) for k in (0..2*n)] for n in (0..10)] # _G. C. Greubel_, Dec 04 2019

%o (GAP) Flat(List([0..10], n-> List([0..2*n], k-> n*(n-2) - (k-n)*(k-n-2) ))); # _G. C. Greubel_, Dec 04 2019

%Y Cf. A152439.

%K sign,tabf

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Dec 03 2008

%E Keyword tabf by _Michel Marcus_, Apr 08 2013

%E New name from _G. C. Greubel_, Dec 04 2019