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Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)

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G. C. Greubel, <a href="/A094728/b094728_1.txt">Table of n, a(n) for n = 1..5050</a>

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Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21, 144

T(n,0)=A000290(n); T(n,1)=A005563(n-1) for n>1; T(n,2)=A028347(n) for n>2; T(n,3)=A028560(n-3) for n>3; T(n,4)=A028566(n-4) for n>4;

T(n,n-1)=A005408(n); T(n,n-2)=A008586(n-1) for n>1; T(n,n-3)=A016945(n-2) for n>2; T(n,n-4)=A008590(n-2) for n>3; T(n,n-5)=A017329(n-3) for n>4; T(n,n-6)=A008594(n-3) for n>5; T(n,n-8)=A008598(n-2) for n>7;

T(A005408(k),k) = A000567(k);

(T(n,k) mod 4) <> 2, see A042965, A016825.

row sums give A002412;All numbers m occur A034178(m) times.

(T(n,k) mod 4) <> 2, see A042965, A016825;

all numbers m occur A034178(m) times;

T(n,k) = A004736(n,k)*A094727(n,k).

G. C. Greubel, <a href="/A094728/b094728_1.txt">Table of n, a(n) for n = 1..5050</a>

Row polynomials: T(n,x) = n^2*sum(Sum_{m=0..n} x^m, - Sum_{m=0..n)-sum(} m^2*x^m,m = Sum_{k=0..n) = sum(-1} T(n,k)*x^k,k=0..n-1), n >= 1.

T(n, k) = A004736(n,k)*A094727(n,k).

T(n, 0) = A000290(n).

T(n, 1) = A005563(n-1) for n>1.

T(n, 2) = A028347(n) for n>2.

T(n, 3) = A028560(n-3) for n>3.

T(n, 4) = A028566(n-4) for n>4.

T(n, n-1) = A005408(n).

T(n, n-2) = A008586(n-1) for n>1.

T(n, n-3) = A016945(n-2) for n>2.

T(n, n-4) = A008590(n-2) for n>3.

T(n, n-5) = A017329(n-3) for n>4.

T(n, n-6) = A008594(n-3) for n>5.

T(n, n-8) = A008598(n-2) for n>7.

T(A005408(k), k) = A000567(k).

Sum_{k=0..n} T(n, k) = A002412(n) (row sums).

From G. C. Greubel, Mar 12 2024: (Start)

Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).

Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).

Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)).

1;

4, 3;

9, 8, 5;

16, 15, 12, 7;

25, 24, 21, 16, 9;

36, 35, 32, 27, 20, 11;

49, 48, 45, 40, 33, 24, 13;

64, 63, 60, 55, 48, 39, 28, 15;

81, 80, 77, 72, 65, 56, 45, 32, 17, etc. - _Philippe Deléham_, Mar 07 2013;

... etc. - Philippe Deléham, Mar 07 2013

Table[n^2 - k^2, {n, 12}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Nov 25 2015 *)

(Magma) [n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

(SageMath) flatten([[n^2-k^2 for k in range(n)] for n in range(1, 16)]) # G. C. Greubel, Mar 12 2024

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.

Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.

Cf. A028566, A034178, A042965, A094727, A120070, A128624.

Cf. A000384 (alternating row sums), A002412 (row sums).

nonn,tabl,easy

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