A094728 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

Offset: 1

Author

Reinhard Zumkeller, May 24 2004

Keywords

• nonn
• tabl
• easy

Comments

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

All numbers m occur A034178(m) times.

The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

Examples

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5050

Crossrefs

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).

Programs

Magma

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

Mathematica

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

SageMath

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

Formula

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

G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025