A172370 Mirrored triangle A120072 read by rows.

3, 5, 8, 7, 3, 15, 9, 16, 21, 24, 11, 5, 1, 2, 35, 13, 24, 33, 40, 45, 48, 15, 7, 39, 3, 55, 15, 63, 17, 32, 5, 56, 65, 8, 77, 80, 19, 9, 51, 4, 3, 21, 91, 6, 99, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 23, 11, 7, 5, 95, 1, 119, 1, 5, 35, 143, 25, 48, 69, 88, 105, 120, 133, 144

Offset: 2

Author

Paul Curtz, Feb 01 2010

Keywords

• nonn
• easy
• tabl

Comments

A table of numerators of 1/n^2 - 1/m^2 extended to negative m looks as follows, stacked such that values of common m are aligned

and the central column of -1 is defined for m=0:

.............................0..-1...0...3...8..15..24..35..48..63..80..99. A005563

.........................0..-3..-1..-3...0...5...3..21...2..45..15..77...6. A061037

.....................0..-5..-8..-1..-8..-5...0...7..16...1..40..55...8..91. A061039

.................0..-7..-3.-15..-1.-15..-3..-7...0...9...5..33...3..65..21. A061041

.............0..-9.-16.-21.-24..-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043

.........0.-11..-5..-1..-2.-35..-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045

.....0.-13.-24.-33.-40.-45.-48..-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047

.0.-15..-7.-39..-3.-55.-15.-63..-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049

The row-reversed variant of A120072 appears (negated) after the leftmost 0.

Equals A061035 with the first column removed. - Georg Fischer, Jul 26 2023

Examples

			The table starts
   3
   5   8
   7   3  15
   9  16  21  24
  11   5   1   2  35
  13  24  33  40  45  48
  15   7  39   3  55  15  63
  17  32   5  56  65   8  77  80
  19   9  51   4   3  21  91   6  99
		

Links

• G. C. Greubel, Rows n = 2..100 of triangle, flattened

Crossrefs

Lower diagonal gives: A070262, A061037(n+2).

Cf. A061035, A172157, A165795.

Programs

Magma

[[Numerator(1/(n-k)^2 -1/n^2): k in [1..n-1]]: n in [2..20]]; // G. C. Greubel, Sep 20 2018

Mathematica

Table[Numerator[1/(n-k)^2 -1/n^2], {n, 2, 20}, {k, 1, n-1}]//Flatten (* G. C. Greubel, Sep 20 2018 *)

PARI

for(n=2,20, for(k=1,n-1, print1(numerator(1/(n-k)^2 -1/n^2), ", "))) \\ G. C. Greubel, Sep 20 2018

Formula

T(n,m) = numerator of 1/(n-m)^2 - 1/n^2, n >= 2, 1 <= m < n. - R. J. Mathar, Nov 23 2010

Extensions

Comment rewritten and offset set to 2 by R. J. Mathar, Nov 23 2010