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

2,1

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Cf. A061035, A172157, A165795.

Sequence in context: A073334 A368741 A021740 * A304026 A110641 A121729

Adjacent sequences: A172367 A172368 A172369 * A172371 A172372 A172373

Keyword

nonn,easy,tabl

Author

Paul Curtz, Feb 01 2010

Extensions

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

Status

approved