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A172370
Mirrored triangle A120072 read by rows.
7
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
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).
Sequence in context: A375338 A368741 A021740 * A304026 A110641 A121729
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