A061035 -id:A061035 - cp's OEIS frontend

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

Paul Curtz, Feb 01 2010

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

			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

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

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

Cf. A061035, A172157, A165795.

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

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

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

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

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

A061048 Denominator of 1/49 - 1/n^2.

Original entry on oeis.org

1, 3136, 3969, 4900, 5929, 7056, 8281, 196, 11025, 12544, 14161, 15876, 17689, 19600, 441, 23716, 25921, 28224, 30625, 33124, 35721, 784, 41209, 44100, 47089, 50176, 53361, 56644, 1225, 63504, 67081, 70756, 74529, 78400, 82369

Offset: 7

N. J. A. Sloane, May 26 2001

G. C. Greubel, Table of n, a(n) for n = 7..1000

Cf. A061035-A061050.

Cf. A061047 (numerator).

Table[Denominator[1/7^2 - 1/n^2], {n, 7, 50}] (* G. C. Greubel, Jul 07 2017 *)

for(n=7,50, print1(denominator(1/7^2 - 1/n^2), ", ")) \\ G. C. Greubel, Jul 07 2017

A174233 Triangle T(n,k) read by rows: the numerator of 1/n^2 - 1/(k-n)^2, 0 <= k < 2n.

Original entry on oeis.org

0, -1, 0, -3, -1, -3, 0, -5, -8, -1, -8, -5, 0, -7, -3, -15, -1, -15, -3, -7, 0, -9, -16, -21, -24, -1, -24, -21, -16, -9, 0, -11, -5, -1, -2, -35, -1, -35, -2, -1, -5, -11, 0, -13, -24, -33, -40, -45, -48, -1, -48, -45, -40, -33, -24, -13, 0, -15, -7, -39, -3, -55, -15, -63

Offset: 1

Paul Curtz, Mar 13 2010

A value of -1 is substituted at the poles where k=n.
The triangle is created by selecting the first 2, 4, 6 etc elements of the array shown in A172370, equivalent to attaching the initial elements of the rows of A172157 to the rows of A174190.
If the first column of zeros is removed from the triangle, each row is left-right symmetric with respect to the center value.

			The triangle starts
  0,  -1;
  0,  -3,  -1,  -3;
  0,  -5,  -8,  -1,  -8,  -5;
  0,  -7,  -3, -15,  -1, -15,  -3,  -7;
  0,  -9, -16, -21, -24,  -1, -24, -21, -16,  -9;
  0, -11,  -5,  -1,  -2, -35,  -1, -35,  -2,  -1,  -5, -11;
  0, -13, -24, -33, -40, -45, -48,  -1, -48, -45, -40, -33, -24, -13;

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A175779, A172370, A061035, A174190, A120072.

A173233 := proc(n,k) if k = n then -1 ; else 1/n^2-1/(k-n)^2 ; numer(%) ; end if; end proc: # R. J. Mathar, Jan 06 2011

T[n_, n_] := -1; T[n_, k_] := 1/n^2 - 1/(k - n)^2; Table[Numerator[T[n, k]], {n, 1, 20}, {k, 0, 2 n - 1}]//Flatten (* G. C. Greubel, Sep 19 2018 *)

A061036 Triangle T(m,n) = denominator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

Original entry on oeis.org

1, 1, 4, 1, 36, 9, 1, 144, 16, 16, 1, 400, 225, 100, 25, 1, 900, 144, 12, 9, 36, 1, 1764, 1225, 784, 441, 196, 49, 1, 3136, 576, 1600, 64, 576, 64, 64, 1, 5184, 3969, 324, 2025, 1296, 81, 324, 81, 1, 8100, 1600, 4900, 225, 100, 400, 900, 25, 100, 1, 12100, 9801

Offset: 1

N. J. A. Sloane, May 26 2001

Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

			Triangle 1/m^2-1/n^2, m >= 1, 1<=n<=m, (i.e. with rows reversed) begins
0
3/4, 0
8/9, 5/36, 0
15/16, 3/16, 7/144, 0
24/25, 21/100, 16/225, 9/400, 0
35/36, 2/9, 1/12, 5/144, 11/900, 0

J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 77.

J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
Eric Weisstein's World of Physics, Balmer Formula
Reinhard Zumkeller, Rows n=..100 of triangle, flattened

Cf. A061035. Rows give A061037-A061050.

Cf. A126252.

import Data.Ratio ((%), denominator)
a061036 n k = a061036_tabl !! (n-1) !! (k-1)
a061036_row = map denominator . balmer where
   balmer n = map (subtract (1 % n ^ 2) . (1 %) . (^ 2)) [n, n-1 .. 1]
a061036_tabl = map a061036_row [1..]
-- Reinhard Zumkeller, Apr 12 2012

t[m_, n_] := Denominator[1/m^2 - 1/n^2]; Table[t[m, n], {n, 1, 12}, {m, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 17 2012 *)

More terms from Naohiro Nomoto, Jul 15 2001

A175779 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2 - 1/n^2.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 8, 1, 0, 7, 3, 15, 1, 0, 9, 16, 21, 24, 1, 0, 11, 5, 1, 2, 35, 1, 0, 13, 24, 33, 40, 45, 48, 1, 0, 15, 7, 39, 3, 55, 15, 63, 1, 0, 17, 32, 5, 56, 65, 8, 77, 80

Offset: 0

Paul Curtz, Dec 04 2010

T(n,0) is set to zero at the pole m=0. T(n,n) is otherwise set to 1 at the pole n=m.
This is the triangle A061035 augmented by a diagonal of 1's.
Essentially the same information is in A120072, A166492, A172157 and A174233.

			The triangle starts in row n=0 with columns 0<=m<=n as:
.1.
.0..1.
.0..3..1.
.0..5..8..1.
.0..7..3.15..1.
.0..9.16.21.24..1.
.0.11..5..1..2.35..1.
.0.13.24.33.40.45.48..1.
.0.15..7.39..3.55.15.63..1.
.0.17.32..5.56.65..8.77.80..1.
.0.19..9.51..4..3.21.91..6.99..1.

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A172157, A166925, A171522 (denominators)

T[n_, n_] := 1; T[n_, k_] := 1/(n - k)^2 - 1/n^2; Table[Numerator[T[n, k]], {n, 0, 20}, {k, 0, n}] // Flatten  (* G. C. Greubel, Sep 19 2018 *)

A174190 Triangle T(n,m) = numerator of 1/n^2-1/(n-m)^2, read by rows.

Original entry on oeis.org

0, 0, -3, 0, -5, -8, 0, -7, -3, -15, 0, -9, -16, -21, -24, 0, -11, -5, -1, -2, -35, 0, -13, -24, -33, -40, -45, -48, 0, -15, -7, -39, -3, -55, -15, -63, 0, -17, -32, -5, -56, -65, -8, -77, -80, 0, -19, -9, -51, -4, -3, -21, -91, -6, -99, 0, -21, -40, -57, -72, -85, -96

Offset: 1

Paul Curtz, Mar 11 2010

The triangle contains the initial values of the array described in A172370.

Ignoring details of column indexing, these are the negated values of A061035.

			The triangle starts in column n=1, rows 0<=m<n as
0;
0,-3;
0,-5,-8;
0,-7,-3,-15;
0,-9,-16,-21,-24;

Comments compactified with reference to A172370, formula and example added - R. J. Mathar, Nov 23 2010

A225233 Triangle read by rows: T(n, k) = (2n + 2 - k)k, for 0 <= k <= n.

Original entry on oeis.org

0, 0, 3, 0, 5, 8, 0, 7, 12, 15, 0, 9, 16, 21, 24, 0, 11, 20, 27, 32, 35, 0, 13, 24, 33, 40, 45, 48, 0, 15, 28, 39, 48, 55, 60, 63, 0, 17, 32, 45, 56, 65, 72, 77, 80, 0, 19, 36, 51, 64, 75, 84, 91, 96, 99, 0, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 0, 23, 44, 63, 80, 95, 108, 119, 128, 135, 140, 143, 0, 25

Offset: 0

R. J. Mathar, May 03 2013

The entries of a row n appear on the diagonal of a square array of dimension n + 1 while filling it with numbers from 0 to n^2 - 1 first along top row and left column, then along 2nd row and 2nd column, 3rd row and 3rd column etc. up to the (single) entry in the n-th row and n-th column. [This may be the preferred order if a set of matrices M is built with requirements on the product M*M.] This vaguely is an alternative to the boustrophedonic re-arrangement of a finite array.
The triangle may also be generated by reading half of each second antidiagonal of the array A003991.
The numbers appear in reverse order as the numerators in the triangle A061035 before they are reduced with the denominators by cancellation of common factors. - Paul Curtz, May 03 2013

			Triangle starts:
[0] 0;
[1] 0,  3;
[2] 0,  5,  8;
[3] 0,  7, 12, 15;
[4] 0,  9, 16, 21, 24;
[5] 0, 11, 20, 27, 32, 35;
[6] 0, 13, 24, 33, 40, 45, 48;
[7] 0, 15, 28, 39, 48, 55, 60, 63;
[8] 0, 17, 32, 45, 56, 65, 72, 77, 80;
[9] 0, 19, 36, 51, 64, 75, 84, 91, 96, 99.
.
The row n = 3, for example, is created by reading the 4 X 4 square array downwards its main diagonal.
0,  1,  3,  5;
2,  7,  8, 10;
4,  9, 12, 13;
6, 11, 14, 15;

Cf. A016061 (row sums), A045944 (central), A005563 (main diagonal).
Cf. A005408 (column k=2), A008586 (column k=3), A016945 (column k=4).
Cf. A003991, A061035.

T := proc(n, k) option remember; if k = 0 then 0 elif k = 1 then 2*n+1 else
T(n, k-1) + T(n-k+1, 1) fi end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jun 02 2021

Offset set to 0 and edited by Peter Luschny, Jun 02 2021

cp's OEIS Frontend

A172370 Mirrored triangle A120072 read by rows.

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A061048 Denominator of 1/49 - 1/n^2.

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A174233 Triangle T(n,k) read by rows: the numerator of 1/n^2 - 1/(k-n)^2, 0 <= k < 2n.

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A061036 Triangle T(m,n) = denominator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

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A175779 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2 - 1/n^2.

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A174190 Triangle T(n,m) = numerator of 1/n^2-1/(n-m)^2, read by rows.

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A225233 Triangle read by rows: T(n, k) = (2*n + 2 - k)*k, for 0 <= k <= n.

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A225233 Triangle read by rows: T(n, k) = (2n + 2 - k)k, for 0 <= k <= n.