A061050 -id:A061050 - cp's OEIS frontend

A061046 Denominator of 1/36 - 1/n^2.

36, 9, 12, 144, 900, 1, 1764, 576, 324, 225, 4356, 48, 6084, 441, 300, 2304, 10404, 81, 12996, 3600, 196, 1089, 19044, 192, 22500, 1521, 2916, 7056, 30276, 75, 34596, 9216, 484, 2601, 44100, 1296, 49284, 3249, 2028, 14400, 60516, 147, 66564, 17424, 8100, 4761, 79524, 256, 86436, 5625, 3468, 24336, 101124, 729, 108900, 28224, 4332, 7569, 125316, 400, 133956, 8649, 15876, 36864, 152100, 363, 161604, 41616, 6348, 11025, 181476, 5184

Offset: 1

N. J. A. Sloane, May 26 2001

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

See A061045 for numerators and further information. Cf. A061035-A061050.

import Data.Ratio ((%), denominator)
a061046 = denominator . (1 % 36 -) . recip . (^ 2) . fromIntegral
-- Reinhard Zumkeller, Jan 06 2014

Denominator[1/36-1/Range[80]^2] (* Harvey P. Dale, Feb 06 2012 *)

for(n=6,50, print1(denominator(1/6^2 - 1/n^2), ", ")) \\ G. C. Greubel, Jul 07 2017

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

A165441 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2.

Original entry on oeis.org

1, 4, 4, 9, 1, 9, 16, 36, 36, 16, 25, 16, 1, 16, 25, 36, 100, 144, 144, 100, 36, 49, 9, 225, 1, 225, 9, 49, 64, 196, 12, 400, 400, 12, 196, 64, 81, 64, 441, 144, 1, 144, 441, 64, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 25, 81, 64, 1225, 1, 1225, 64, 81, 25, 121

Offset: 1

Paul Curtz, Sep 19 2009

A synopsis of the denominators of the transitions in the Rydberg-Ritz spectrum of hydrogenic atoms.

			.1,   4,   9,   16,   25,   36,   49,   64,   81, ... A000290
.4,   1,  36,   16,  100,    9,  196,   64,  324, ... A061038
.9,  36,   1,  144,  225,   12,  441,  576,   81, ... A061040
16,  16, 144,    1,  400,  144,  784,   64, 1296, ... A061042
25, 100, 225,  400,    1,  900, 1225, 1600, 2025, ... A061044
36,   9,  12,  144,  900,    1, 1764,  576,  324, ... A061046
49, 196, 441,  784, 1225, 1764,    1, 3136, 3969, ... A061048
64,  64, 576,   64, 1600,  576, 3136,    1, 5184, ... A061050
81, 324,  81, 1296, 2025,  324, 3969, 5184,    1, ...

Alois P. Heinz, Table of n, a(n) for n = 1..1035

T:= (k,n)-> denom(1/min (n,k)^2 -1/max (n, k)^2):
seq(seq(T(k, d-k), k=1..d-1), d=2..12);

T[n_, k_] := Denominator[1/Min[n, k]^2 - 1/Max[n, k]^2];
Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 04 2020 *)

T(n,k) = A165727(n,k).

Edited by R. J. Mathar, Feb 27 2010, Mar 03 2010

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

A165727 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2 with T(0,n) = T(k,0) = 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 4, 0, 0, 9, 1, 9, 0, 0, 16, 36, 36, 16, 0, 0, 25, 16, 1, 16, 25, 0, 0, 36, 100, 144, 144, 100, 36, 0, 0, 49, 9, 225, 1, 225, 9, 49, 0, 0, 64, 196, 12, 400, 400, 12, 196, 64, 0, 0, 81, 64, 441, 144, 1, 144, 441, 64, 81, 0, 0, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 0

Offset: 0

Paul Curtz, Sep 25 2009

A synopsis of the denominators of the transitions in the Rydberg-Ritz spectrum of hydrogenic atoms.

			0,  0,   0,   0,    0,    0,    0,    0,    0,    0, ... A000004
0,  1,   4,   9,   16,   25,   36,   49,   64,   81, ... A000290
0,  4,   1,  36,   16,  100,    9,  196,   64,  324, ... A061038
0,  9,  36,   1,  144,  225,   12,  441,  576,   81, ... A061040
0, 16,  16, 144,    1,  400,  144,  784,   64, 1296, ... A061042
0, 25, 100, 225,  400,    1,  900, 1225, 1600, 2025, ... A061044
0, 36,   9,  12,  144,  900,    1, 1764,  576,  324, ... A061046
0, 49, 196, 441,  784, 1225, 1764,    1, 3136, 3969, ... A061048
0, 64,  64, 576,   64, 1600,  576, 3136,    1, 5184, ... A061050
0, 81, 324,  81, 1296, 2025,  324, 3969, 5184,    1, ...

Alois P. Heinz, Table of n, a(n) for n = 0..1034

Cf. A165441 (top row and left column removed)

T:= (k,n)-> `if` (n=0 or k=0, 0, denom (1/min (n,k)^2 -1/max (n, k)^2)):
seq (seq (T (k, d-k), k=0..d), d=0..11);

Edited by R. J. Mathar, Feb 27 2010, Mar 03 2010

A152018 Denominator of 1/n^2-1/(3n)^2 or of 8/(9n^2).

Original entry on oeis.org

9, 9, 81, 18, 225, 81, 441, 72, 729, 225, 1089, 162, 1521, 441, 2025, 288, 2601, 729, 3249, 450, 3969, 1089, 4761, 648, 5625, 1521, 6561, 882, 7569, 2025, 8649, 1152, 9801, 2601, 11025, 1458, 12321, 3249, 13689, 1800, 15129, 3969, 16641, 2178, 18225

Offset: 1

Paul Curtz, Nov 20 2008

nonn

The associated terms of the n-th main series of the Hydrogen energy spectrum are A000290(3), A061038(6), A061040(9), A061042(12), A061044(15), A061046(18), A061048(21), A061050(24), etc.

All numbers are multiples of 9.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1).

Cf. A143025 with a similar principle of construction.

Cf. A291050.

Denominator/@(8/(9Range[50]^2))  (* Harvey P. Dale, Mar 15 2011 *)

Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Sep 14 2022

Stratified definition, corrected indices, extended, R. J. Mathar, Dec 10 2008

cp's OEIS Frontend

A061046 Denominator of 1/36 - 1/n^2.

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

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A165441 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2.

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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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A165727 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2 with T(0,n) = T(k,0) = 0.

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A152018 Denominator of 1/n^2-1/(3n)^2 or of 8/(9n^2).

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