cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A120072 Numerator triangle for hydrogen spectrum rationals.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

Frequencies or energies of the spectral lines of the hydrogen (H) atom are given, according to quantum theory, by r(m,n)*3.287*PHz (1 Peta Hertz= 10^15 s^{-1}) or r(m,n)*13.599 eV (electron Volts), respectively. The wave lengths are lambda(m,n) = (1/r(m,n))* 91.196 nm (all decimals rounded). See the W. Lang link for more details.
The spectral series for n=1,2,...,7, m>=n+1, are named after Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, respectively.
The corresponding denominator triangle is A120073.
The rationals are r(m,n):= a(m,n)/A120073(m,n) = A120070(m,n)/(m^2*n^2) = 1/ n^2 - 1/m^2 and they are given in lowest terms.

Examples

			For the rational triangle see W. Lang link.
Numerator triangle begins as:
   3;
   8,  5;
  15,  3,  7;
  24, 21, 16,  9;
  35,  2,  1,  5, 11;
  48, 45, 40, 33, 24, 13;
  63, 15, 55,  3, 39,  7, 15;
  80, 77,  8, 65, 56,  5, 32, 17;
  99,  6, 91, 21,  3,  4, 51,  9, 19;

Links

Crossrefs

Row sums give A120074.
Row sums of r(m, n) triangle give A120076(m)/A120077(m), m>=2.
Cf. A120070, A120073, A120075, A126252.

Programs

  • Magma
    [Numerator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023
  • Mathematica
    Table[1/n^2 - 1/m^2, {m,2,12}, {n,m-1}]//Flatten//Numerator (* Jean-François Alcover, Sep 16 2013 *)
  • SageMath
    def A120072(n,k): return numerator(1/k^2 - 1/n^2)
flatten([[A120072(n,k) for k in range(1,n)] for n in range(2,19)]) # G. C. Greubel, Apr 24 2023

Formula

a(m,n) = numerator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.
The g.f.s for the columns n=1,..,10 of triangle r(m,n) = a(m, n) / A120073(m, n), m >= 2, 1 <= n <= m-1, are given in the W. Lang link.

A120073 Denominator triangle for hydrogen spectrum rationals.

Original entry on oeis.org

4, 9, 36, 16, 16, 144, 25, 100, 225, 400, 36, 9, 12, 144, 900, 49, 196, 441, 784, 1225, 1764, 64, 64, 576, 64, 1600, 576, 3136, 81, 324, 81, 1296, 2025, 324, 3969, 5184, 100, 25, 900, 400, 100, 225, 4900, 1600, 8100, 121, 484, 1089, 1936, 3025, 4356, 5929, 7744, 9801, 12100
Offset: 2

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

The corresponding numerator triangle is A120072.
See A120072 and A120070 for more details.

Examples

			For the rational triangle see W. Lang link.
Denominator triangle begins as:
    4;
    9,  36;
   16,  16, 144;
   25, 100, 225,  400;
   36,   9,  12,  144,  900;
   49, 196, 441,  784, 1225, 1764;
   64,  64, 576,   64, 1600,  576, 3136;
   81, 324,  81, 1296, 2025,  324, 3969, 5184;
  100,  25, 900,  400,  100,  225, 4900, 1600, 8100;

Links

Crossrefs

Row sums give A120075.
Cf. A120070, A120072, A120074, A120076, A120077, A126252.

Programs

  • Magma
    [Denominator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023
  • Mathematica
    Table[(1/n^2 - 1/m^2)//Denominator, {m,2,15}, {n,m-1}]//Flatten (* Jean-François Alcover, Sep 16 2013 *)
  • SageMath
    def A120073(n,k): return denominator(1/k^2 - 1/n^2)
flatten([[A120073(n,k) for k in range(1,n)] for n in range(2,19)]) # G. C. Greubel, Apr 24 2023

Formula

a(m,n) = denominator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.

A120077 Denominators of row sums of rational triangle A120072/A120073.

Original entry on oeis.org

4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 2167695039654144, 1548353599752960, 221193371393280, 117011293467045120
Offset: 2

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

The first 19 terms coincide with A007407(n), for n>=2. However a(20) = 2167695039654144 and A007407(20) = 10838475198270720 = 5*a(20). Also a(21) = 1548353599752960 and A007407(21) = 221193371393280 = a(21)/7. From n = 22 up to at least n = 100 (checked) both sequences coincide again.
See the W. Lang link under A120072 for more details.
The corresponding numerators are given by A120076.
The n for which a(n) differs from A007407(n) are given by A309829. - Jeppe Stig Nielsen, Aug 18 2019

Examples

			The rationals A120076(m)/a(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ... ).

Links

Crossrefs

Cf. A007407, A120070, A120072, A120073, A120074, A120075, A120076, A309829.

Programs

  • Magma
    A120077:= func< n | Denominator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120077(n): n in [2..30]]; // G. C. Greubel, Apr 25 2023
  • Mathematica
    Table[Denominator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 25 2023 *)
  • PARI
    a(n) = denominator(sum(j=1,n-1,1/j^2-1/n^2)) \\ Jeppe Stig Nielsen, Aug 18 2019
  • PARI
    a(n) = denominator(sum(j=1,n,1/j^2) - 1/n) \\ Jeppe Stig Nielsen, Aug 18 2019
  • SageMath
    def A120077(n): return denominator(harmonic_number(n,2) - 1/n)
[A120077(n) for n in range(2,31)] # G. C. Greubel, Apr 25 2023

Formula

a(n) = denominator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link under A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).

Extensions

a(21)-a(23) from Jeppe Stig Nielsen, Aug 18 2019

A120075 Row sums of triangle A120073 (denominator triangle for H atom spectrum).

Original entry on oeis.org

4, 45, 176, 750, 1101, 4459, 6080, 13284, 16350, 46585, 33954, 109850, 92463, 142705, 198400, 432344, 255096, 761349, 500355, 824866, 925529, 2007555, 1044616, 2612500, 2158130, 3301641, 2848741
Offset: 2

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Links

Crossrefs

Cf. A120070, A120072, A120073, A120074, A120076, A120077.

Programs

  • Magma
    A120073:= func< n,k | Denominator(1/k^2 - 1/n^2) >;
[(&+[A120073(n,k): k in [1..n-1]]): n in [2..50]]; // G. C. Greubel, Apr 24 2023
  • Mathematica
    A120075[n_]:= Sum[Denominator[1/k^2 -1/n^2], {k,n-1}];
Table[A120075[n], {n,2,50}] (* G. C. Greubel, Apr 24 2023 *)
  • SageMath
    def A120073(n,k): return denominator(1/k^2 - 1/n^2)
[sum(A120073(n,k) for k in range(1,n)) for n in range(2,51)] # G. C. Greubel, Apr 24 2023

Formula

a(n) = Sum_{k=1..n-1} A120073(n,k), for n >= 2.

A120076 Numerators of row sums of rational triangle A120072/A120073.

Original entry on oeis.org

3, 37, 169, 4549, 4769, 241481, 989549, 9072541, 1841321, 225467009, 227698469, 38801207261, 39076419341, 196577627041, 790503882349, 229526961468061, 230480866420061, 83512167402400421, 3351610394325821
Offset: 2

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

The corresponding denominators are given by A120077.
See the W. Lang link under A120072 for more details.

Examples

			The rationals a(m)/A120077(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ...).

Links

Crossrefs

Cf. A120070, A120072, A120073, A120074, A120075, A120077.

Programs

  • Magma
    A120076:= func< n | Numerator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120076(n): n in [2..30]]; // G. C. Greubel, Apr 24 2023
  • Mathematica
    Table[Numerator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 24 2023 *)
  • SageMath
    def A120076(n): return numerator(harmonic_number(n,2) - 1/n)
[A120076(n) for n in range(2,31)] # G. C. Greubel, Apr 24 2023

Formula

a(n) = numerator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m >= 2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link in A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).
Showing 1-5 of 5 results.

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