A120070 -id:A120070 - cp's OEIS frontend

A098850 a(n) = n*(n + 18).

0, 19, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 360, 403, 448, 495, 544, 595, 648, 703, 760, 819, 880, 943, 1008, 1075, 1144, 1215, 1288, 1363, 1440, 1519, 1600, 1683, 1768, 1855, 1944, 2035, 2128, 2223, 2320, 2419, 2520, 2623, 2728, 2835, 2944, 3055

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A098832, A102928.

a(n-9), n>=10, ninth column (used for the n=9 series of the hydrogen atom) of triangle A120070.

seq(n*(n+18),n=0..52); # Emeric Deutsch, Mar 06 2005

LinearRecurrence[{3, -3, 1}, {0, 19, 40}, 25] (* G. C. Greubel, Jul 29 2016 *)

a(n)=n*(n+18) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (n+9)^2 - 9^2 = n*(n + 18), n>=0.
G.f.: x*(19 - 17*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 17 (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Jul 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(19 + x)*exp(x). (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(18)/18 = A001008(18)/A102928(18) = 14274301/73513440, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1632341/44108064. (End)

More terms from Emeric Deutsch, Mar 06 2005

A120071 a(n) = n*(n+20).

Original entry on oeis.org

0, 21, 44, 69, 96, 125, 156, 189, 224, 261, 300, 341, 384, 429, 476, 525, 576, 629, 684, 741, 800, 861, 924, 989, 1056, 1125, 1196, 1269, 1344, 1421, 1500, 1581, 1664, 1749, 1836, 1925, 2016, 2109, 2204, 2301, 2400, 2501, 2604, 2709, 2816, 2925, 3036, 3149, 3264

Offset: 0

Wolfdieter Lang, Jul 20 2006

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n-10), n >= 11, tenth column (used for the n=10 series of the hydrogen atom) of triangle A120070.
For n*(n+18) see A098850.
Cf. A001008, A001477, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A102928, A120061, A132759, A132760, A132761, A132762, A132765.

seq(sum(4*k-n, k=7..n), n=6..51); # Zerinvary Lajos, Feb 17 2008

s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,21,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

a(n)=n*(n+20) \\ Charles R Greathouse IV, Jan 21 2015

a(n) = (n+10)^2 - 10^2 = n*(n+20), n >= 0.
G.f.: x*(21-19*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 19 (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(20)/20 = A001008(20)/A102928(20) = 11167027/62078016, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 155685007/4655851200. (End)
From Elmo R. Oliveira, Jan 12 2025: (Start)
E.g.f.: exp(x)*x*(21 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

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

Wolfdieter Lang, Jul 20 2006

The corresponding numerator triangle is A120072.

See A120072 and A120070 for more details.

			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;

G. C. Greubel, Rows n = 2..50 of the triangle, flattened
Wofdieter Lang, First ten rows, rationals and more.

Row sums give A120075.

Cf. A120070, A120072, A120074, A120076, A120077, A126252.

[Denominator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023

Table[(1/n^2 - 1/m^2)//Denominator, {m,2,15}, {n,m-1}]//Flatten (* Jean-François Alcover, Sep 16 2013 *)

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

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

A144433 Multiples of 8 interleaved with the sequence of odd numbers >= 3.

Original entry on oeis.org

8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59, 240, 61, 248, 63, 256, 65, 264

Offset: 1

Paul Curtz, Oct 04 2008

For n >= 2, these are the numerators of 1/n^2 - 1/(n+1)^2: A061037(4), A061039(5), A061041(6), A061043(7), A061045(8), A061047(9), A061049(10), etc.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A120070.

[5+3/2*(-1)^(n-1)*(n-1)+3*(-1)^(n-1)+5/2*(n-1): n in [1..70]]; // Vincenzo Librandi, Jul 30 2011

A144433:=n->(n+1)*4^(n mod 2); seq(A144433(n), n=1..100); # Wesley Ivan Hurt, Nov 27 2013

Table[(n + 1)* 4^Mod[n, 2], {n, 100}] (* Wesley Ivan Hurt, Nov 27 2013 *)

x='x+O('x^50); Vec( x*(8+3*x-x^3)/((1-x)^2*(1+x)^2)) \\ G. C. Greubel, Sep 19 2018

a(2*n+1) = A008590(n+1), a(2*n) = A005408(n).
a(2*n+1) + a(2*n+2) = A017281(n+1).
From R. J. Mathar, Apr 01 2009: (Start)
a(n) = 2*a(n-2) - a(n-4).
G.f.: x*(8+3*x-x^3)/((1-x)^2*(1+x)^2). (End)
a(n) = (n + 1) * 4^(n mod 2). - Wesley Ivan Hurt, Nov 27 2013

Edited by R. J. Mathar, Apr 01 2009

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

Wolfdieter Lang, Jul 20 2006

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

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

Jeppe Stig Nielsen, Table of n, a(n) for n = 2..1150

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

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

Table[Denominator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 25 2023 *)

a(n) = denominator(sum(j=1,n-1,1/j^2-1/n^2)) \\ Jeppe Stig Nielsen, Aug 18 2019

a(n) = denominator(sum(j=1,n,1/j^2) - 1/n) \\ Jeppe Stig Nielsen, Aug 18 2019

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

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).

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

A120081 Denominators of expansion for original Debye function (n=3).

Original entry on oeis.org

1, 8, 20, 1, 1680, 1, 90720, 1, 4435200, 1, 207567360, 1, 6538371840000, 1, 423437414400, 1, 67580611338240000, 1, 35763659520196608000, 1, 6155242080686899200000, 1, 117509166994931712000000, 1, 15244417230585693025075200000, 1, 1799300365026394374144000000

Offset: 0

Wolfdieter Lang, Jul 20 2006

Numerators are given in A120080.

See A120070 for the definition of the Debye function D(x)=D(3,x) and references and links.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000367, A002445, A120080.

[Denominator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // G. C. Greubel, May 01 2023

max = 26; Denominator[CoefficientList[Integrate[Normal[Series[(3*(t^3/(Exp[t] -1)))/x^3, {t, 0, max}]], {t, 0, x}], x]] (* Jean-François Alcover, Oct 04 2011 *)
Table[Denominator[3*BernoulliB[n]/((n+3)*n!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

def A120081(n): return denominator(3*bernoulli(n)/((n+3)*factorial(n)))
[A120081(n) for n in range(51)] # G. C. Greubel, May 01 2023

a(n) = denominator(r(n)), with r(n) = [x^n]( 1 - 3*x/8 + Sum_{k >= 1} (3*B(2*k)/((2*k+3)*(2*k)!))*x^(2*k) ), where B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator( 3*Bernoulli(n)/((n+3)*n!) ), n >= 0. - G. C. Greubel, May 01 2023

A120074 Row sums of triangle A120072 (numerator triangle for H atom spectrum).

Original entry on oeis.org

3, 13, 25, 70, 54, 203, 197, 340, 303, 825, 445, 1378, 892, 1221, 1565, 3128, 1545, 4389, 2427, 3592, 3688, 7843, 3589, 8420, 6191, 9097, 7135, 15834, 5774, 19375, 12493, 14814, 14147, 19647, 12264, 33078

Offset: 2

Wolfdieter Lang, Jul 20 2006

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A120070, A120072, A120073, A120075, A120076, A120077.

A120072:= func< n,k | Numerator(1/k^2 - 1/n^2) >;
[(&+[A120072(n,k): k in [1..n-1]]): n in [2..50]]; // G. C. Greubel, Apr 24 2023

Table[Sum[1/n^2 - 1/m^2 //Numerator, {n,m-1}], {m,2,40}]  (* Jean-François Alcover, Sep 16 2013 *)

def A120072(n,k): return numerator(1/k^2 - 1/n^2)
[sum(A120072(n,k) for k in range(1,n)) for n in range(2,51)] # G. C. Greubel, Apr 24 2023

a(n) = Sum_{k=1..n-1} A120072(n,k) for n >= 2.

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

Wolfdieter Lang, Jul 20 2006

G. C. Greubel, Table of n, a(n) for n = 2..1000

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

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

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 *)

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

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

Wolfdieter Lang, Jul 20 2006

The corresponding denominators are given by A120077.

See the W. Lang link under A120072 for more details.

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

G. C. Greubel, Table of n, a(n) for n = 2..1000

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

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

Table[Numerator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 24 2023 *)

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

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).

cp's OEIS Frontend

A098850 a(n) = n*(n + 18).

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A120071 a(n) = n*(n+20).

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A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

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A120073 Denominator triangle for hydrogen spectrum rationals.

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A144433 Multiples of 8 interleaved with the sequence of odd numbers >= 3.

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

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