A105371 -id:A105371 - cp's OEIS frontend

A171522 Denominator of 1/n^2-1/(n+2)^2.

0, 9, 16, 225, 144, 1225, 576, 3969, 1600, 9801, 3600, 20449, 7056, 38025, 12544, 65025, 20736, 104329, 32400, 159201, 48400, 233289, 69696, 330625, 97344, 455625, 132496, 613089, 176400, 808201, 230400, 1046529, 295936, 1334025, 374544, 1677025, 467856

Offset: 0

Paul Curtz, Dec 11 2009

This is the third column in the table of denominators of the hydrogenic spectra (the main diagonal A147560):
0,   0,   0,   0,   0,   0,   0,   0... A000004
1,   4,   9,  16,  25,  36,  49,  64... A000290
1,  36,  16, 100,   9, 196,  64, 324... A061038
1, 144, 225,  12, 441, 576,  81, 900... A061040
1, 400, 144, 784,  64,1296, 400,1936... A061042
1, 900 1225,1600,2025, 100,3025,3600... A061044
1,1764, 576, 324, 225,4356,  48,6084... A061046
1,3136,3969,4900,5929,7056,8281, 196... A061048.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A105371. Bisections: A060300, A069075.

A171522 := proc(n) if n = 0 then 0 else lcm(n+2,n) ; %^2 ; end if ; end:
seq(A171522(n),n=0..70) ; # R. J. Mathar, Dec 15 2009

a[n_] := If[EvenQ[n], (n*(n+2))^2/4, (n*(n+2))^2]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jun 13 2017 *)

concat(0, Vec(x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5) + O(x^100))) \\ Colin Barker, Nov 05 2014

a(n) = (A066830(n+1))^2.
a(n) = -((-5+3*(-1)^n)*n^2*(2+n)^2)/8. - Colin Barker, Nov 05 2014
G.f.: x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5). - Colin Barker, Nov 05 2014

Edited and extended by R. J. Mathar, Dec 15 2009

A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 5, 0, -13, -34, -55, -55, 0, 144, 377, 610, 610, 0, -1597, -4181, -6765, -6765, 0, 17711, 46368, 75025, 75025, 0, -196418, -514229, -832040, -832040, 0, 2178309, 5702887, 9227465, 9227465, 0, -24157817, -63245986, -102334155, -102334155

Offset: 0

Paul Curtz, May 04 2008

sign

Obeys also the recurrence a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5), so the sequence is identical to its fifth differences (cf. A135356). a(n) = A138110(0,n): if A138110 is interpreted as an array with five rows, this is the top row.
The first differences are represented by A100334(n-1).
The 2nd differences are represented by A103311(n).
The 3rd differences are essentially represented by -A138003(n-2).
The 4th differences are represented by -A105371(n).
A102312 contains the absolute values of the terms which occur in pairs, for example a(5)=a(6)=5=A102312(1), a(10)=a(11)= -55 = -A102312(2).
Inverse BINOMIAL transform yields two zeros followed by A105384. - R. J. Mathar, Jul 04 2008

Index entries for linear recurrences with constant coefficients, signature (3, -4, 2, -1).

Cf. A138003, A103311, A105371.

CoefficientList[Series[x^3/(1-3x+4x^2-2x^3+x^4),{x,0,45}],x] (* or *) LinearRecurrence[{3,-4,2,-1},{0,0,0,1},45] (* Harvey P. Dale, Jun 22 2011 *)

O.g.f.: x^3/(1-3x+4x^2-2x^3+x^4). - R. J. Mathar, Jul 04 2008

Edited and extended by R. J. Mathar, Jul 04 2008

A138003 Binomial transform of 1, 1, 0, -1, -1 (periodically continued).

Original entry on oeis.org

1, 2, 3, 3, 0, -8, -21, -34, -34, 0, 89, 233, 377, 377, 0, -987, -2584, -4181, -4181, 0, 10946, 28657, 46368, 46368, 0, -121393, -317811, -514229, -514229, 0, 1346269, 3524578, 5702887, 5702887, 0, -14930352, -39088169, -63245986, -63245986

Offset: 0

Paul Curtz, May 01 2008

Shares many elements with A103311, as already indicated by the similarity of the two generating functions. First differences are essentially in A105371. - R. J. Mathar, May 02 2008

The longer of the two recurrences ensures that the sequence (like A133476) equals its 5th differences. - R. J. Mathar, May 02 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,2,-1).

Cf. A129929.

LinearRecurrence[{3,-4,2,-1},{1,2,3,3},50] (* Paolo Xausa, Dec 05 2023 *)

a=[1,2,3,3];for(i=1,99,a=concat(a,3*a[#a]-4*a[#a-1]+2*a[#a-2]-a[#a-3]));a \\ Charles R Greathouse IV, Jun 02 2011

From R. J. Mathar, May 02 2008: (Start)
O.g.f.: (x^2-x+1)/(x^4-2*x^3+4*x^2-3*x+1).
a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5).
a(n) = 3a(n-1)-4a(n-2)+2a(n-3)-a(n-4). (End)

Edited by R. J. Mathar, May 02 2008

A138110 Table T(d,n) read column by column: the n-th term in the sequence of the d-th differences of A138112, d=0..4.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 2, 0, -2, -1, 3, 2, -2, -3, 0, 5, 0, -5, -3, 3, 5, -5, -8, 0, 8, 0, -13, -8, 8, 13, -13, -21, 0, 21, 13, -34, -21, 21, 34, 0, -55, 0, 55, 34, -34, -55, 55, 89, 0, -89, 0, 144, 89, -89, -144, 144, 233, 0, -233, -144, 377, 233, -233, -377, 0, 610, 0, -610, -377, 377

Offset: 0

Paul Curtz, May 04 2008

Ignoring signs, the sequence contains A000045(2)=1 ten times and each of the following Fibonacci numbers A000045(i>2) four times.

			All 5 rows of the table T(d,n) are:
.0,.0,.0,.1,.3,.5,.5,..0,-13,-34,-55,-55,...0,.144,...
.0,.0,.1,.2,.2,.0,-5,-13,-21,-21,..0,.55,.144,.233,...
.0,.1,.1,.0,-2,-5,-8,.-8,..0,.21,.55,.89,..89,...0,...
.1,.0,-1,-2,-3,-3,.0,..8,.21,.34,.34,..0,.-89,-233,...
-1,-1,-1,-1,.0,.3,.8,.13,.13,..0,-34,-89,-144,-144,...

Cf. A102312 (A049666), A099100, A134489, A134490, A134491.

T(0,n)=A138112(n). T(d,n)= T(d-1,n+1)-T(d-1,n), d=1..4.
T(1,n)=A100334(n-1). T(2,n)=A103311(n). T(3,n) = -A138003(n-2). T(4,n)= -A105371(n).
sum_(d=0..4) T(d,n)=0 (columns sum to zero).

Edited by R. J. Mathar, Jul 04 2008

A171373 Binomial transform of A171372.

Original entry on oeis.org

1, 6, 16, 36, 76, 152, 292, 552, 1052, 2052, 4104, 8344, 17044, 34664, 69904, 139808, 278108, 552268, 1098148, 2189908, 4379816, 8776356, 17596496, 35263836, 70598516, 141197032, 282208592, 563931612, 1127077552, 2253369432, 4506738864, 9015534644

Offset: 0

Paul Curtz, Dec 07 2009

nonn

The recurrence shows that the sequence and its successive differences are identical to their fifth differences (see A135356).

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 2).

LinearRecurrence[{5,-10,10,-5,2},{1,6,16,36,76},40] (* Harvey P. Dale, Dec 09 2013 *)

a(n+1)-2*a(n) = 4*A105371(n-1) = 4*A138110(4,n).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+2*a(n-5).
G.f.: (1+x-4*x^2+6*x^3+x^4)/((1-2*x)*(x^4-2*x^3+4*x^2-3*x+1)).

Edited and extended by R. J. Mathar, Dec 15 2009, Mar 02 2010

A171372 a(n) = Numerator of 1/(2n)^2 - 1/(3n)^2 for n > 0, a(0) = 1.

Original entry on oeis.org

1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5

Offset: 0

Paul Curtz, Dec 07 2009

The diagonal of a table of numerators of the Rydberg-Ritz spectrum of hydrogen:
  1,  1,  1,   1,  1,   1,  1,   1,  1,   1,   1, ... A000012
  0,  5,  3,  21,  2,  45, 15,  77,  6, 117,  35, ... A061037
  0,  9,  5,  33,  3,  65, 21, 105,  1, 153,  45, ... A061041
  0, 13,  7,   5,  4,  85,  1, 133, 10,   7,  55, ... A061045
  0, 17,  9,  57,  5, 105, 33, 161,  3, 225,  65, ... A061049
  0, 21, 11,  69,  6,   1, 39, 189, 14, 261,   3, ...
  0, 25, 13,   1,  7, 145,  5, 217,  1,  11,  85, ...
  0, 29, 15,  93,  8, 165, 51,   5, 18, 333,  95, ...
  0, 33, 17, 105,  9, 185, 57, 273,  5, 369, 105, ...
  0, 37, 19,  13, 10, 205,  7, 301, 22,   5, 115, ...
  0, 41, 21, 129, 11,   9, 69, 329,  3, 441,   1, ...
In that respect, constructed similar to A144437.

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

Cf. A171373 (binomial transform), A171408, A105371.

[1] cat [Numerator(5/(6*n)^2): n in [1..100]]; // G. C. Greubel, Sep 20 2018

Table[If[n==0,1,Numerator[5/(6*n)^2]], {n,0,100}] (* G. C. Greubel, Sep 20 2018 *)

concat([1], vector(100, n, numerator(5/(6*n)^2))) \\ G. C. Greubel, Sep 20 2018

a(n) = numerator of 5/(6*n)^2 .
Period 5: repeat [1,5,5,5,5].
G.f.: (1 + 5*x + 5*x^2 + 5*x^3 + 5*x^4)/((1-x)*(1 + x + x^2 + x^3 + x^4)).
a(n) = 1 + 4*sign(n mod 5). - Wesley Ivan Hurt, Sep 26 2018
a(n) = (21-8*cos(2*n*Pi/5)-8*cos(4*n*Pi/5))/5. - Wesley Ivan Hurt, Sep 27 2018

Edited by R. J. Mathar, Dec 15 2009

A171408 a(n) = A171373(n+1) - 2*A171373(n).

Original entry on oeis.org

4, 4, 4, 4, 0, -12, -32, -52, -52, 0, 136, 356, 576, 576, 0, -1508, -3948, -6388, -6388, 0, 16724, 43784, 70844, 70844, 0, -185472, -485572, -785672, -785672, 0, 2056916, 5385076, 8713236, 8713236, 0, -22811548, -59721408, -96631268, -96631268, 0, 252983944, 662320564

Offset: 0

Paul Curtz, Dec 08 2009

sign

The least significant digits have a period of length 20. Another period (not the same but of the same length, as this a sequence contains negative numbers) is defined reading the sequence modulo 10.

Index entries for linear recurrences with constant coefficients, signature (3,-4,2,-1)

Cf. A138112, A105371.

a(n)=4*A105371(n-1), n>0.

a(n)= 3*a(n-1) -4*a(n-2) +2*a(n-3) -a(n-4). G.f.: 4*(1-2*x+2*x^2)/(1-3*x+4*x^2-2*x^3+x^4).

Edited and extended by R. J. Mathar, Mar 02 2010

cp's OEIS Frontend

A171522 Denominator of 1/n^2-1/(n+2)^2.

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A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

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A138003 Binomial transform of 1, 1, 0, -1, -1 (periodically continued).

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A138110 Table T(d,n) read column by column: the n-th term in the sequence of the d-th differences of A138112, d=0..4.

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A171373 Binomial transform of A171372.

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A171372 a(n) = Numerator of 1/(2*n)^2 - 1/(3*n)^2 for n > 0, a(0) = 1.

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A171408 a(n) = A171373(n+1) - 2*A171373(n).

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A171372 a(n) = Numerator of 1/(2n)^2 - 1/(3n)^2 for n > 0, a(0) = 1.