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 1-10 of 31 results. Next

A144454 First trisection of A061039.

Original entry on oeis.org

0, 1, 8, 5, 8, 35, 16, 7, 80, 11, 40, 143, 56, 65, 224, 85, 32, 323, 40, 133, 440, 161, 176, 575, 208, 75, 728, 87, 280, 899, 320, 341, 1088, 385, 136, 1295, 152, 481, 1520, 533, 560, 1763, 616, 215, 2024, 235, 736, 2303, 800, 833, 2600, 901, 312, 2915, 336, 1045
Offset: 1

Views

Author

Paul Curtz, Oct 07 2008

Keywords

Comments

Numerator of (n^2-1)/(9n^2). Denominator is A147650(n).
Terms alternate between even and odd. The sequence modulo 9 reads (0, 1, 8, 5, 8, 8, 7, 7, 8, 2, 4, 8, 2, 2, 8, 4, 5, ...) (Is there a meaning to the interpretation as the constant 0.1858877824822845...?) The first appearance of 3 (mod 9) is at a(26)=75, the second at a(55)=336. The first appearance of 6 (mod 9) is at a(28)=87, the second at a(53)=312.
a(n) also gives the numerator of (n^2 - 1)/(3*((2*n)^2 - 1)) =: r(n-1), with denominators A300295(n-1), for n >= 1. For the proof see a comment in A300295; also for details on r(n) with the Jolley reference. - Wolfdieter Lang, Mar 15 2018
a(n) is also the numerator of Sum_{k=0..n} (1/((2*k-3)(2*k-1)*(2*k+1))). This summation is an offset adjusted form of formula 209 in Jolley's "Summation of Series". The closed form is given in the Lang comment above. - Gary Detlefs, Mar 15 2018

Examples

			The rationals (n^2 - 1)/(9*n^2) begin: 0/1, 1/12, 8/81, 5/48, 8/75, 35/324, 16/147, 7/64, 80/729, 11/100, 40/363, 143/1296, 56/507, 65/588, ... - _Wolfdieter Lang_, Mar 15 2018

References

  • L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 40, 41.

Links

Crossrefs

Cf. A061039, A147650, A300295.

Programs

  • Maple
    P:=n-> sum(1/((2*k-3)*(2*k-1)*(2*k+1)), k = 0 n); seq(numer(P(i))i,=1..50) # Gary Detlefs, Mar 15 2018
  • Mathematica
    Numerator[Table[((n-1)(n+1))/(9n^2),{n,60}]] (* or *) LinearRecurrence[ {0,0,0,0, 0,0,0,0,3,0,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,0,1}, {0,1,8,5,8,35,16,7,80,11,40, 143,56,65,224,85,32,323,40,133,440,161,176,575,208,75,728}, 60] (* Harvey P. Dale, Jan 16 2013 *)
  • PARI
    concat(0, Vec(x^2*(1 + 8*x + 5*x^2 + 8*x^3 + 35*x^4 + 16*x^5 + 7*x^6 + 80*x^7 + 11*x^8 + 37*x^9 + 119*x^10 + 41*x^11 + 41*x^12 + 119*x^13 + 37*x^14 + 11*x^15 + 83*x^16 + 7*x^17 + 16*x^18 + 35*x^19 + 8*x^20 + 5*x^21 + 8*x^22 + x^23 - x^25) / ((1 - x)^3*(1 + x + x^2)^3*(1 + x^3 + x^6)^3) + O(x^60))) \\ Colin Barker, Mar 15 2018
  • PARI
    a(n) = numerator(1/9-1/(3*n)^2); \\ Altug Alkan, Mar 15 2018
  • Sage
    [numerator((1 - 1/n^2)/9) for n in (1..100)] # G. C. Greubel, Mar 07 2022

Formula

a(n) = A061039(3*n).
For n > 27, a(n) = 3*a(n-9) - 3*a(n-18) + a(n-27). - Harvey P. Dale, Jan 16 2013
a(n) = (n^2 - 1)/9 if n == 1 (mod 9) or == 8 (mod 9). For other n: a(n) = (n^2 - 1)/3 if n == 1 (mod 3) or == 2(mod 3), and a(n) = n^2 - 1 if n == 0 (mod 3). The proof uses the first comment and gcd(n^2-1, n^2) = 1. - Wolfdieter Lang, Mar 15 2018
G.f.: x^2*(1 + 8*x + 5*x^2 + 8*x^3 + 35*x^4 + 16*x^5 + 7*x^6 + 80*x^7 + 11*x^8 + 37*x^9 + 119*x^10 + 41*x^11 + 41*x^12 + 119*x^13 + 37*x^14 + 11*x^15 + 83*x^16 + 7*x^17 + 16*x^18 + 35*x^19 + 8*x^20 + 5*x^21 + 8*x^22 + x^23 - x^25) / ((1 - x)^3*(1 + x + x^2)^3*(1 + x^3 + x^6)^3). - Colin Barker, Mar 15 2018

Extensions

Edited and extended by R. J. Mathar, Oct 24 2008

A144448 First bisection of A061039.

Original entry on oeis.org

0, 16, 40, 8, 112, 160, 8, 280, 352, 16, 520, 616, 80, 832, 952, 40, 1216, 1360, 56, 1672, 1840, 224, 2200, 2392, 32, 2800, 3016, 40, 3472, 3712, 440, 4216, 4480, 176, 5032, 5320, 208, 5920, 6232, 728, 6880, 7216, 280, 7912, 8272, 320, 9016, 9400, 1088, 10192, 10600, 136, 11440, 11872, 152
Offset: 1

Views

Author

Paul Curtz, Oct 06 2008

Keywords

Comments

From Paschen spectrum of hydrogen.
All numbers are multiples of 8.

Links

Crossrefs

Cf. A061039, A178978.

Programs

  • Mathematica
    Table[Numerator[1/3^2 - 1/(2*n+1)^2], {n,100}] (* G. C. Greubel, Mar 06 2022 *)
  • Sage
    [numerator(1/9 -1/(2*n+1)^2) for n in (1..100)] # G. C. Greubel, Mar 06 2022

Formula

a(n) = A061039(2*n+1).
From G. C. Greubel, Mar 06 2022: (Start)
a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81).
a(n) = 8*A178978(n). (End)

Extensions

Formula index corrected, extended by R. J. Mathar, Dec 02 2008

A146322 a(n) = A061039(n) mod 9.

Original entry on oeis.org

0, 7, 7, 1, 4, 1, 8, 1, 4, 5, 7, 7, 8, 4, 1, 8, 1, 4, 7, 7, 7, 7, 4, 1, 8, 1, 4, 2, 7, 7, 4, 4, 1, 8, 1, 4, 2, 7, 7, 2, 4, 1, 8, 1, 4, 4, 7, 7, 5, 4, 1, 8, 1, 4, 4, 7, 7, 7, 4, 1, 8, 1, 4, 8, 7, 7, 5, 4, 1, 8, 1, 4, 1, 7, 7, 3, 4, 1, 8, 1, 4, 6, 7, 7, 1, 4, 1, 8, 1, 4, 5, 7, 7, 8, 4, 1, 8, 1, 4, 7, 7, 7, 1, 4, 1, 8, 1, 4, 8, 7, 7, 4, 4, 1, 8, 1, 4, 2, 7, 7
Offset: 3

Views

Author

Paul Curtz, Oct 30 2008

Keywords

Comments

Is the number 0.77141814577... rational? Note groups of 4,1,8 at positions 7, 16, 25, 34, 43 etc.; also 7,7 positions 4, 13, 21, 31, 40, 49 etc.
From Paschen spectrum of hydrogen.
The number 6 appears at positions n=84, 159, 327, 402 etc.; the number 3 at n=78, 165, 321 etc; the number 0 at n=3, 240, 246 etc. - R. J. Mathar, Feb 28 2009
Starting with a(7) the pattern {4, 1, 8, 1, 4, b(n), 7, 7, c(n)} is repeated with b(n) and c(n) containing numbers zero to nine. - G. C. Greubel, Mar 08 2022

Links

Crossrefs

Cf. A061039.

Programs

  • Mathematica
    Mod[Numerator[1/9 - 1/(Range[3, 150])^2], 9] (* G. C. Greubel, Mar 08 2022 *)
  • Sage
    [numerator(1/9 - 1/n^2)%9 for n in (3..150)] # G. C. Greubel, Mar 08 2022

Formula

a(n) = A061039(n) mod 9.

A144453 a(n) = A061039(8*n+5).

Original entry on oeis.org

16, 160, 16, 832, 1360, 224, 2800, 3712, 176, 5920, 7216, 320, 10192, 11872, 1520, 15616, 17680, 736, 22192, 24640, 336, 29920, 32752, 3968, 38800, 42016, 560, 48832, 52432, 2080, 60016, 64000, 7568, 72352, 76720, 3008, 85840, 90592, 3536, 100480, 105616
Offset: 0

Views

Author

Paul Curtz, Oct 07 2008

Keywords

Comments

Numerators of 16*(n+1)*(4*n+1)/(9*(8*n+5)^2), so all numbers are multiples of 16 because the denominator is always odd.
Interpreted modulo 9, all numbers from 1 to 8 appear: a(20) is the first entry = 3 (mod 9), a(26) is the first entry = 2 (mod 9), a(80) is the first entry = 6 (mod 9).

Links

Crossrefs

Cf. A020806, A141425, A146537.

Programs

  • Mathematica
    Numerator[1/9 - 1/(8*Range[0,100] +5)^2] (* G. C. Greubel, Mar 07 2022 *)
  • Sage
    [numerator(1/9 - 1/(8*n+5)^2) for n in (0..100)] # G. C. Greubel, Mar 07 2022

Formula

a(n) = A061039(8*n+5).
a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81) for n>83. - Colin Barker, Oct 10 2016

Extensions

Edited and extended by R. J. Mathar, Oct 24 2008

A145909 First 6-fold decimation of A061039. First bisection of A144454.

Original entry on oeis.org

0, 8, 8, 16, 80, 40, 56, 224, 32, 40, 440, 176, 208, 728, 280, 320, 1088, 136, 152, 1520, 560, 616, 2024, 736, 800, 2600, 312, 336, 3248, 1160, 1240, 3968, 1408, 1496, 4760, 560, 592, 5624, 1976, 2080, 6560, 2296, 2408, 7568, 880, 920
Offset: 0

Views

Author

Paul Curtz, Oct 24 2008

Keywords

Links

Crossrefs

Cf. A061039, A144454.

Programs

Formula

a(n) = A061039(6*n+3).

A146763 Rank of terms ending in 0 in A061039.

Original entry on oeis.org

0, 4, 10, 14, 20, 24, 30, 34, 40, 44, 50, 54, 60, 64, 70, 74, 80, 84, 90, 94, 100, 104, 110, 114, 120, 124, 130, 134, 140, 144, 150, 154, 160, 164, 170, 174, 180, 184, 190, 194, 200, 204, 210, 214, 220, 224, 230, 234, 240, 244, 250, 254, 260, 264, 270, 274
Offset: 0

Views

Author

Paul Curtz, Nov 02 2008

Keywords

Comments

From Paschen spectrum of hydrogen.
Numbers that are congruent to 0 or 4 mod 10. - Philippe Deléham, Oct 18 2011

Links

Crossrefs

Cf. A001622, A020714, A030308, A061039.

Programs

  • Magma
    [5*n - (n mod 2): n in [0..60]]; // G. C. Greubel, Mar 10 2022
  • Mathematica
    Select[Range[0, 100], MemberQ[{0,4}, Mod[#, 10]] &] (* K G Teal, Dec 02 2014 *)
  • Sage
    [5*n - (n%2) for n in (0..60)] # G. C. Greubel, Mar 10 2022

Formula

a(n) = 10*n - 6 - a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=4 and b(k) = 5*2^k = A020714(k) for k>0. - Philippe Deléham, Oct 18 2011
From Colin Barker, May 14 2012: (Start)
a(n) = (-1 + (-1)^n + 10*n)/2.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: x*(4+6*x)/((1-x)^2*(1+x)). (End)
E.g.f.: 1/2 (exp(-x) - (1 - 10*x)*exp(x)). - G. C. Greubel, Mar 10 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/20 - log(phi)/(4*sqrt(5)) + log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 17 2023

A144450 Second bisection of A061039.

Original entry on oeis.org

7, 1, 55, 91, 5, 187, 247, 35, 391, 475, 7, 667, 775, 11, 1015, 1147, 143, 1435, 1591, 65, 1927, 2107, 85, 2491, 2695, 323, 3127, 3355, 133, 3835, 4087, 161, 4615, 4891, 575, 5467, 5767, 75, 6391, 6715, 87, 7387, 7735, 899, 8455, 8827, 341, 9595, 9991, 385, 10807, 11227, 1295, 12091, 12535
Offset: 1

Views

Author

Paul Curtz, Oct 06 2008

Keywords

Comments

Related to the Paschen spectrum of hydrogen. Contains only odd numbers. The sequence read modulo 9 is "full" and contains all numbers from 0 to 8.

Links

Crossrefs

Cf. A061039, A144448.

Programs

  • Mathematica
    Numerator[1/9 - 1/(2*Range[2, 100])^2] (* G. C. Greubel, Mar 06 2022 *)
  • Sage
    [numerator(1/9 -1/(2*n+2)^2) for n in (1..100)] # G. C. Greubel, Mar 06 2022

Formula

a(n) = A061039(2*n+2).
a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81). - G. C. Greubel, Mar 06 2022

Extensions

Formula index corrected, extended by R. J. Mathar, Dec 02 2008

A146762 Terms of A061039 that are multiple of 10, in the order in which they appear.

Original entry on oeis.org

0, 40, 160, 280, 520, 80, 40, 1360, 1840, 2200, 2800, 40, 440, 4480, 5320, 5920, 6880, 280, 320, 9400, 10600, 11440, 12760, 1520, 560, 16120, 17680, 18760, 20440, 800, 2600, 24640, 26560, 27880, 29920, 1160, 1240, 34960, 37240, 38800, 41200, 4760, 560
Offset: 1

Views

Author

Paul Curtz, Nov 02 2008

Keywords

Comments

Previous name was: Numbers in A061039 ending with 0.

Crossrefs

Cf. A008592, A061039.

Programs

  • PARI
    lista(nn) = my(list=List(), x); for (n=3, nn, if ((((x=numerator(1/9-1/n^2))) % 10) == 0, listput(list, x));); Vec(list); \\ Michel Marcus, Jan 28 2023

Extensions

More terms from Arkadiusz Wesolowski, Dec 31 2011
New name from Michel Marcus, Jan 27 2023

A061047 Numerator of 1/49 - 1/n^2.

Original entry on oeis.org

0, 15, 32, 51, 72, 95, 120, 3, 176, 207, 240, 275, 312, 351, 8, 435, 480, 527, 576, 627, 680, 15, 792, 851, 912, 975, 1040, 1107, 24, 1247, 1320, 1395, 1472, 1551, 1632, 5, 1800, 1887, 1976, 2067, 2160, 2255, 48, 2451, 2552, 2655, 2760, 2867
Offset: 7

Views

Author

N. J. A. Sloane, May 26 2001

Keywords

Comments

a(n) = (n+7)^2-49 = n*(n+14) = A098848(n), except a(7p). The corresponding series of atomic transitions is named Hansen-Strong. It comes after Lyman (1906-1914), Balmer (1885), Paschen (1908), Brackett (1922), Pfund (1924) and Humphreys series (1952 not 1953, justified later). - Paul Curtz, Oct 07 2008

Links

Crossrefs

Cf. A061035-A061050.
Cf. A061039, A061043, A045944, A098848.

Programs

  • Magma
    [Numerator(1/49-1/n^2): n in [7..60]]; // Vincenzo Librandi, Sep 07 2016
  • Mathematica
    Table[Numerator[1/49-1/n^2],{n,7,70}] (* Harvey P. Dale, Apr 26 2016 *)
  • PARI
    a(n) = numerator(1/49 - 1/n^2); \\ Michel Marcus, Aug 15 2013

Extensions

Edited by M. F. Hasler, Nov 17 2014

A033567 a(n) = (2*n-1)*(4*n-1).

Original entry on oeis.org

1, 3, 21, 55, 105, 171, 253, 351, 465, 595, 741, 903, 1081, 1275, 1485, 1711, 1953, 2211, 2485, 2775, 3081, 3403, 3741, 4095, 4465, 4851, 5253, 5671, 6105, 6555, 7021, 7503, 8001, 8515, 9045, 9591, 10153, 10731, 11325, 11935, 12561, 13203, 13861, 14535, 15225
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n+1) = A005563(1), A061037(3), A061039(5), A061041(7), A061043(9), A061045(11), A061047(13), A061049(15). Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, ... spectra of hydrogen. - Paul Curtz, Oct 08 2008
Sequence found by reading the segment [1, 3] together with the line from 3, in the direction 3, 21, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011

Links

Crossrefs

Cf. A045944, A014634.

Programs

  • Magma
    [(2*n-1)*(4*n-1): n in [0..50]]; // G. C. Greubel, Sep 19 2018
  • Mathematica
    Table[(2*n - 1)*(4*n - 1), {n, 0, 50}] (* G. C. Greubel, Jul 06 2017 *)
LinearRecurrence[{3,-3,1},{1,3,21},50] (* Harvey P. Dale, Aug 25 2019 *)
  • PARI
    vector(60, n, n--; (2*n-1)*(4*n-1)) \\ Michel Marcus, Apr 12 2015

Formula

a(n) = a(n-1) + 16*n - 14 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Jul 06 2017: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-2).
E.g.f.: (1 + 2*x + 8*x^2)*exp(x).
G.f.: (1 + 15*x^2)/(1 - x)^3. (End)
From Amiram Eldar, Jan 03 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + Pi/4 - log(2)/2.
Sum_{n>=0} (-1)^n/a(n) = 1 + (sqrt(2)-1)*Pi/4 + log(sqrt(2)-1)/sqrt(2). (End)

Extensions

More terms from Michel Marcus, Apr 12 2015
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