A061043 -id:A061043 - cp's OEIS frontend

A061044 Denominator of 1/25 - 1/n^2.

1, 900, 1225, 1600, 2025, 100, 3025, 3600, 4225, 4900, 225, 6400, 7225, 8100, 9025, 80, 11025, 12100, 13225, 14400, 625, 16900, 18225, 19600, 21025, 180, 24025, 25600, 27225, 28900, 1225, 32400, 34225, 36100, 38025, 1600, 42025

Offset: 5

N. J. A. Sloane, May 26 2001

Reinhard Zumkeller, Table of n, a(n) for n = 5..10000
A. H. Pfund, The emission of nitrogen and hydrogen in the infrared, Journal of the Optical Society of America, Vol. 9, Issue 3, (1924) pp. 193-196.

See A061041 for comments, references, links.

Cf. A061043 (numerator).

import Data.Ratio ((%), denominator)
a061044 = denominator . (1 % 25 -) . recip . (^ 2) . fromIntegral
-- Reinhard Zumkeller, Jan 06 2014

Table[Denominator[1/5^2 - 1/n^2], {n, 7, 50}] (* G. C. Greubel, Jul 07 2017 *)

a(n) = denominator(1/25 - 1/n^2); \\ Michel Marcus, Aug 15 2013

A143025 Period length 4: repeat [1, 8, 2, 8].

Original entry on oeis.org

1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8

Offset: 0

Paul Curtz, Oct 13 2008

Numerator of 1/n^2-1/(3n)^2 if n>0.
This can be generated from the transitions between principal quantum numbers n and 3n in the Hydrogen series: A005563(2), A061037(6), A061039(9), A061041(12), A061043(15), A061045(18), A061047(21), A061049(24),... (The mention of A005563(2) is somewhat a fluke to maintain the periodic pattern.)
Related to the continued fraction of (12*sqrt(55)-72)/19 = 0.89444115.. = 0+1/(1+1/(8+1/(2+...))). - R. J. Mathar, Jun 27 2011

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

Cf. A045944, A144437.

&cat [[1, 8, 2, 8]^^30]; // Wesley Ivan Hurt, Jul 10 2016

seq(op([1, 8, 2, 8]), n=0..50); # Wesley Ivan Hurt, Jul 10 2016

PadRight[{}, 120, {1, 8, 2, 8}] (* Harvey P. Dale, Jul 01 2015 *)

a(n)=[1,8,2,8][n%4+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n+4) = a(n).
G.f.: (1+8*x+2*x^2+8*x^3)/(1-x^4).
From Wesley Ivan Hurt, Jul 10 2016: (Start)
a(n) = (19 - 13*I^(2*n) - I^(-n) - I^n)/4, where I = sqrt(-1).
a(n) = (19 - 2*cos(n*Pi/2) - 13*cos(n*Pi))/4. (End)

Partially edited by R. J. Mathar, Dec 10 2008

A349487 a(n) = A132739((n-5)*(n+5)).

Original entry on oeis.org

11, 24, 39, 56, 3, 96, 119, 144, 171, 8, 231, 264, 299, 336, 3, 416, 459, 504, 551, 24, 651, 704, 759, 816, 7, 936, 999, 1064, 1131, 48, 1271, 1344, 1419, 1496, 63, 1656, 1739, 1824, 1911, 16, 2091, 2184, 2279, 2376, 99, 2576, 2679, 2784, 2891, 24, 3111

Offset: 6

Simon Strandgaard, Nov 19 2021

nonn

Shares 614 initial terms with A061043. First difference is A061043(620)=615 vs. a(620)=123.

			a(9)  = A132739(( 9-5)*( 9+5)) = A132739(56) = 56,
a(10) = A132739((10-5)*(10+5)) = A132739(75) = 3,
a(11) = A132739((11-5)*(11+5)) = A132739(96) = 96.

Cf. A061043, A098603, A132739.

Table[Last@Select[Divisors[(n - 5)*(n + 5)], Mod[#, 5] != 0 &], {n, 6,
   56}] (* Giorgos Kalogeropoulos, Nov 19 2021 *)
Table[(n - 5)*(n + 5)/5^IntegerExponent[(n - 5)*(n + 5), 5], {n, 6, 56}] (* Amiram Eldar, Nov 22 2021 *)

A132739(n)=n/5^valuation(n, 5);
a(n) = A132739((n-5)*(n+5));
[a(n)|n<-[6..25]]

def A349487(n):
    a, b = divmod(n*n-25, 5)
    while b == 0:
        a, b = divmod(a,5)
    return 5*a+b # Chai Wah Wu, Dec 05 2021

p (6..25).map { |n| x = (n-5)*(n+5); x /= 5 while (x % 5) == 0; x }

a(n) = A132739(A098603(n-5)).

A177083 A006093(k)-fold repetition of A001248(k), k=1,2,3,..

Original entry on oeis.org

4, 9, 9, 25, 25, 25, 25, 49, 49, 49, 49, 49, 49, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169

Offset: 1

Paul Curtz, Dec 09 2010

Consider the initial terms of numerator sequences (dropping initial zeros) of
3; A005563=N(1) ,
5,3; A061037=N(2) ,
7,16,1; A061039=N(3) ,
9,5,33,3; A061041=N(4) ,
11,24,39,56,3; A061043=N(5) ,
13,7,5,4,85,1; A061045=N(6) ,
15,32,51,72,95,120,3; A061047=N(7) ,
17,9,57,5,105,33,161,3; A061049=N(8) ,
19,40,7,88,115,16,175,208,1; N(9),
21,11,69,6,1,39,189,14,261,3; N(10),
23,48,75,104,135,168,203,240,279,320,3; N(11)
One must add the following associated (minimum) squares (taken from squared entries in A172038) to these values to reach the next possible square not larger than the entry itself:
1;    N(1)
4,1;    N(2)
9,9,0;     N(3)
16,4,16,1;    N(4)
25,25,25,25,1;   N(5)
36,9,4,0,36,0;    N(6)
49,49,49,49,49,49,1;   N(7)
64,16,64,4,64,16,64,1, ; N(8)
Only if the index of N(.) is a prime we obtain a string of equal consecutive terms in these complementary rows: 4, 9, 25, 49, 121, 169..
The current sequence lists the consecutive complementary squares, A001248, in the rows with prime index, including their multiplicity (which is A006093).
This generates a link between the primes and the Rydberg-Ritz spectrum of the hydrogen atom.

Cf. A072055, A135177.

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A061044 Denominator of 1/25 - 1/n^2.

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A143025 Period length 4: repeat [1, 8, 2, 8].

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A349487 a(n) = A132739((n-5)*(n+5)).

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A177083 A006093(k)-fold repetition of A001248(k), k=1,2,3,..

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