A061047 -id:A061047 - cp's OEIS frontend

A061048 Denominator of 1/49 - 1/n^2.

1, 3136, 3969, 4900, 5929, 7056, 8281, 196, 11025, 12544, 14161, 15876, 17689, 19600, 441, 23716, 25921, 28224, 30625, 33124, 35721, 784, 41209, 44100, 47089, 50176, 53361, 56644, 1225, 63504, 67081, 70756, 74529, 78400, 82369

Offset: 7

N. J. A. Sloane, May 26 2001

G. C. Greubel, Table of n, a(n) for n = 7..1000

Cf. A061035-A061050.

Cf. A061047 (numerator).

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

for(n=7,50, print1(denominator(1/7^2 - 1/n^2), ", ")) \\ G. C. Greubel, Jul 07 2017

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

A146951 Numbers that are congruent to 0 or 6 mod 10.

Original entry on oeis.org

0, 6, 10, 16, 20, 26, 30, 36, 40, 46, 50, 56, 60, 66, 70, 76, 80, 86, 90, 96, 100, 106, 110, 116, 120, 126, 130, 136, 140, 146, 150, 156, 160, 166, 170, 176, 180, 186, 190, 196, 200, 206, 210, 216, 220, 226, 230, 236, 240, 246, 250, 256, 260, 266, 270, 276, 280

Offset: 0

Paul Curtz, Nov 03 2008

Rank of terms of A061047 ending in with 0.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A020714, A030308, A061047.

I:=[0, 6, 10]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // Vincenzo Librandi, May 18 2012

CoefficientList[Series[x*(6+4*x)/((1-x)^2*(1+x)),{x,0,50}],x] (* Vincenzo Librandi, May 18 2012 *)

a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=6 and b(k) = 5*2^k = A020714(k) for k > 0. - Philippe Deléham, Oct 18 2011
From Colin Barker, May 15 2012: (Start)
a(n) = 1/2 - (-1)^n/2 + 5*n.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: x*(6+4*x)/((1-x)^2*(1+x)). (End)
E.g.f.: 5*x*exp(x) + sinh(x). - Stefano Spezia, May 14 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(5)/8 - sqrt(1-2/sqrt(5))*Pi/20 - log(phi)/(4*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Sep 15 2022

Replaced definition by a comment from Philippe Deléham, Oct 18 2011. Afer the change this becomes a list, but it is better to keep the offset as 0. - N. J. A. Sloane, Sep 08 2022

A146952 a(n) = A146950(n)/40.

Original entry on oeis.org

0, 3, 6, 12, 17, 26, 33, 45, 54, 69, 80, 2, 111, 132, 3, 171, 188, 215, 234, 264, 285, 318, 341, 377, 402, 9, 468, 510, 11, 584, 615, 663, 696, 747, 782, 836, 873, 930, 969, 3, 1070, 1133, 24, 1242, 1287, 1356, 1403, 1475, 1524, 1599, 1650, 1728, 1781, 38

Offset: 1

Paul Curtz, Nov 03 2008

nonn

A061047(n) is divisible by 10 iff n is 3 or 7 mod 10. In this case, A061047(n) is also divisible by 40. - Andrew Howroyd, Jan 03 2020

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A061047, A146950.

a(n)={my(t=5*n + 3 - n%2); numerator(1/49 - 1/t^2)/40} \\ Andrew Howroyd, Jan 03 2020

a(n) = A061047(5*n + 3 - (n mod 2)). - Andrew Howroyd, Jan 03 2020

Offset corrected and terms a(11) and beyond from Andrew Howroyd, Jan 03 2020

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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A061048 Denominator of 1/49 - 1/n^2.

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

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A146951 Numbers that are congruent to 0 or 6 mod 10.

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A146952 a(n) = A146950(n)/40.

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

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