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 1-3 of 3 results.

A171733 a(2n)=A165568(n). a(2n+1)=A165563(n).

Original entry on oeis.org

-1, 1, 1, 7, 31, 41, 137, 151, 391, 409, 889, 911, 1751, 1777, 3121, 3151, 5167, 5201, 8081, 8119, 12079, 12121, 17401, 17447, 24311, 24361, 33097, 33151, 44071, 44129, 57569, 57631, 73951, 74017, 93601, 93671, 116927, 117001, 144361, 144439, 176359, 176441, 213401, 213487, 255991
Offset: 0

Views

Author

Paul Curtz, Dec 17 2009

Keywords

Links

Crossrefs

Cf. A035287.

Programs

  • PARI
    A171733(n)=if( bittest(n,0), n, -1-n) +(n\=2)^2 +2*n^3 +n^4

Formula

a(n) = n^3/8 +n^2/16 -15/32 +n^4/16 +(-1)^n*( n^3/8+3*n^2/16-n-17/32 ).
G.f.: ( 1-2*x-4*x^2+2*x^3-18*x^4+2*x^5-4*x^6-2*x^7+x^8 ) / ( (1+x)^4*(-1+x)^5 ).
a(n)= +a(n-1) +4*a(n-2) -4*a(n-3) -6*a(n-4) +6*a(n-5) +4*a(n-6) -4*a(n-7) -a(n-8) +a(n-9).

A165568 a(n) = -1 - 2*n + n^2 + 2*n^3 + n^4.

Original entry on oeis.org

-1, 1, 31, 137, 391, 889, 1751, 3121, 5167, 8081, 12079, 17401, 24311, 33097, 44071, 57569, 73951, 93601, 116927, 144361, 176359, 213401, 255991, 304657, 359951, 422449, 492751, 571481, 659287, 756841, 864839, 984001, 1115071, 1258817, 1416031, 1587529, 1774151
Offset: 0

Views

Author

Paul Curtz, Sep 22 2009

Keywords

Comments

Consider the Lyman spectrum of Hydrogen A005563(n)/A000290(n+1) = n*(n+2)/(n+1)^2 = 0/1, 3/4, 8/9, 15/16, ...
The first differences of these fractions are 3/4, 5/36, 7/144, 9/400, 11/900, 13/1764, 15/3136, ... = (2n+1)/(n*(n+1))^2.
Adding numerator and denominator of these first differences yields 1 + 2n + n^2 + 2n^3 + n^4 = A165563(n) = 3+4, 5+36, 7+144, ... = 1 + 2n + n^2*(n+1)^2 = A144396(n) + A035287(n+1) = A005408(n) + A035287(n+1).
Subtracting numerator from denominator, on the other hand, yields this sequence here: a(n) = A035287(n+1) - A005408(n).

Links

Programs

Formula

a(-1-n) = A165563(n). A165563(-1-n) = a(n).
a(n) = A165563(n) - 2 - 4*n = A165563(n) - A016825(n).
a(n) + A165563 + a(n) = 2*n^2*(1+n)^2 = 2*A035287(n+1).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24.
G.f.: (-1 + 6*x + 16*x^2 + 2*x^3 + x^4)/(1-x)^5.

Extensions

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

A171677 Period 9:repeat 7,5,7,4,2,4,1,8,1.

Original entry on oeis.org

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

Views

Author

Paul Curtz, Dec 15 2009

Keywords

Comments

Represents also the decimal expansion of 252474727/333333333.
Contains the same set of numbers as A141425.

Links

Crossrefs

Cf. A165568.

Programs

  • Mathematica
    PadRight[{},80,{7,5,7,4,2,4,1,8,1}] (* Harvey P. Dale, Aug 27 2019 *)

Formula

a(n) = A165563(n+1) mod 9.
G.f.: ( -7-5*x-7*x^2-4*x^3-2*x^4-4*x^5-x^6-8*x^7-x^8 ) / ( (x-1) *(1+x+x^2) *(x^6+x^3+1) ). - R. J. Mathar, Mar 08 2011

Extensions

More terms from Jinyuan Wang, Feb 26 2020
Showing 1-3 of 3 results.

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