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.

A051048 Sqrt[a(n)a(n+1)+1] of A051047.

Original entry on oeis.org

2, 5, 31, 449, 6271, 87361, 1216799, 16947841, 236052991, 3287794049, 45793063711, 637815097921, 8883618307199, 123732841202881, 1723376158533151, 24003533378261249, 334326091137124351
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A051047.

Formula

G.f.: x(-4x^4+57x^3-14x^2-25x+2)/[(1-x)(1-14x+x^2)].

A045899 Numbers k such that k+1 and 3*k+1 are perfect squares.

Original entry on oeis.org

0, 8, 120, 1680, 23408, 326040, 4541160, 63250208, 880961760, 12270214440, 170902040408, 2380358351280, 33154114877520, 461777249934008, 6431727384198600, 89582406128846400, 1247721958419651008, 17378525011746267720, 242051628206028097080, 3371344269872647091408
Offset: 1

Views

Author

Andrej Dujella (duje(AT)math.hr)

Keywords

Comments

Essentially the same as A051047.
It appears that a(n) = A046175(n)-A046174(n), that is, the triangular index of the n-th pentagonal triangular number minus its pentagonal index. - Jonathan Vos Post, Feb 28 2011
Sequence lists the nonnegative x solutions when (x + 1)*(3*x + 1) is a square. Positive x solutions when (x - 1)*(3*x - 1) is a square are in A011922. - Bruno Berselli, Feb 20 2018

Links

Crossrefs

Cf. A051047, A067900. A076139, A076140, A123480, A217855.
Cf. A046184, A245031.

Programs

  • Mathematica
    f[n_] := FullSimplify[((Sqrt[3] + 2)*(7 + 4*Sqrt[3])^n - (Sqrt[3] - 2) (7 - 4 Sqrt[3])^n - 4)/6]; Array[f, 18, 0] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 23 2006 *)
Rest[CoefficientList[Series[-8*x^2/((x - 1)*(x^2 - 14*x + 1)), {x,0,50}], x]] (* G. C. Greubel, Jun 07 2017 *)
LinearRecurrence[{15,-15,1},{0,8,120},20] (* Harvey P. Dale, Jul 14 2024 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(-8*x^2/((x - 1)*(x^2 - 14*x + 1)))) \\ G. C. Greubel, Jun 07 2017

Formula

a(n) = A046184(n+1) - 1.
a(n) = 14*a(n-1) - a(n-2) + 8.
a(n) = ((2 + sqrt(3))*(7 + 4*sqrt(3))^n + (2 - sqrt(3))*(7 - 4*sqrt(3))^n - 4)/6. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 23 2006
a(n) = 8*A076139(n-1) = 4*A217855(n-1) = 2*A123480(n-1) = 8/3*A076140(n-1). - Peter Bala, Dec 31 2012
From Colin Barker, Jul 30 2013: (Start)
G.f.: -8*x^2 / ((x - 1)*(x^2 - 14*x + 1)).
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3). (End)
E.g.f.: (-4*exp(x) + (2 + sqrt(3))*exp((7-4*sqrt(3))*x) + (2 - sqrt(3))*exp((7+4*sqrt(3))*x))/6. - Ilya Gutkovskiy, Apr 28 2016

A046184 Indices of octagonal numbers which are also squares.

Original entry on oeis.org

1, 9, 121, 1681, 23409, 326041, 4541161, 63250209, 880961761, 12270214441, 170902040409, 2380358351281, 33154114877521, 461777249934009, 6431727384198601, 89582406128846401, 1247721958419651009, 17378525011746267721, 242051628206028097081
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

The equation a(t)*(3*a(t)-2) = m*m is equivalent to the Pell equation (3*a(t)-1)*(3*a(t)-1) - 3*m*m = 1. - Paul Weisenhorn, May 12 2009
As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> infinity} a(n)/a(n-1) = (2 + sqrt(3))^2 = 7 + 4 * sqrt(3). - Ant King, Nov 16 2011
Also numbers n such that the octagonal number N(n) is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 11 2014
Also nonnegative integers y in the solutions to 2*x^2 - 6*y^2 + 4*x + 4*y + 2 + 2 = 0, the corresponding values of x being A251963. - Colin Barker, Dec 11 2014

Links

Crossrefs

Cf. A028230, A036428, A251963.
Cf. A006253.

Programs

  • Magma
    I:=[1, 9, 121]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Nov 17 2011
  • Mathematica
    LinearRecurrence[ {15, -15, 1}, {1, 9, 121}, 17 ] (* Ant King, Nov 16 2011 *)
CoefficientList[Series[x (1-6x+x^2)/((1-x)(1-14x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Sep 01 2021 *)
  • PARI
    Vec(x*(1-6*x+x^2) / ((1-x)*(1-14*x+x^2)) + O(x^100)) \\ Colin Barker, Dec 11 2014

Formula

{n: A000567(n) in A000290}.
Nearest integer to (1/6) * (2+sqrt(3))^(2n-1). - Ralf Stephan, Feb 24 2004
a(n) = A045899(n-1) + 1 = A051047(n+1) + 1 = A003697(2n-2). - N. J. A. Sloane, Jun 12 2004
a(n) = A001835(n)^2. - Lekraj Beedassy, Jul 21 2006
From Paul Weisenhorn, May 12 2009: (Start)
With A=(2+sqrt(3))^2=7+4*sqrt(3) the equation x*x-3*m*m=1 has solutions
x(t) + sqrt(3)*m(t) = (2+sqrt(3))*A^t and the recurrences
x(t+2) = 14*x(t+1) - x(t) with = 2, 26, 362, 5042
m(t+2) = 14*m(t+1) - m(t) with = 1, 15, 209, 2911
a(t+2) = 14*a(t+1) - a(t) - 4 with = 1, 9, 121, as above. (End)
From Ant King, Nov 15 2011: (Start)
a(n) = 14*a(n-1) - a(n-2) - 4.
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).
a(n) = (1/6)*( (2+sqrt(3))^(2n-1) + (2-sqrt(3))^(2n-1) + 2 ).
a(n) = ceiling( (1/6)*(2 + sqrt(3))^(2n-1) ).
a(n) = (1/6)*( (tan(5*Pi/12))^(2n-1) + (tan(Pi/12))^(2n-1) + 2 ).
a(n) = ceiling ( (1/6)*(tan(5*Pi/12))^(2n-1) ).
G.f.: x*(1-6*x+x^2) / ((1-x)*(1-14*x+x^2)). (End)
a(n) = A006253(2n-2). - Andrey Goder, Oct 17 2021
Showing 1-3 of 3 results.

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