A014176 -id:A014176 - cp's OEIS frontend

A098318 Decimal expansion of [5, 5, ...] = (5 + sqrt(29))/2.

5, 1, 9, 2, 5, 8, 2, 4, 0, 3, 5, 6, 7, 2, 5, 2, 0, 1, 5, 6, 2, 5, 3, 5, 5, 2, 4, 5, 7, 7, 0, 1, 6, 4, 7, 7, 8, 1, 4, 7, 5, 6, 0, 0, 8, 0, 8, 2, 2, 3, 9, 4, 4, 1, 8, 8, 4, 0, 1, 9, 4, 3, 3, 5, 0, 0, 8, 3, 2, 2, 9, 8, 1, 4, 1, 3, 8, 2, 9, 3, 4, 6, 4, 3, 8, 3, 1, 6, 8, 9, 0, 8, 3, 9, 9, 1, 7, 7, 4, 2, 2, 0

Offset: 1

Eric W. Weisstein, Sep 02 2004

The "metallic" constants A001622, A014176 etc. are defined inserting a = 1, 2, 3, 4, ... into (a+sqrt(a^2+4))/2. [Stakhov & Aranson] - R. J. Mathar, Feb 14 2011

This is the length/width ratio of a 5-extension rectangle; see A188640 where the metallic constants are defined for rational numbers. - Clark Kimberling, Apr 09 2011

			5.19258240356725201562535524577016477814756...

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. Stakhov and Samuil Aranson, Hyperbolic Fibonacci and Lucas functions, Golden Fibonacci Goniometry, Bodnar's Geometry, ..., Appl. Math. 2 (1) (2011) 74-84.
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A001622, A014176, A098316, A098317, A010716 (continued fraction).

Cf. A052918, A049310.

SetDefaultRealField(RealField(100)); (5 + Sqrt(29))/2; // G. C. Greubel, Jun 30 2019

r=5; t=(r+(4+r^2)^(1/2))/2; FullSimplify[t]
N[t,130]
RealDigits[N[t,130]][[1]]
ContinuedFraction[t,120] (* Clark Kimberling, Apr 09 2011 *)

(5 + sqrt(29))/2 \\ Charles R Greathouse IV, Jul 24 2013

numerical_approx((5+sqrt(29))/2, digits=100) # G. C. Greubel, Jun 30 2019

5 plus the constant in A085551. - R. J. Mathar, Sep 02 2008
c^n = A052918(n-2) + A052918(n-1) * c, where c = (5 + sqrt(29))/2. - Gary W. Adamson, Oct 09 2023
Equals lim_{n->infinity} S(n, sqrt(29))/ S(n-1, sqrt(29)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A276862 First differences of the Beatty sequence A003151 for 1 + sqrt(2).

Original entry on oeis.org

2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2

Offset: 1

Clark Kimberling, Sep 24 2016

Conjectures: Equals both A245219 and A097509. - Michel Dekking, Feb 18 2020

Theorem: If the initial term of A097509 is omitted, it is identical to the present sequence. For proof, see A097509. The argument may also imply that A082844 is also the same as these two sequences, apart from the initial terms. - Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Mar 02 2021. Postscript from the same authors, Sep 09 2021: We have proved that the present sequence, A276862 (indexed from 1) matches the characterization of {c_{i-1}} given by (8) of our "Solutions" page.

Clark Kimberling, Table of n, a(n) for n = 1..9999 [Offset adapted by _Georg Fischer_, Mar 07 2020]
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A003151, A006337, A014176, A082844, A097509, A245219, A276879.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021

[Floor((n+1)*(1 + Sqrt(2))) - Floor(n*(1+Sqrt(2))): n in [1..100]]; // G. C. Greubel, Aug 16 2018

z = 500; r = 1+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A003151 *)
Differences[b] (* A276862 *)
Last@SubstitutionSystem[{2 -> {2, 3}, 3 -> {2, 3, 2}}, {2}, 5] (* John Keith, Apr 21 2021 *)

vector(100, n, floor((n+1)*(1 + sqrt(2))) - floor(n*(1+sqrt(2)))) \\ G. C. Greubel, Aug 16 2018

from math import isqrt
def A276862(n): return 1-isqrt(m:=n*n<<1)+isqrt(m+(n<<2)+2) # Chai Wah Wu, Aug 03 2022

a(n) = floor((n+1)*r) - floor(n*r) = A003151(n+1)-A003151(n), where r = 1 + sqrt(2), n >= 1.
a(n) = 1 + A006337(n) for n >+ 1. - R. J. Mathar, Sep 30 2016
Fixed point of the morphism 2 -> 2,3; 3 -> 2,3,2. - John Keith, Apr 21 2021

Corrected by Michel Dekking, Feb 18 2020

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

Original entry on oeis.org

1, 2, 4, 1, 8, 4, 16, 12, 1, 32, 32, 6, 64, 80, 24, 1, 128, 192, 80, 8, 256, 448, 240, 40, 1, 512, 1024, 672, 160, 10, 1024, 2304, 1792, 560, 60, 1, 2048, 5120, 4608, 1792, 280, 12, 4096, 11264, 11520, 5376, 1120, 84, 1, 8192, 24576, 28160, 15360

Offset: 1

Clark Kimberling, Feb 18 2012

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is the triangle given by (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012
The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and  along (first layer) skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. - Zagros Lalo, Jul 31 2018
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.414213562373095... (A014176: Decimal expansion of the silver mean, 1+sqrt(2)), when n approaches infinity. - Zagros Lalo, Jul 31 2018

			First seven rows:
1
2
4...1
8...4
16..12..1
32..32..6
64..80..24..1
(2, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, ...) begins:
    1
    2,   0
    4,   1,  0
    8,   4,  0, 0
   16,  12,  1, 0, 0
   32,  32,  6, 0, 0, 0
   64,  80, 24, 1, 0, 0, 0
  128, 192, 80, 8, 0, 0, 0, 0

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358.

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
S. Halici, On some Pell polynomials , Acta Universitatis Apulensis, No. 29/2012, pp. 105-112.
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n

Cf. A028297, A207537, A133156, A038207, A099089.
Cf. A053117, A097610, A091894.
Cf. A013609, A038207.
Cf. A128099.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)
t[n_, k_] := t[n, k] = 2^(n - 2 k) * (n -  k)!/((n - 2 k)! k!) ; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]} ]  // Flatten (* Zagros Lalo, Jul 31 2018 *)

u(n,x) = u(n-1,x)+(x+1)*v(n-1,x), v(n,x) = u(n-1,x)+v(n-1,x), where u(1,x) = 1, v(1,x) = 1. Also, A207538 = |A133156|.
From Philippe Deléham, Mar 04 2012: (Start)
With 0<=k<=n:
Mirror image of triangle in A099089.
Skew version of A038207.
Riordan array (1/(1-2*x), x^2/(1-2*x)).
G.f.: 1/(1-2*x-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190958(n+1), A127357(n), A090591(n), A089181(n+1), A088139(n+1), A045873(n+1), A088138(n+1), A088137(n+1), A099087(n), A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively.
T(n,k) = 2*T(n-1,k) + T(-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n, k) = 0 if k<0 or if k>n. (End)
T(n,k) = A013609(n-k, n-2*k+1). - Johannes W. Meijer, Sep 05 2013
From Tom Copeland, Feb 11 2016: (Start)
A053117 is a reflected, aerated and signed version of this entry. This entry belongs to a family discussed in A097610 with parameters h1 = -2 and h2 = -y.
Shifted o.g.f.: G(x,t) = x / (1 - 2 x - t x^2).
The compositional inverse of G(x,t) is Ginv(x,t) = -[(1 + 2x) - sqrt[(1+2x)^2 + 4t x^2]] / (2tx) = x - 2 x^2 + (4-t) x^3 - (8-6t) x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0).
(End)

A384277 Decimal expansion of the smallest zero of the Laguerre polynomial of degree 3.

Original entry on oeis.org

4, 1, 5, 7, 7, 4, 5, 5, 6, 7, 8, 3, 4, 7, 9, 0, 8, 3, 3, 1, 1, 5, 3, 3, 8, 7, 3, 1, 2, 8, 2, 7, 4, 4, 7, 3, 5, 4, 6, 6, 1, 7, 4, 1, 2, 6, 9, 3, 1, 1, 8, 4, 6, 5, 0, 9, 3, 9, 6, 5, 9, 5, 4, 3, 2, 2, 3, 2, 5, 0, 1, 9, 9, 3, 6, 9, 1, 3, 3, 1, 4, 9, 5, 7, 1, 9, 6

Offset: 0

A.H.M. Smeets, May 24 2025

			0.41577455678347908331153387312827447354661741269311...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 3.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 3.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | this sequence, A384278, A384279          | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[3, #] &, 1], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

Smallest root of x^3 - 9 x^2 + 18 x - 6 = 0.

A384279 Decimal expansion of the largest zero of the Laguerre polynomial of degree 3.

Original entry on oeis.org

6, 2, 8, 9, 9, 4, 5, 0, 8, 2, 9, 3, 7, 4, 7, 9, 1, 9, 6, 8, 6, 6, 4, 1, 5, 7, 6, 5, 5, 1, 2, 1, 3, 1, 6, 5, 7, 4, 9, 3, 5, 2, 0, 8, 6, 6, 2, 4, 6, 6, 0, 0, 7, 0, 0, 8, 7, 0, 8, 3, 2, 7, 9, 7, 5, 9, 3, 6, 4, 4, 5, 2, 8, 7, 2, 5, 9, 2, 0, 2, 3, 8, 4, 7, 9, 6, 1

Offset: 1

A.H.M. Smeets, May 26 2025

			6.28994508293747919686641576551213165749352086624660...

Paolo Xausa, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 3.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 3.

Cf. A384590.
There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, this sequence          | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[3, #] &, 3], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

largest root of x^3 - 9 x^2 + 18 x - 6 = 0.

A384278 Decimal expansion of the second smallest zero of the Laguerre polynomial of degree 3.

Original entry on oeis.org

2, 2, 9, 4, 2, 8, 0, 3, 6, 0, 2, 7, 9, 0, 4, 1, 7, 1, 9, 8, 2, 2, 0, 5, 0, 3, 6, 1, 3, 5, 9, 5, 9, 3, 8, 6, 8, 9, 5, 9, 8, 6, 1, 7, 2, 1, 0, 6, 0, 2, 8, 0, 8, 3, 4, 0, 3, 5, 2, 0, 1, 2, 4, 8, 0, 8, 4, 0, 3, 0, 4, 5, 1, 3, 3, 7, 1, 6, 6, 4, 4, 6, 5, 6, 3, 1, 8

Offset: 1

A.H.M. Smeets, May 24 2025

			2.29428036027904171982205036135959386895986172106028...

Paolo Xausa, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 3.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 3.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, this sequence, A384279          | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[3, #] &, 2], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

Second smallest root of x^3 - 9 x^2 + 18 x - 6 = 0.

A384280 Decimal expansion of the smallest zero of the Laguerre polynomial of degree 4.

Original entry on oeis.org

3, 2, 2, 5, 4, 7, 6, 8, 9, 6, 1, 9, 3, 9, 2, 3, 1, 1, 8, 0, 0, 3, 6, 1, 4, 5, 9, 1, 0, 4, 3, 6, 7, 4, 7, 9, 7, 4, 3, 7, 5, 7, 2, 2, 4, 4, 7, 4, 2, 9, 5, 7, 6, 7, 1, 8, 8, 4, 5, 1, 8, 5, 3, 8, 0, 6, 9, 6, 8, 6, 7, 8, 7, 0, 7, 7, 0, 4, 0, 0, 9, 8, 6, 8, 5, 8, 5

Offset: 0

A.H.M. Smeets, May 26 2025

			0.32254768961939231180036145910436747974375722447429...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279          |
   4 | this sequence, A384281

First[RealDigits[Root[LaguerreL[4, #] &, 1], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

Smallest root of x^4 - 16 x^3 + 72 x^2 - 96 x + 24 = 0.

A384281 Decimal expansion of the second smallest zero of the Laguerre polynomial of degree 4.

Original entry on oeis.org

1, 7, 4, 5, 7, 6, 1, 1, 0, 1, 1, 5, 8, 3, 4, 6, 5, 7, 5, 6, 8, 6, 8, 1, 6, 7, 1, 2, 5, 1, 7, 9, 4, 7, 0, 2, 3, 6, 7, 3, 8, 7, 4, 5, 1, 5, 5, 3, 1, 0, 7, 2, 5, 0, 1, 7, 8, 2, 7, 8, 2, 6, 6, 0, 9, 9, 8, 4, 5, 6, 0, 5, 7, 4, 4, 2, 1, 9, 7, 1, 6, 4, 1, 4, 0, 1, 3

Offset: 1

A.H.M. Smeets, May 26 2025

			1.74576110115834657568681671251794702367387451553107...

Paolo Xausa, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279          |
   4 | A384280, this sequence

First[RealDigits[Root[LaguerreL[4, #] &, 2], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

Second smallest root of x^4 - 16 x^3 + 72 x^2 - 96 x + 24 = 0.

A006275 Pierce expansion of sqrt(2) - 1.

Original entry on oeis.org

2, 5, 7, 197, 199, 7761797, 7761799, 467613464999866416197, 467613464999866416199, 102249460387306384473056172738577521087843948916391508591105797

Offset: 0

N. J. A. Sloane

From Peter Bala, Nov 22 2012: (Start)
For x in the open interval (0,1) define the map f(x) = 1 - x*floor(1/x). The n-th term (n >= 0) in the Pierce expansion of x is given by floor(1/f^(n)(x)), where f^(n)(x) denotes the n-th iterate of the map f, with the convention that f^(0)(x) = x. The present sequence is the case x = sqrt(2) - 1.
The Pierce expansion of (sqrt(2) - 1)^(3^n) is [a(0)*a(2)*...*a(2*n), a(2*n+1), a(2*n+2), ...] = [sqrt(a(2*n+1) - 1), a(2*n+1), a(2*n+2), ...]. The Pierce expansion of (sqrt(2) - 1)^(2*3^n) is [a(2*n+1), a(2*n+2), ...]. Some examples of the associated alternating series are given below.
(End)

			Let c(0)=6, c(n+1) = c(n)^3-3*c(n); then this sequence is 2, c(0)-1, c(0)+1, c(1)-1, c(1)+1, c(2)-1, c(2)+1, ...
From _Peter Bala_, Nov 22 2012: (Start)
Let x = sqrt(2) - 1. We have the alternating series expansions
x = 1/2 - 1/(2*5) + 1/(2*5*7) - 1/(2*5*7*197) + ...
x^3 = 1/14 - 1/(14*197) + 1/(14*197*199) - ...
x^9 = 1/2786 - 1/(2786*7761797) + 1/(2786*7761797*7761799) - ...,
where 2786 = 2*7*199, and also
x^2 = 1/5 - 1/(5*7) + 1/(5*7*197) - 1/(5*7*197*199) + ...
x^6 = 1/197 - 1/(197*199) + 1/(197*199*7761797) - ...
x^18 = 1/7761797 - 1/(7761797*7761799) + ....
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..14
T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly, Vol. 36, No. 10 (1929), pp. 523-525.
Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
Eric Weisstein's World of Mathematics, Pierce Expansion.

Cf. A014176, A006276.

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Sqrt[2] - 1, 7!], 10] (* G. C. Greubel, Nov 14 2016 *)

my(r=1+quadgen(8)); for(n=1, 10, print1(floor(r), ", "); r=r/(r-floor(r)));

Let u(0)=1+sqrt(2) and u(n+1)=u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n)=floor(u(n)). - Benoit Cloitre, Mar 09 2004
From Peter Bala, Nov 22 2012: (Start)
a(2*n+2) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n) + 1.
a(2*n+1) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n) - 1. (End)
sqrt(2) - 1 = a(0)/a(1) + (a(0)*a(2))/(a(1)*a(3)) + (a(0)*a(2)*a(4))/(a(1)*a(3)*a(5)) + ... = 2/5 + (2*7)/(5*197) + (2*7*199)/(5*197*7761797) + ... . - Peter Bala, Dec 03 2012

More terms from James Sellers, May 19 2000

A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

Original entry on oeis.org

1, 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, 229, 65, 293, 82, 365, 101, 445, 122, 533, 145, 629, 170, 733, 197, 845, 226, 965, 257, 1093, 290, 1229, 325, 1373, 362, 1525, 401, 1685, 442, 1853, 485, 2029, 530, 2213, 577, 2405, 626, 2605, 677

Offset: 0

Paul Curtz, Aug 15 2015

Using (n+sqrt(4+n^2))/2, after the integer 1 for n=0, the reduced metallic means are b(1) = (1+sqrt(5))/2, b(2) = 1+sqrt(2), b(3) = (3+sqrt(13))/2, b(4) = 2+sqrt(5), b(5) = (5+sqrt(29))/2, b(6) = 3+sqrt(10), b(7) = (7+sqrt(53))/2, b(8) = 4+sqrt(17), b(9) = (9+sqrt(85))/2, b(10) = 5+sqrt(26), b(11) = (11+sqrt(125))/2 = (11+5*sqrt(5))/2, ... . The last value yields the radicals in a(n) or A013946.
b(2) = 2.41, b(3) = 3.30, b(4) = 4.24, b(5) = 5.19 are "good" approximations of fractal dimensions corresponding to dimensions 3, 4, 5, 6: 2.48, 3.38, 4.33 and 5.45 based on models. See "Arbres DLA dans les espaces de dimension supérieure: la théorie des peaux entropiques" in Queiros-Condé et al. link. DLA: beginning of the title of the Witten et al. link.
Consider the symmetric array of the half extended Rydberg-Ritz spectrum of the hydrogen atom:
   0,    1/0,     1/0,     1/0,     1/0,      1/0,      1/0,     1/0, ...
-1/0,      0,     3/4,     8/9,   15/16,    24/25,    35/36,   48/49, ...
-1/0,   -3/4,       0,    5/36,    3/16,   21/100,      2/9,  45/196, ...
-1/0,   -8/9,   -5/36,       0,   7/144,   16/225,     1/12,  40/441, ...
-1/0, -15/16,   -3/16,  -7/144,       0,    9/400,    5/144,  33/784, ...
-1/0, -24/25, -21/100, -16/225,  -9/400,        0,   11/900, 24/1225, ...
-1/0, -35/36,    -2/9,   -1/12,  -5/144,  -11/900,        0, 13/1764, ...
-1/0, -48/49, -45/196, -40/441, -33/784, -24/1225, -13/1764,       0, ... .
The numerators are almost A165795(n).
Successive rows: A000007(n)/A057427(n), A005563(n-1)/A000290(n), A061037(n)/A061038(n), A061039(n)/A061040(n), A061041(n)/A061042(n), A061043(n)/A061044(n), A061045(n)/A061046(n), A061047(n)/A061048(n), A061049(n)/A061050(n).
A144433(n) or A195161(n+1) are the numerators of the second upper diagonal (denominators: A171522(n)).
c(n+1) = a(n) + a(n+1) = 6, 7, 15, 18, 34, 39, 63, 70, 102, 111, ... .
c(n+3) - c(n+1) = 9, 11, 19, 21, 29, 31, ... = A090771(n+2).
The final digit of a(n) is neither 4 nor 8. - Paul Curtz, Jan 30 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Diogo Queiros-Condé, Jean Chaline, and Jacques Dubois, Le monde des fractales La Nature trans-échelles, 478p., ellipses, Paris, 2015, page 220.
T. A. Witten, Jr. and L. M. Sander, Diffusion-Limited Aggregation, a Kinetic Critical Phenomenom, Phys. Rev. Lett., Vol. 47 (Nov 09 1981), pp. 1400-1403.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000007, A000290, A001622, A002522, A005563, A010685, A010698, A013946, A014176, A057427, A061035-A061050, A078370, A087475, A090771, A098316, A098317, A098318, A144433, A168077, A171621, A176398, A176439, A176458, A176522, A176537, A195161.

Cf. A069894.

[Numerator(1+n^2/4): n in [0..60]]; // Vincenzo Librandi, Aug 15 2015

A261327:=n->numer((4 + n^2)/4); seq(A261327(n), n=0..60); # Wesley Ivan Hurt, Aug 15 2015

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 5, 2, 13, 5, 29}, 60] (* Vincenzo Librandi, Aug 15 2015 *)
a[n_] := (n^2 + 4) / 4^Mod[n + 1, 2]; Table[a[n], {n, 0, 52}] (* Peter Luschny, Mar 18 2022 *)

vector(60, n, n--; numerator(1+n^2/4)) \\ Michel Marcus, Aug 15 2015

Vec((1+5*x-x^2-2*x^3+2*x^4+5*x^5)/(1-x^2)^3 + O(x^60)) \\ Colin Barker, Aug 15 2015

a(n)=if(n%2,n^2+4,(n/2)^2+1) \\ Charles R Greathouse IV, Oct 16 2015

[(n*n+4)//4**((n+1)%2) for n in range(60)] # Gennady Eremin, Mar 18 2022

[numerator(1+n^2/4) for n in (0..60)] # G. C. Greubel, Feb 09 2019

a(n) = numerator(1 + n^2/4). (Previous name.) See A010685 (denominators).
a(2*k) = 1 + k^2.
a(2*k+1) = 5 + 4*k*(k+1).
a(2*k+1) = 4*a(2*k) + 4*k + 1.
a(4*k+2) = A069894(k). - Paul Curtz, Jan 30 2019
a(-n) = a(n).
a(n+2) = a(n) + A144433(n) (or A195161(n+1)).
a(n) = A168077(n) + period 2: repeat 1, 4.
a(n) = A171621(n) + period 2: repeat 2, 8.
From Colin Barker, Aug 15 2015: (Start)
a(n) = (5 - 3*(-1)^n)*(4 + n^2)/8.
a(n) = n^2/4 + 1 for n even;
a(n) = n^2 + 4 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
G.f.: (1 + 5*x - x^2 - 2*x^3 + 2*x^4 + 5*x^5)/ (1 - x^2)^3. (End)
E.g.f.: (5/8)*(x^2 + x + 4)*exp(x) - (3/8)*(x^2 - x + 4)*exp(-x). - Robert Israel, Aug 18 2015
Sum_{n>=0} 1/a(n) = (4*coth(Pi)+tanh(Pi))*Pi/8 + 1/2. - Amiram Eldar, Mar 22 2022

New name by Peter Luschny, Mar 18 2022

cp's OEIS Frontend

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