A040001 -id:A040001 - cp's OEIS frontend

A040031 Continued fraction for sqrt(38).

6, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6

Offset: 0

N. J. A. Sloane

			6.1644140029689764502501923... = 6 + 1/(6 + 1/(12 + 1/(6 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010492 (decimal expansion), A040001.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[38],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)

{ allocatemem(932245000); default(realprecision, 38000); x=contfrac(sqrt(38)); for (n=0, 20000, write("b040031.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Stefano Spezia, Jul 27 2025: (Start)
a(n) = 6*A040001(n).
G.f.: 6*(1 + x + x^2)/(1 - x^2). (End)

A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 0, 7, 0, 5, 1, 2, 3, 6, 7, 8, 6, 6, 3, 2, 0, 9, 0, 20, 0, 35, 0, 34, 0, 25, 0, 7, 1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4, 2, 0, 11, 0, 29, 0, 63, 0, 115, 0, 176, 0, 238, 0, 255, 0, 230, 0, 169, 0, 92, 0, 41, 0, 9

Offset: 0

Alois P. Heinz, Mar 16 2018

			              .______.
              | /\/\ |  ,  rectangle area: 12, above path area: 5,
T(3,-1) = 1:  |/____\|  ,  below path area: 7, difference: (5-7) = 2 * (-1).
.
                 /\
                /  \
T(3,0) = 2:    /    \   /\/\/\  .
.
                /\         /\
T(3,1) = 2:    /  \/\   /\/  \  .
.
Triangle T(n,k) begins:
:                                   1                                    ;
:                                   1                                    ;
:                                   2                                    ;
:                               1,  2,  2                                ;
:                           2,  0,  7,  0,  5                            ;
:                   1,  2,  3,  6,  7,  8,  6,  6,  3                    ;
:           2,  0,  9,  0, 20,  0, 35,  0, 34,  0, 25,  0,  7            ;
:  1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4  ;

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Counting lattice paths

Row sums give A000108.
Column k=0 gives A300952.
Cf. A002620, A026741, A040001, A129182, A239927, A300322, A300996.

Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * T(n,k) = A300996(n).
T(n,-floor((n-1)^2/4)) = A040001(n).
T(n, floor((n-1)^2/4)) = A026741(n+1) for n > 2.
T(n,k) = 0 iff n is even and k is odd or abs(k) > floor(n*(n-1)/6).

A327767 Period 2: repeat [1, -2].

Original entry on oeis.org

1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2

Offset: 1

Michael Somos, Sep 24 2019

			G.f. = x - 2*x^2 + x^3 - 2*x^4 + x^5 - 2*x^6 + x^7 - 2*x^8 + ...

Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000034, A040001, A134451, A168361, A280193.

&cat [[1, -2]^^50]; // Vincenzo Librandi, Feb 29 2020

a[ n_] := If[ n < 1, 0, -2 + 3 Mod[n, 2]];
a[ n_] := Which[ n < 1, 0, OddQ[n], 1, True, -2];
a[ n_] := SeriesCoefficient[ (x - 2*x^2) / (1 - x^2), {x, 0, n}];
PadRight[{}, 100, {1, -2}] (* Vincenzo Librandi, Feb 29 2020 *)

{a(n) = if( n<1, 0, -(1 + 3*(-1)^n)/2)};

PARI
```
{a(n) = if( n<1, 0, -2 + 3*(n%2))};
```

{a(n) = if( n<1, 0, [-2, 1][n%2 + 1])};

{a(n) = if( n<0, 0, polcoeff( (x - 2*x^2) / (1 - x^2) + x * O(x^n), n))};

G.f.: x * (1 - 2*x) / (1 - x^2) = x / (1 + 2*x / (1 - 3*x / (2 - x))).
E.g.f.: (exp(x) - 1)*(3/exp(x) - 1)/2.
a(n) is multiplicative with a(2^e) = -2 if e>0, a(p^e) = 1 otherwise.
Moebius transform is length 2 sequence [1, -3].
a(n) = -(1 + 3*(-1)^n)/2 if n>=1.
a(2*n) = -2, a(2*n + 1) = 1, a(0) = 0.
a(n) = -(-1)^n * A134451(n) for all n in Z.
a(n) = a(n+2) = -(-1)^n * A000034(n-1) = -A168361(n+1) for n>=1.
Dirichlet g.f.: zeta(s)*(1-3/2^s). - Amiram Eldar, Jan 03 2023

A067881 Factorial expansion of sqrt(3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 0, 4, 2, 5, 10, 8, 1, 5, 6, 8, 5, 13, 18, 0, 7, 20, 9, 6, 14, 2, 7, 7, 18, 11, 0, 12, 20, 10, 31, 28, 27, 34, 29, 18, 13, 8, 28, 14, 9, 12, 39, 5, 15, 8, 5, 0, 7, 21, 54, 13, 16, 20, 24, 18, 12, 14, 6, 53, 21, 42, 47, 14, 46, 14, 42, 71, 41, 63, 24, 28, 32, 61, 35

Offset: 1

Benoit Cloitre, Mar 10 2002

			sqrt(3) = 1 + 1/2! + 1/3! + 1/4! + 2/5! + 5/6! + 0/7! + 4/8! + 2/9! + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for factorial base representation

Cf. A002194 (decimal expansion), A040001 (continued fraction).

Cf. A009949 (sqrt(2)), A068446 (sqrt(5)), A320839 (sqrt(7)).

SetDefaultRealField(RealField(250));  [Floor(Sqrt(3))] cat [Floor(Factorial(n)*Sqrt(3)) - n*Floor(Factorial((n-1))*Sqrt(3)) : n in [2..80]]; // G. C. Greubel, Dec 10 2018

Digits:=200: a:=n->`if`(n=1,floor(sqrt(3)),floor(factorial(n)*sqrt(3))-n*floor(factorial(n-1)*sqrt(3))): seq(a(n),n=1..90); # Muniru A Asiru, Dec 11 2018

With[{b = Sqrt[3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 10 2018 *)

default(realprecision, 250); {b = sqrt(3); a(n) = if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b))};
for(n=1, 80, print1(a(n), ", ")) \\ G. C. Greubel, Dec 10 2018

apply( A067881(n)=if(n>1,sqrt(precision(3., n*log(n/2.5)\2.3+2))*(n-1)!%1*n\1,1), [1..79]) \\ M. F. Hasler, Dec 14 2018

b=sqrt(3);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 10 2018

a(1) = 1; for n > 1, a(n) = floor(n!*sqrt(3)) - n*floor((n-1)!*sqrt(3)).

A082953 a(n) = A000252(n) / A070732(n).

Original entry on oeis.org

1, 2, 4, 8, 16, 8, 36, 32, 36, 32, 100, 32, 144, 72, 64, 128, 256, 72, 324, 128, 144, 200, 484, 128, 400, 288, 324, 288, 784, 128, 900, 512, 400, 512, 576, 288, 1296, 648, 576, 512, 1600, 288, 1764, 800, 576, 968, 2116

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003

From Jianing Song, Apr 20 2019: (Start)
a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].
a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000252, A065464, A070732.
Cf. A000010, A062570, A040001.
Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).

A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
seq(A082953(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

Array[Times @@ Map[EulerPhi, {#, 2 #}] &, 47] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = eulerphi(n)*eulerphi(2*n); \\ Michel Marcus, Jun 04 2025

a(n) = phi(n)*phi(2*n) = A000010(n)*A062570(n). - Vladeta Jovovic, May 02 2005
Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - R. J. Mathar, Apr 14 2011
a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - Jianing Song, Apr 20 2019
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 =  0.171299... . - Amiram Eldar, Oct 30 2022
a(n) = gcd(n,2)*phi(n)^2 = A040001(n)*A127473(n). - Ridouane Oudra, Jun 04 2025

A133081 An interpolation operator, companion to A133080.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Sep 09 2007

A133081 * [1,2,3,...] = A133090: (1, 1, 5, 3, 9, 5, 13, 7, 17, ...).

A133080: diagonal and subdiagonal are switched.

			First few rows of the triangle:
  1;
  1, 0;
  0, 1, 1;
  0, 0, 1, 0;
  0, 0, 0, 1, 1;
  ...

Cf. A133080, A133090, A040001 (row sums).

row(n) = vector(n, k, if (k==n-1, 1, if (k==n, n%2)));
lista(nn) = my(list=List()); for (n=1, nn, my(v=row(n)); for (k=1, #v, listput(list, v[k]))); Vec(list); \\ Michel Marcus, Mar 06 2022

Infinite lower triangular matrix, (1,0,1,0,...) in the main diagonal and (1,1,1,...) in the subdiagonal.

More terms from Michel Marcus, Mar 06 2022

A181655 Expansion of (1+2x-x^3+x^4)/(1-4x^2+3x^4).

Original entry on oeis.org

1, 2, 4, 7, 14, 22, 44, 67, 134, 202, 404, 607, 1214, 1822, 3644, 5467, 10934, 16402, 32804, 49207, 98414, 147622, 295244, 442867, 885734, 1328602, 2657204, 3985807, 7971614, 11957422, 23914844, 35872267, 71744534, 107616802, 215233604

Offset: 0

Paul Barry, Nov 03 2010

Row sums of A181654.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Cf. A060816, A198643 (bisections).

CoefficientList[Series[(1+2x-x^3+x^4)/(1-4x^2+3x^4),{x,0,40}],x] (* or *) Join[{1},LinearRecurrence[{0,4,0,-3},{2,4,7,14},40]] (* Harvey P. Dale, Jan 11 2012 *)

A181655(n)=if(bitand(n,1), 3^(n\2)*5\2, n, 3^(n\2-1)*5-1, 1) \\ M. F. Hasler, Apr 06 2019

G.f.: (1+2*x-x^3+x^4)/((1-x^2)*(1-3*x^2)).
a(n) = 5*A038754(n+1)/6 - A040001(n)/2. - R. J. Mathar, May 14 2016
a(2n-1) = A060816(n-1), a(2n) = A198643(n-1); n >= 1. a(n+1) = 2*a(n) if n is odd. - M. F. Hasler, Apr 06 2019

A184324 The number of disconnected k-regular simple graphs on 2k+4 vertices.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 11, 13, 18, 21, 26, 33, 40, 49, 61, 73, 89, 110, 131, 158, 192, 230, 274, 331, 392, 468, 557, 660, 780, 927, 1088, 1284, 1511, 1775, 2076, 2438, 2843, 3323, 3873, 4510, 5238, 6095, 7057, 8182, 9466, 10945, 12626, 14578, 16780, 19323, 22211

Offset: 0

Jason Kimberley, Jan 11 2011

			The a(0)=1 graph is 4K_1. The a(1)=1 graph is 3K_2. The a(2)=2 graphs are C_3+C_5 and C_4+C_4.

This sequence is the third highest diagonal of D=A068933: that is a(n)=D(2k+4, k).

Cf. A184325(k) = D(4k+5, 2k) and A184326(k) = D(2k+6, k).

A184324 := func< n | n eq 0 select 1 else (n+1)mod 2 + A008483(n+3) >; // see A008483 for its MAGMA code.

a(0)=1. For k>0, a(k) = (k+1) mod 2 + A008483(k+3).
For k>=0, a(k) = A040001(k) + A165652(k+3).
Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(2k+4, k) = C(k+1, k) C(k+3, k) + A000217(C(k+2,k)), from the disconnected Euler transform. C(k+1, k)=1 because K_{k+1} is connected and the unique k-regular graph on k+1 vertices. For k > 1, since D(k+3,k)=0, then C(k+3,k) = E(k+3,k) = E(k+3,2) = A008483(k + 3). Also, for k >0, since D(k+2,k)=0, then C(k+2,k) = E(k+2,k) = E(k+2,1) = (k+1) mod 2. With the examples below and A165652(n)=0 for n < 6 = offset, QED.

A226555 Numerators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n.

Original entry on oeis.org

1, 5, 4, 13, 7, 25, 10, 33, 17, 45, 16, 69, 19, 65, 38, 81, 25, 109, 28, 125, 55, 105, 34, 177, 53, 125, 68, 181, 43, 241, 46, 193, 89, 165, 100, 301, 55, 185, 106, 321, 61, 349, 64, 293, 167, 225, 70, 433, 109, 341, 140, 349, 79, 433, 162, 465, 157, 285, 88

Offset: 1

Reinhard Zumkeller, Jun 10 2013

			.   n         A226314(n,k) / A054531(n,k), 1<=k<=n<=12           row sums
.  --   -------------------------------------------------------- --------
.   1:  1                                                           1
.   2:  1/2 2                                                       5/2
.   3:  1/3 2/3 3                                                   4
.   4:  1/4 3/2 3/4 4                                              13/2
.   5:  1/5 2/5 3/5 4/5 5                                           7
.   6:  1/6 4/3 5/2 5/3 5/6 6                                      25/2
.   7:  1/7 2/7 3/7 4/7 5/7 6/7 7                                  10
.   8:  1/8 5/4 3/8 7/2 5/8 7/4 7/8 8                              33/2
.   9:  1/9 2/9 7/3 4/9 5/9 8/3 7/9 8/9 9                          17
.  10:  1/10 6/5 3/10 7/5 9/2 8/5 7/10 9/5 9/10 10                 45/2
.  11:  1/11 2/11 3/11 4/11 5/11 6/11 7/11 8/11 9/11 10/11 11      16
.  12:  1/12 7/6 9/4 10/3 5/12 11/2 7/12 11/3 11/4 11/6 11/12 12   69/2 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A040001 (denominators).

import Data.Ratio ((%), numerator); import Data.Function (on)
a226555 n = numerator $ sum $
            zipWith ((%) `on` toInteger) (a226314_row n) (a054531_row n)

A257088 a(2n) = 4n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

Original entry on oeis.org

1, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128

Offset: 0

Michael Somos, Apr 16 2015

			G.f. = 1 + x + 4*x^2 + 3*x^3 + 8*x^4 + 5*x^5 + 12*x^6 + 7*x^7 + 16*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A001082, A005408, A007310, A008574, A040001, A215947.

CF. A257083 (partial sums), A246695.

import Data.List (transpose)
a257088 n = a257088_list !! n
a257088_list = concat $ transpose [a008574_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], n, True, 2 n];
a[ n_] := SeriesCoefficient[ (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4), {x, 0, n}];

PARI
```
{a(n) = if( n<1, n==0, n%2, n, 2*n)};
```

{a(n) = if( n<0, 0, polcoeff( (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4) + x * O(x^n), n))};

Euler transform of length 4 sequence [ 1, 3, -1, -1].
a(n) is multiplicative with a(2^e) = 2^(e+1) if e>0, otherwise a(p^e) = p^e.
G.f.: (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4).
G.f.: (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^2)^3).
MOBIUS transform of A215947 is [1, 4, 3, 8, 5, ...].
a(n) = n * A040001(n) if n>0.
a(n) + a(n-1) = A007310(n) if n>0.
a(n) = A001082(n+1) - A001082(n) if n>0.
Binomial transform with a(0)=0 is A128543 if n>0.
a(2*n) = A008574(n). a(2*n + 1) = A005408(n).
a(n) = A022998(n) if n>0. - R. J. Mathar, Apr 19 2015
From Amiram Eldar, Jan 28 2025: (Start)
Dirichlet g.f.: (1+2^(1-s)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (3/4) * n^2. (End)
a(n) = gcd(n^n, 2*n). - Mia Boudreau, Jun 27 2025

cp's OEIS Frontend

A040031 Continued fraction for sqrt(38).

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A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

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A327767 Period 2: repeat [1, -2].

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A067881 Factorial expansion of sqrt(3) = Sum_{n>=1} a(n)/n!.

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A082953 a(n) = A000252(n) / A070732(n).

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A133081 An interpolation operator, companion to A133080.

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A181655 Expansion of (1+2x-x^3+x^4)/(1-4x^2+3x^4).

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A257088 a(2n) = 4n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.