A010709 -id:A010709 - cp's OEIS frontend

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A098317 Decimal expansion of phi^3 = 2 + sqrt(5).

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

Offset: 1

Eric W. Weisstein, Sep 02 2004

This sequence is also the decimal expansion of ((1+sqrt(5))/2)^3. - Mohammad K. Azarian, Apr 14 2008
This is the length/width ratio of a 4-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 10 2011
Its continued fraction is [4, 4, ...] (see A010709). - Robert G. Wilson v, Apr 10 2011

			4.23606797749978969640917366873127623544061835961152572427...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
Alexey Stakhov, The mathematics of harmony: from Euclid to contemporary mathematics and computer science, World Scientific, Singapore, 2009, p. 657.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A001622, A014176, A098316, A098318.

Cf. A001076, A049310.

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

RealDigits[N[2+Sqrt[5],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

sqrt(5)+2 \\ Charles R Greathouse IV, Mar 11 2012

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

2 plus the constant in A002163. - R. J. Mathar, Sep 02 2008
Equals 3 + 4*sin(Pi/10) = 1 + 4*cos(Pi/5) = 1 + 4*sin(3*Pi/10) = 3 + 4*cos(2*Pi/5) = 1 + csc(Pi/10). - Arkadiusz Wesolowski, Mar 11 2012
Equals lim_{n -> infinity} F(n+3)/F(n) = lim_{n -> infinity} (1 + 2*F(n+1)/F(n)) = 2 + sqrt(5), with F(n) = A000045(n). - Arkadiusz Wesolowski, Mar 11 2012
Equals exp(arcsinh(2)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals Sum_{n>=1} n/phi^n = phi/(phi-1)^2 = phi^3. - Richard R. Forberg, Jun 29 2014
Equals 1 + 2*phi, with phi = A001622, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Dec 10 2022
c^n = A001076(n-1) + c * A001076(n); where c = 2 + sqrt(5). - Gary W. Adamson, Oct 09 2023
Equals lim_{n -> infinity} = S(n, 2*(-1 + 2*phi))/S(n-1, 2*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). See also the above limit formula with Fibonacci numbers. - Wolfdieter Lang, Nov 15 2023

Title expanded to include observation from Mohammad K. Azarian by Charles R Greathouse IV, Mar 11 2012

A047522 Numbers that are congruent to {1, 7} mod 8.

Original entry on oeis.org

1, 7, 9, 15, 17, 23, 25, 31, 33, 39, 41, 47, 49, 55, 57, 63, 65, 71, 73, 79, 81, 87, 89, 95, 97, 103, 105, 111, 113, 119, 121, 127, 129, 135, 137, 143, 145, 151, 153, 159, 161, 167, 169, 175, 177, 183, 185, 191, 193, 199, 201, 207, 209, 215, 217, 223, 225, 231, 233

Offset: 1

N. J. A. Sloane

Also n such that Kronecker(2,n) = mu(gcd(2,n)). - Jon Perry and T. D. Noe, Jun 13 2003
Also n such that x^2 == 2 (mod n) has a solution. The primes are given in sequence A001132. - T. D. Noe, Jun 13 2003
As indicated in the formula, a(n) is related to the even triangular numbers. - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
Cf. property described by Gary Detlefs in A113801: more generally, these a(n) are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h,n natural numbers). Therefore a(n)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 8). Also a(n)^2 - 1 == 0 (mod 16). - Bruno Berselli, Nov 17 2010
A089911(3*a(n)) = 2. - Reinhard Zumkeller, Jul 05 2013
S(a(n+1)/2, 0) = (1/2)*(S(a(n+1), sqrt(2)) - S(a(n+1) - 2, sqrt(2)))  = T(a(n+1), sqrt(2)/2) = cos(a(n+1)*Pi/4) = sqrt(2)/2 = A010503, identically for n >= 0, where S is the Chebyshev polynomial (A049310) here extended to fractional n, evaluated at x = 0. (For T see A053120.) - Wolfdieter Lang, Jun 04 2023

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001132, A014494, A056575, A010709, A074378, A047336, A056020, A005408, A047209, A007310, A090771, A175885, A091998, A175886, A175887, A058529, A047621, A179260, A352125.
Cf. A077221 (partial sums).
Cf. A000217, A089911, A113801.
Cf. A010503, A049310, A053120.

a047522 n = a047522_list !! (n-1)
a047522_list = 1 : 7 : map (+ 8) a047522_list
-- Reinhard Zumkeller, Jan 07 2012

Select[Range[1, 191, 2], JacobiSymbol[2, # ]==1&]

a(n)=4*n-2+(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = sqrt(8*A014494(n)+1) = sqrt(16*ceiling(n/2)*(2*n+1)+1) = sqrt(8*A056575(n)-8*(2n+1)*(-1)^n+1). - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
1 - 1/7 + 1/9 - 1/15 + 1/17 - ... = (Pi/8)*(1 + sqrt(2)). [Jolley] - Gary W. Adamson, Dec 16 2006
From R. J. Mathar, Feb 19 2009: (Start)
a(n) = 4n - 2 + (-1)^n = a(n-2) + 8.
G.f.: x(1+6x+x^2)/((1+x)(1-x)^2). (End)
a(n) = 8*n - a(n-1) - 8. - Vincenzo Librandi, Aug 06 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 8*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
E.g.f.: 1 + (4*x - 1)*cosh(x) + (4*x - 3)*sinh(x). - Stefano Spezia, May 13 2021
E.g.f.: 1 + (4*x - 3)*exp(x) + 2*cosh(x). - David Lovler, Jul 16 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2+sqrt(2)) (A179260).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/8)*cosec(Pi/8) (A352125). (End)

A074378 Even triangular numbers halved.

Original entry on oeis.org

0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333, 390, 410, 473, 495, 564, 588, 663, 689, 770, 798, 885, 915, 1008, 1040, 1139, 1173, 1278, 1314, 1425, 1463, 1580, 1620, 1743, 1785, 1914, 1958, 2093, 2139, 2280, 2328, 2475

Offset: 0

W. Neville Holmes, Sep 04 2002

Set of integers k such that k + (1 + 2 + 3 + 4 + ... + x) = 3*k, where x is sufficiently large. For example, 203 is a term because 203 + (1 + 2 + 3 + 4 + ... +28) = 609 and 609 = 3*203. - Gil Broussard, Sep 01 2008
Set of all m such that 16*m+1 is a perfect square. - Gary Detlefs, Feb 21 2010
Integers of the form Sum_{k=0..n} k/2. - Arkadiusz Wesolowski, Feb 07 2012
Numbers of the form h*(4*h + 1) for h = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018
Numbers whose distance to nearest square equals their distance to nearest oblong; that is, numbers k such that A053188(k) = A053615(k). - Lamine Ngom, Oct 27 2020
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5)) = 1 + q^3 + q^5 + q^14 + q^18 + .... - Peter Bala, Dec 30 2024

David A. Corneth, Table of n, a(n) for n = 0..9999
Neville Holmes, More Geometric Integer Sequences
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A011848, A014493, A014494, A074377, A035294, A274757.
Cf. A010709, A047522. [Vincenzo Librandi, Feb 14 2009]
Cf. A266883 (numbers n such that 16*n-15 is a square).
Cf. A016687, A153799.
Cf. A053615, A053188.

f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

Maple

a:=n->(2*n+1)*floor((n+1)/2): seq(a(n),n=0..50); # Muniru A Asiru, Feb 01 2019

Mathematica

1/2 * Select[PolygonalNumber@ Range[0, 100], EvenQ] (* Michael De Vlieger, Jun 01 2017, Version 10.4 *) Select[Accumulate[Range[0,100]],EvenQ]/2 (* Harvey P. Dale, Feb 15 2025 *)

PARI

a(n)=(2*n+1)*(n-n\2)

Formula

Sum_{n>=0} q^a(n) = (Prod_{n>0} (1-q^n))*(Sum_{n>=0} A035294(n)*q^n).

a(n) = n*(n + 1)/4 where n*(n + 1)/2 is even.

G.f.: x*(3 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)^2).

From Benoit Jubin, Feb 05 2009: (Start)

a(n) = (2*n + 1)*floor((n + 1)/2).

a(2*k) = k*(4*k+1); a(2*k+1) = (k+1)*(4*k+3). (End)

a(2*n) = A007742(n), a(2*n-1) = A033991(n). - Arkadiusz Wesolowski, Jul 20 2012

a(n) = (4*n + 1 - (-1)^n)*(4*n + 3 - (-1)^n)/4^2. - Peter Bala, Jan 21 2019

a(n) = (2*n+1)*(n+1)*(1+(-1)^(n+1))/4 + (2*n+1)*(n)*(1+(-1)^n)/4. - Eric Simon Jacob, Jan 16 2020

From Amiram Eldar, Jul 03 2020: (Start)

Sum_{n>=1} 1/a(n) = 4 - Pi (A153799).

Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(2) - 4 (See A016687). (End)

a(n) = A014494(n)/2 = A274757(n)/3 = A266883(n) - 1. - Hugo Pfoertner, Dec 31 2024

A040002 Continued fraction for sqrt(5).

Original entry on oeis.org

2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

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Author

N. J. A. Sloane

Keywords

nonn
cofr
easy
cons

Comments

Decimal expansion of 11/45. - Natan Arie Consigli, Jan 19 2016

Examples

2.236067977499789696409173668... = 2 + 1/(4 + 1/(4 + 1/(4 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009

References

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.

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

Links

Harry J. Smith, Table of n, a(n) for n = 0..20000

G. Xiao, Contfrac

Index entries for continued fractions for constants

Index of eventually constant sequences

Index of divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A002163 (decimal expansion), A001077/A001076 (convergents), A248235 (Egyptian fraction).

Cf. Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).

Essentially the same as A010709.

Programs

Maple

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

Mathematica

ContinuedFraction[Sqrt[5],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *) PadRight[{2},120,{4}] (* Harvey P. Dale, Jul 06 2019 *)

PARI

{ allocatemem(932245000); default(realprecision, 26000); x=contfrac(sqrt(5)); for (n=0, 20000, write("b040002.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

Formula

a(0) = 2, a(n) = 4 n>0. - Natan Arie Consigli, Jan 19 2016

From Elmo R. Oliveira, Feb 16 2024: (Start)

G.f.: 2*(1+x)/(1-x).

E.g.f.: 4*exp(x) - 2.

a(n) = 2*A040000(n). (End)

A010722 Constant sequence: the all 6's sequence.

Original entry on oeis.org

6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

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Author

N. J. A. Sloane

Keywords

nonn
easy

Comments

Continued fraction expansion of 3+sqrt(10). - Bruno Berselli, Mar 15 2011

Decimal expansion of Sum_{n >= 0} n/binomial(2*n+1, n) = 2/3. - Bruno Berselli, Sep 14 2015

Decimal expansion of 2/3. - Franklin T. Adams-Watters, Feb 23 2019

References

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

Links

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1014.

Tanya Khovanova, Recursive Sequences.

Index to divisibility sequences.

Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A145429: decimal expansion of Sum_{n >= 0} n/binomial(2*n, n).

Cf. A000012, A007395, A010701, A010709, A010716.

First differences of A008588.

Programs

PARI

a(n)=6 \\ Charles R Greathouse IV, Sep 28 2015

Formula

G.f.: 6/(1-x). - Bruno Berselli, Mar 15 2011

E.g.f.: 6*e^x. - Vincenzo Librandi, Jan 27 2012

a(n) = floor(1/(-n + csc(1/n))). - Clark Kimberling, Mar 10 2020

A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 1, 6, 0, 0, 0, 0, 4, 0, 5, 7, 0, 0, 0, 0, 4, 0, 1, 1, 8, 0, 0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 0, 0, 4, 0, 0, 0, 16, 6, 10, 0, 0, 0, 0, 4, 0, 0, 0, 32, 5, 7, 11, 0, 0, 0, 0, 4, 0, 0, 0, 80, 1, 1, 1, 12

Offset: 0

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Author

Alois P. Heinz, Jun 06 2015

Keywords

nonn
tabl
look

Examples

Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 6, 5, 1, 0, 0, 0, 0, 0, 0, 0, ... 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ... 9, 6, 5, 1, 0, 0, 0, 0, 0, 0, ...

Links

Alois P. Heinz, Antidiagonals n = 0..115, flattened

J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8

Wikipedia, Arithmetic derivative

Crossrefs

Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.

Rows n=0,1,4,8 give: A000004, A000007, A010709, A129150.

Row 15: A090636, Row 28: A090637, Row 63: A090635, Row 81: A129151, Row 128: A369638, Row 1024: A214800, Row 15625: A129152.

Main diagonal gives A185232.

Antidiagonal sums give A258652.

Cf. also A328383.

Programs

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]): A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end: seq(seq(A(n, h-n), n=0..h), h=0..14);

Mathematica

d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0; A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]]; Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* Jean-François Alcover, Apr 27 2017, translated from Maple *)

Formula

A(n,k) = A003415^k(n).

A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 4, 0, 0, 0, 1, 4, 5, 0, 0, 0, 0, 4, 3, 6, 0, 0, 0, 0, 4, 2, 7, 7, 0, 0, 0, 0, 4, 1, 4, 4, 8, 0, 0, 0, 0, 4, 0, 4, 4, 12, 9, 0, 0, 0, 0, 4, 0, 4, 4, 20, 12, 10, 0, 0, 0, 0, 4, 0, 4, 4, 32, 20, 11, 11, 0, 0, 0, 0, 4, 0, 4, 4, 80, 32, 5, 5, 12

Offset: 0

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Author

Alois P. Heinz, Jun 12 2015

Keywords

nonn
tabl

Examples

Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, ... 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, ... 6, 7, 4, 4, 4, 4, 4, 4, 4, 4, ... 7, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... 8, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ... 9, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...

Links

Alois P. Heinz, Antidiagonals n = 0..31, flattened

Crossrefs

Columns k=0-10 give: A001477, A258851, A258852, A258853, A258854, A258855, A258856, A258857, A258858, A258859, A258860.

Rows n=0,1,4,8 give: A000004, A000007, A010709, A258848.

Antidiagonal sums give A258847.

Main diagonal gives A258849.

Cf. A000720, A258975, A259016.

Programs

Maple

with(numtheory): d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]): A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end: seq(seq(A(n, h-n), n=0..h), h=0..14);

Mathematica

d[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; d[0] = 0; A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]]; Table[Table[A[n, h-n], {n, 0, h}], {h, 0, 14}] // Flatten (* Jean-François Alcover, Apr 24 2016, adapted from Maple *)

Formula

A(n,k) = A258851^k(n).

A(A259016(n,k),k) = n.

A(A258975(n),n) = 1.

A023003 Number of partitions of n into parts of 4 kinds.

Original entry on oeis.org

1, 4, 14, 40, 105, 252, 574, 1240, 2580, 5180, 10108, 19208, 35693, 64960, 116090, 203984, 353017, 602348, 1014580, 1688400, 2778517, 4524760, 7296752, 11658920, 18468245, 29015700, 45235414, 70005376, 107585845, 164245380, 249162620, 375704920, 563251038

Offset: 0

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Author

David W. Wilson

Keywords

nonn
easy

Comments

a(n) is Euler transform of A010709. - Alois P. Heinz, Oct 17 2008

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)

Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, 2016, hal-01285685v2.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.

P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.

N. J. A. Sloane, Transforms

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

4th column of A144064.

Programs

Julia

# DedekindEta is defined in A000594. A023003List(len) = DedekindEta(len, -4) A023003List(33) |> println # Peter Luschny, Mar 10 2018

Maple

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*4, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^4,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *) CoefficientList[1/QPochhammer[x]^4 + O[x]^40, x] (* Jean-François Alcover, Jan 31 2016 *)

PARI

\ps100 for(n=0,100,print1((polcoeff(1/eta(x)^4,n,x)),","))

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^4.

a(0)=1, a(n) = (1/n) * Sum_{k=0..n-1} 4*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011

a(n) ~ exp(2 * Pi * sqrt(2*n/3)) / (2^(7/4) * 3^(5/4) * n^(7/4)) * (1 - (35*sqrt(3)/(16*Pi) + Pi/(3*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017

G.f.: exp(4*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

From Peter Bala, Oct 05 2023: (Start)

The even bisection of the g.f. A(x) is (A(x) + A(-x))/2 = 1 + 14*x^2 + 105*x^4 + 574*x^6 + ... = Product_{n >= 1} (1 + x^(2*n))^14 / (1 - x^(8*n))^4 = F(x^2)*A(x^8), where F(x) = Product_{n >= 1} (1 + x^n)^14 is the g.f. of A022579.

The odd bisection of the g.f. is (A(x) - A(-x))/2 = 4*x + 40*x^3 + 252*x^5 + 1240*x^7 + ... = (4*x) * Product_{n >= 1} (1 + x^(2*n))^2 * (1 - x^(8*n))^4 / (1 - x^(2*n))^8. (End)

A123932 a(0) = 1, a(n) = 4 for n > 0.

Original entry on oeis.org

1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

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Author

Philippe Deléham, Nov 28 2006

Keywords

nonn
cofr
easy

Comments

Continued fraction for sqrt(5)-1.

a(n) = number of permutations of length n+3 having only one ascent such that the first element of the permutation is 3. - Ran Pan, Apr 20 2015

Also, decimal expansion of 13/90. - Bruno Berselli, Apr 24 2015

Column 1 of A327331 and of A327333. - Omar E. Pol, Nov 25 2019

Links

Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Essentially the same as A113311, A040002 and A010709.

Programs

Magma

[4^n mod 12: n in [0..40]]; // Vincenzo Librandi, Apr 23 2015

Mathematica

ContinuedFraction[Sqrt[5] - 1, 120] (* Michael De Vlieger, Apr 20 2015 *)

Maxima

makelist(if n=0 then 1 else 4, n, 0, 100); /* Bruno Berselli, Apr 24 2015 */

PARI

a(n)=(n>=0)+3*(n>0) \\ Jaume Oliver Lafont, Mar 26 2009

Sage

[power_mod(4, n, 12) for n in range(0, 84)] # Zerinvary Lajos, Nov 25 2009

Formula

G.f.: (1 + 3*x) / (1 - x).

a(n) = 4 - 3*0^n .

a(n) = 4^n mod 12. - Zerinvary Lajos, Nov 25 2009

E.g.f.: 4*exp(x) - 3. - Elmo R. Oliveira, Aug 06 2024

A259016 A(n,k) = k-th pi-based antiderivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 3, 3, 0, 5, 5, 5, 4, 0, 11, 11, 11, 4, 5, 0, 10, 10, 10, 4, 11, 6, 0, 29, 29, 29, 4, 10, 13, 7, 0, 78, 78, 78, 4, 29, 41, 6, 8, 0, 141, 141, 141, 4, 78, 35, 13, 19, 9, 0, 266, 266, 266, 4, 141, 38, 41, 15, 23, 10, 0, 147, 147, 147, 4, 266, 163, 35, 14, 83, 29, 11

Offset: 0

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Author

Alois P. Heinz, Jun 16 2015

Keywords

nonn
tabl

Examples

A(5,3) = 29 -> 10 -> 11 -> 5. A(5,4) = 78 -> 127 -> 31 -> 11 -> 5. Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, 2, 3, 5, 11, 10, 29, 78, 141, 266, ... 2, 3, 5, 11, 10, 29, 78, 141, 266, 147, ... 3, 5, 11, 10, 29, 78, 141, 266, 147, 194, ... 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... 5, 11, 10, 29, 78, 141, 266, 147, 194, 1181, ... 6, 13, 41, 35, 38, 163, 138, 253, 346, 1383, ... 7, 6, 13, 41, 35, 38, 163, 138, 253, 346, ... 8, 19, 15, 14, 43, 191, 201, 217, 1113, 1239, ... 9, 23, 83, 431, 3001, 27457, 10626, 112087, 87306, 172810, ...

Links

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Crossrefs

Columns k=0-3 give: A001477, A258861, A258862, A258995.

Rows n=0,1,4,7,8,9 give: A000004, A258975, A010709, A259168, A259169, A259170.

Cf. A000040, A000720, A258850, A258851.

Programs

Maple

with(numtheory): d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]): A:= proc() local t, A; t, A:= proc()-1 end, proc()-1 end; proc(n, k) local h; while A(n, k) = -1 do t(k):= t(k)+1; h:= (d@@k)(t(k)); if A(h, k) = -1 then A(h, k):= t(k) fi od; A(n, k) end end(): seq(seq(A(n, h-n), n=0..h), h=0..12);

Mathematica

d[n_] := If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]]; A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]]; Table[A[n, k-n], {k, 0, 12}, {n, 0, k}] // Flatten (* Jean-François Alcover, Mar 20 2017 *)

Formula

A(n,k) = min { m >= 0 : A258851^k(m) = n }.

A258850(A(n,k),k) = n.

A(n,k) <= A000040^k(n) for n>0.

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A098317 Decimal expansion of phi^3 = 2 + sqrt(5).

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A047522 Numbers that are congruent to {1, 7} mod 8.

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Haskell

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A074378 Even triangular numbers halved.

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A040002 Continued fraction for sqrt(5).

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A010722 Constant sequence: the all 6's sequence.

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A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.