A040001 -id:A040001 - cp's OEIS frontend

A168077 a(2n) = A129194(2n)/2; a(2n+1) = A129194(2n+1).

0, 1, 1, 9, 4, 25, 9, 49, 16, 81, 25, 121, 36, 169, 49, 225, 64, 289, 81, 361, 100, 441, 121, 529, 144, 625, 169, 729, 196, 841, 225, 961, 256, 1089, 289, 1225, 324, 1369, 361, 1521, 400, 1681, 441, 1849, 484, 2025, 529, 2209, 576, 2401, 625, 2601

Offset: 0

Paul Curtz, Nov 18 2009

From Paul Curtz, Mar 26 2011: (Start)
Successive A026741(n) * A026741(n+p):
  p=0:  0, 1,  1,  9,  4, 25,  9,     a(n),
  p=1:  0, 1,  3,  6, 10, 15, 21,  A000217,
  p=2:  0, 3,  2, 15,  6, 35, 12,  A142705,
  p=3:  0, 2,  5,  9, 14, 20, 27,  A000096,
  p=4:  0, 5,  3, 21,  8, 45, 15,  A171621,
  p=5:  0, 3,  7, 12, 18, 25, 33,  A055998,
  p=6:  0, 7,  4, 27, 10, 55, 18,
  p=7:  0, 4,  9, 15, 22, 30, 39,  A055999,
  p=8:  0, 9,  5, 33, 12, 65, 21,  (see A061041),
  p=9:  0, 5, 11, 18, 26, 35, 45,  A056000. (End)
The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (-4 * log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016
Multiplicative because both A129194 and A040001 are. - Andrew Howroyd, Jul 26 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A040001, A168068, A181318.

I:=[0,1,1,9,4,25]; [n le 6 select I[n] else 3*Self(n-2)-3*Self(n-4)+Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 10 2016

a := proc(n): n^2*(5-3*(-1)^n)/8 end: seq(a(n), n=0..46); # Johannes W. Meijer, Jul 03 2016

LinearRecurrence[{0,3,0,-3,0,1},{0,1,1,9,4,25},60] (* Harvey P. Dale, May 14 2011 *)
f[n_] := Numerator[(n/2)^2]; Array[f, 60, 0] (* Robert G. Wilson v, Dec 18 2012 *)
CoefficientList[Series[x(1+x+6x^2+x^3+x^4)/((1-x)^3(1+x)^3), {x,0,60}], x] (* Vincenzo Librandi, Jul 10 2016 *)

concat(0, Vec(x*(1+x+6*x^2+x^3+x^4)/((1-x)^3*(1+x)^3) + O(x^60))) \\ Altug Alkan, Jul 04 2016

a(n) = lcm(4, n^2)/4; \\ Andrew Howroyd, Jul 26 2018

(x*(1+x+6*x^2+x^3+x^4)/(1-x^2)^3).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 20 2019

From R. J. Mathar, Jan 22 2011: (Start)
G.f.: x*(1 + x + 6*x^2 + x^3 + x^4) / ((1-x)^3*(1+x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
a(n) = n^2*(5 - 3*(-1)^n)/8. (End)
a(n) = A026741(n)^2.
a(2*n) = A000290(n); a(2*n+1) = A016754(n).
a(n) - a(n-4) = 4*A064680(n+2). - Paul Curtz, Mar 27 2011
4*a(n) = A061038(n) * A010121(n+2) = A109043(n)^2, n >= 2. - Paul Curtz, Apr 07 2011
a(n) = A129194(n) / A040001(n). - Andrew Howroyd, Jul 26 2018
From Peter Bala, Feb 19 2019: (Start)
a(n) = numerator(n^2/(n^2 + 4)) = n^2/(gcd(n^2,4)) = (n/gcd(n,2))^2.
a(n) = n^2/b(n), where b(n) = [1, 4, 1, 4, ...] is a purely periodic sequence of period 2. Thus a(n) is a quasi-polynomial in n.
O.g.f.: x*(1 + x)/(1 - x)^3 - 3*x^2*(1 + x^2)/(1 - x^2)^3.
Cf. A181318. (End)
From Werner Schulte, Aug 30 2020: (Start)
Multiplicative with a(2^e) = 2^(2*e-2) for e > 0, and a(p^e) = p^(2*e) for prime p > 2.
Dirichlet g.f.: zeta(s-2) * (1 - 3/2^s).
Dirichlet convolution with A259346 equals A000290.
Sum_{n>0} 1/a(n) = Pi^2 * 7 / 24. (End)
Sum_{k=1..n} a(k) ~ (5/24) * n^3. - Amiram Eldar, Nov 28 2022

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

A226314 Triangle read by rows: T(i,j) = j+(i-j)/gcd(i,j) (1<=i<=j).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 3, 4, 1, 2, 3, 4, 5, 1, 4, 5, 5, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 5, 3, 7, 5, 7, 7, 8, 1, 2, 7, 4, 5, 8, 7, 8, 9, 1, 6, 3, 7, 9, 8, 7, 9, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 7, 9, 10, 5, 11, 7, 11, 11, 11, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 8, 3, 9, 5, 10, 13, 11, 9, 12, 11, 13, 13, 14

Offset: 1

N. J. A. Sloane, Jun 09 2013

The triangle of fractions A226314(i,j)/A054531(i,j) is an efficient way to enumerate the rationals [Fortnow].

Sum(A226314(n,k)/A054531(n,k): 1<=k<=n) = A226555(n)/A040001(n). - Reinhard Zumkeller, Jun 10 2013

			Triangle begins:
[1]
[1, 2]
[1, 2, 3]
[1, 3, 3, 4]
[1, 2, 3, 4, 5]
[1, 4, 5, 5, 5, 6]
[1, 2, 3, 4, 5, 6, 7]
[1, 5, 3, 7, 5, 7, 7, 8]
[1, 2, 7, 4, 5, 8, 7, 8, 9]
[1, 6, 3, 7, 9, 8, 7, 9, 9, 10]
...
The resulting triangle of fractions begins:
1,
1/2, 2,
1/3, 2/3, 3,
1/4, 3/2, 3/4, 4,
1/5, 2/5, 3/5, 4/5, 5,
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A002262.

Cf. A037161, A037162, A066657, A066658.

a226314 n k = n - (n - k) `div` gcd n k
a226314_row n = a226314_tabl !! (n-1)
a226314_tabl = map f $ tail a002262_tabl where
   f us'@(_:us) = map (v -) $ zipWith div vs (map (gcd v) us)
     where (v:vs) = reverse us'
-- Reinhard Zumkeller, Jun 10 2013

f:=(i,j) -> j+(i-j)/gcd(i,j);
g:=n->[seq(f(i,n),i=1..n)];
for n from 1 to 20 do lprint(g(n)); od:

A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

Original entry on oeis.org

1, 3, 10, 33, 106, 333, 1030, 3153, 9586, 29013, 87550, 263673, 793066, 2383293, 7158070, 21490593, 64504546, 193579173, 580868590, 1742867913, 5229128026, 15688432653, 47067395110, 141206379633, 423627527506, 1270899359733

Offset: 1

Vladeta Jovovic and Goran Kilibarda, Feb 28 2000

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y. - Ross La Haye, Jan 12 2008
From Paul Barry, Apr 27 2003: (Start)
With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
Binomial transform of A083329.
Second binomial transform of A040001. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000225, A000392, A028243, A000079.
Cf. A036239.
Column k=2 of A288638.
Third column of A294201.

[(3^n-2^n+1)/2: n in [1..30]]; // G. C. Greubel, Oct 06 2017

A053156:=n->(3^n - 2^n + 1)/2: seq(A053156(n), n=1..40); # Wesley Ivan Hurt, Oct 06 2017

LinearRecurrence[{6,-11,6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n,1,50}] (* G. C. Greubel, Oct 06 2017 *)

a(n) = (3^n-2^n+1)/2; \\ Michel Marcus, Nov 30 2015

a(n) = (3^n - 2^n + 1)/2.
a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 12 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)

A066660 Number of divisors of 2n excluding 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 3, 4, 5, 5, 3, 7, 3, 5, 7, 5, 3, 8, 3, 7, 7, 5, 3, 9, 5, 5, 7, 7, 3, 11, 3, 6, 7, 5, 7, 11, 3, 5, 7, 9, 3, 11, 3, 7, 11, 5, 3, 11, 5, 8, 7, 7, 3, 11, 7, 9, 7, 5, 3, 15, 3, 5, 11, 7, 7, 11, 3, 7, 7, 11, 3, 14, 3, 5, 11, 7, 7, 11, 3, 11, 9, 5, 3, 15, 7, 5, 7, 9, 3, 17, 7, 7, 7, 5, 7

Offset: 1

Benoit Cloitre, Jan 11 2002

a(n) is the number of integers of the form (n+k)/(n-k) for k=0,1,2,...,n-1.
Inverse Moebius transform of A040001 (offset 1).
The number of partitions of 2n into exactly two parts (2n-i,i) such that i divides (2n-i). - Wesley Ivan Hurt, Dec 22 2013

			a(4)=3 because (4+0)/(4-0), (4+2)/(4-2), (4+3)/(4-3) are integers.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A000005, A040001, A234306.

[DivisorSigma(0,2*n) -1: n in [1..100]]; // G. C. Greubel, Feb 13 2019

with(numtheory); A066660:=n->tau(2*n)-1; seq(A066660(n), n=1..100); # Wesley Ivan Hurt, Dec 22 2013

Table[DivisorSigma[0, 2 n] - 1, {n, 100}] (* Wesley Ivan Hurt, Dec 22 2013 *)

PARI
```
a(n)=if(n<1,0,sumdiv(n,d,(d>1)+d%2))
```

{a(n)=if(n<1, 0, numdiv(2*n)-1)} /* Michael Somos, Sep 03 2006 */

[sigma(2*n,0) -1 for n in (1..100)] # G. C. Greubel, Feb 13 2019

a(n) = A069930(n) + 1.
If n is an odd prime, then a(n)=3.
Asymptotic formula: 1/n*Sum(i=1, n, a(i)) = C*log(n) + o(log(n)) with C=3/2. [corrected by Vaclav Kotesovec, Feb 13 2019]
Also lim_{n -> infinity} card(iG.f.: Sum_{n>0} x^n(1 - x^(3n))/((1 - x^n)(1 - x^(2n))).
a(n) = d(2n) - 1, where d(n) is the number of divisors of n (A000005). - Wesley Ivan Hurt, Dec 22 2013
a(n) = n - A234306(n). - Antti Karttunen, Dec 22 2013
a(n) = Sum_{i=1..n} floor(2*n/i) - floor((2*n-1)/i). - Wesley Ivan Hurt, Nov 15 2017
Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 5), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

A102341 Areas of 'close-to-equilateral' integer triangles.

Original entry on oeis.org

12, 120, 1848, 25080, 351780, 4890480, 68149872, 949077360, 13219419708, 184120982760, 2564481115560, 35718589344360, 497495864091732, 6929223155685600, 96511629630137568, 1344233586759971040, 18722758603319903340

Offset: 1

Johannes Koelman (Joc_Kay(AT)hotmail.com), Feb 20 2005

A close-to-equilateral integer triangle is defined to be a triangle with integer sides and integer area such that the largest and smallest sides differ in length by unity. The first five close-to-equilateral integer triangles have sides (5, 5, 6), (17, 17, 16), (65, 65, 66), (241, 241, 240) and (901, 901, 902).
After these first five triangles, there are two more (namely (3361,3361,3360,4890480) and (12545,12545,12546,68149872)). - Nícolas V. Calsavara, Jul 13 2023
Then, the next four terms are {three sides a<=b<=c and area}: {46816, 46817, 46817, 949077360}, {174725, 174725, 174726, 13219419708}, {652080, 652081, 652081, 184120982760}, {2433601, 2433601, 2433602, 2564481115560}. Also, if we allow degenerate triangles (area 0), the first case would be {1,1,2,0}. We have 12 cases and a weak conjecture is that the total number of the 'close-to-equilateral' triangles is finite. - Zak Seidov, Feb 23 2005
This is an infinite series; two sides are equal in length to the hypotenuse of almost 30-60 triangles and the third side alternates between that length +/- 1. - Dan Sanders (dan(AT)ified.ca), Oct 22 2005
Heron's formula: a triangle with side lengths (x,y,z) has area A = sqrt(s*(s-x)*(s-y)*(s-z)) where s = (x+y+z)/2. For this sequence we assume integer side-lengths x = y = z +/- 1. Then for A to also be an integer, x+y+z must be even, so we can assume z = 2k for some positive integer k. Now s = (x+y+z)/2 = 3k +/- 1 and A = sqrt((3*k +/- 1)*k*k*(k +/- 1)) = k*sqrt(3*k^2 +/- 4*k + 1). To determine when this is an integer, set 3*k^2 +/- 4*k + 1 = d^2. If we multiply both sides by 3, it is easier to complete the square: (3*k +/- 2)^2 - 1 = 3*d^2. Now we are looking for solutions to the Pell equation c^2 - 3*d^2 = 1 with c = 3*k +/- 2, for which there are infinitely many solutions: use the upper principal convergents of the continued fraction expansion of sqrt(3) (A001075/A001353). - Danny Rorabaugh, Oct 16 2015

			a(2) = 120 because 120 is the area of a triangle with side lengths of 16, 17 and 17.

Danny Rorabaugh, Table of n, a(n) for n = 1..875
Steven Dutch, Almost 30-60 Triples
Project Euler, Problem 94: Almost equilateral triangles
Eric Weisstein's World of Mathematics, Heronian Triangle and Pell Equation

For the continued fraction expansion of sqrt(3), cf. A002530, A002531, A040001.

(2/3) [ A007655(n+2) - (-1)^n*A001353(n+1) ] (conjectured). - Ralf Stephan, May 17 2007
Empirical g.f.: 12*x / ((x^2-14*x+1)*(x^2+4*x+1)). - Colin Barker, Apr 10 2013
a(n) = A001353(n+1)*A195499(n) = A001353(n+1)*A120892(n+1) - Danny Rorabaugh, Oct 16 2015

More terms from Zak Seidov, Feb 23 2005

More terms from Dan Sanders (dan(AT)ified.ca), Oct 22 2005

A290732 Number of distinct values of X(3X-1)/2 mod n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 4, 8, 9, 6, 6, 12, 7, 8, 9, 16, 9, 18, 10, 12, 12, 12, 12, 24, 11, 14, 27, 16, 15, 18, 16, 32, 18, 18, 12, 36, 19, 20, 21, 24, 21, 24, 22, 24, 27, 24, 24, 48, 22, 22, 27, 28, 27, 54, 18, 32, 30, 30, 30, 36

Offset: 1

N. J. A. Sloane, Aug 10 2017

			The values taken by (3*X^2-X)/2 mod n for small n are:
   1, [0]
   2, [0, 1]
   3, [0, 1, 2]
   4, [0, 1, 2, 3]
   5, [0, 1, 2]
   6, [0, 1, 2, 3, 4, 5]
   7, [0, 1, 2, 5]
   8, [0, 1, 2, 3, 4, 5, 6, 7]
   9, [0, 1, 2, 3, 4, 5, 6, 7, 8]
  10, [0, 1, 2, 5, 6, 7]
  11, [0, 1, 2, 4, 5, 7]
  12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], (24-August-2016). See Table 6.

Cf. A000224 (analog for X^2), A014113, A290729, A290730, A290731, A317623.

a:=[]; M:=80;
for n from 1 to M do
q1:={};
for i from 0 to 2*n-1 do q1:={op(q1), i*(3*i-1)/2 mod n}; od;
s1:=sort(convert(q1,list));
a:=[op(a),nops(s1)];
od:
a;

a[n_] := Table[PolynomialMod[X(3X-1)/2, n], {X, 0, 2*n-1}]// Union // Length;
Array[a, 60] (* Jean-François Alcover, Sep 01 2018 *)

a(n)={my(v=vector(n)); for(i=0, 2*n-1, v[i*(3*i-1)/2%n + 1]=1); vecsum(v)} \\ Andrew Howroyd, Oct 27 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p<=3, p^e, 1 + p^(e+1)\(2*p+2)))} \\ Andrew Howroyd, Nov 03 2018

a(3^n) = 3^n. - Hugo Pfoertner, Aug 25 2018
a(n) = A317623(n) * A040001(n). - Andrew Howroyd, Oct 27 2018
Multiplicative with a(2^e) = 2^e, a(3^e) = 3^e, a(p^e) = 1 + floor( p^(e+1)/(2*p+2) ) for prime p >= 5. - Andrew Howroyd, Nov 03 2018

Even terms corrected by Andrew Howroyd, Nov 03 2018

A040003 Continued fraction for sqrt(6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			2.449489742783178098197284074... = 2 + 1/(2 + 1/(4 + 1/(2 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 143.
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. A010464 (decimal expansion).
Equals twice A040001.
Essentially the same as A010694.

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

ContinuedFraction[Sqrt[6], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(6)); for (n=0, 20000, write("b040003.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

a(n-1) = gcd(2^n, 3^n+1) (empirical). - Michel Marcus, Sep 03 2020

G.f.: 2*(1 + x + x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A337336 a(n) = A048673(n^2).

Original entry on oeis.org

1, 5, 13, 41, 25, 113, 61, 365, 313, 221, 85, 1013, 145, 545, 613, 3281, 181, 2813, 265, 1985, 1513, 761, 421, 9113, 1201, 1301, 7813, 4901, 481, 5513, 685, 29525, 2113, 1625, 2965, 25313, 841, 2381, 3613, 17861, 925, 13613, 1105, 6845, 15313, 3785, 1405, 82013, 7321, 10805, 4513, 11705, 1741, 70313, 4141, 44105

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

All terms are odd and neither there are multiples of 3, thus only terms of A007310 occur here.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000290, A003961, A007310, A040001, A048673, A337337, A337338, A337339.

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A337336(n) = A048673(n^2);

a(n) = A048673(A000290(n)) = (1+(A003961(n)^2))/2.

For all n>= 1, A010872(a(n)) = A040001(n).

A013648 Numbers k such that the periodic part of the continued fraction for sqrt(k) contains a single 1.

Original entry on oeis.org

3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 175, 176, 195, 208, 224, 255, 288, 323, 360, 399, 440, 483, 528, 551, 575, 624, 675, 728, 783, 799, 840, 899, 960, 1023, 1035, 1088, 1155, 1224, 1247, 1295, 1368, 1403, 1443, 1520, 1599, 1680, 1763, 1848, 1872

Offset: 1

Clark Kimberling

nonn

All the terms of A005563 are here, as well as some additional terms (with even period > 2 and the digit 1 in central position) (e.g., sqrt(175) = [13,'4, 2, 1, 2, 4, 26']).

Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Robert Macmillan, Continued fractions, Math. Gaz., Vol. 84, No. 499 (2000), pp. 30-35. See p. 34.

Union of A005563 and A102538.

Cf. A040001, A040005, A040011, A040019, A040029, A062196.

Select[ Range@ 1900, !IntegerQ[ Sqrt@ #] && Count[ ContinuedFraction[ Sqrt@ #][[2]], 1] == 1 &] (* Robert G. Wilson v, Jul 03 2011 *)

Additional comments from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001

Incorrect formulas and programs removed by R. J. Mathar, Jan 06 2011

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cp's OEIS Frontend

A168077 a(2n) = A129194(2n)/2; a(2n+1) = A129194(2n+1).

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A187110 Decimal expansion of sqrt(3/8).

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A226314 Triangle read by rows: T(i,j) = j+(i-j)/gcd(i,j) (1<=i<=j).

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A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

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A066660 Number of divisors of 2n excluding 1.

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A102341 Areas of 'close-to-equilateral' integer triangles.

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A290732 Number of distinct values of X*(3*X-1)/2 mod n.

A290732 Number of distinct values of X(3X-1)/2 mod n.