A019679 -id:A019679 - cp's OEIS frontend

A001353 a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

0, 1, 4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244

Offset: 0

N. J. A. Sloane

3*a(n)^2 + 1 is a square. Moreover, 3*a(n)^2 + 1 = (2*a(n) - a(n-1))^2.
Consecutive terms give nonnegative solutions to x^2 - 4*x*y + y^2 = 1. - Max Alekseyev, Dec 12 2012
Values y solving the Pellian x^2 - 3*y^2 = 1; corresponding x values given by A001075(n). Moreover, we have a(n) = 2*a(n-1) + A001075(n-1). - Lekraj Beedassy, Jul 13 2006
Number of spanning trees in 2 X n grid: by examining what happens at the right-hand end we see that a(n) = 3*a(n-1) + 2*a(n-2) + 2*a(n-3) + ... + 2*a(1) + 1, where the final 1 corresponds to the tree ==...=| !. Solving this we get a(n) = 4*a(n-1) - a(n-2).
Complexity of 2 X n grid.
A016064 also describes triangles whose sides are consecutive integers and in which an inscribed circle has an integer radius. A001353 is exactly and precisely mapped to the integer radii of such inscribed circles, i.e., for each term of A016064, the corresponding term of A001353 gives the radius of the inscribed circle. - Harvey P. Dale, Dec 28 2000
n such that 3*n^2 = floor(sqrt(3)*n*ceiling(sqrt(3)*n)). - Benoit Cloitre, May 10 2003
For n>0, ratios a(n+1)/a(n) may be obtained as convergents of the continued fraction expansion of 2+sqrt(3): either as successive convergents of [4;-4] or as odd convergents of [3;1, 2]. - Lekraj Beedassy, Sep 19 2003
Ways of packing a 3 X (2*n-1) rectangle with dominoes, after attaching an extra square to the end of one of the sides of length 3. With reference to A001835, therefore: a(n) = a(n-1) + A001835(n-1) and A001835(n) = 3*A001835(n-1) + 2*a(n-1). - Joshua Zucker and the Castilleja School Math Club, Oct 28 2003
a(n+1) is a Chebyshev transform of 4^n, where the sequence with g.f. G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)). - Paul Barry, Oct 25 2004
This sequence is prime-free, because a(2n) = a(n) * (a(n+1)-a(n-1)) and a(2n+1) = a(n+1)^2 - a(n)^2 = (a(n+1)+a(n)) * (a(n+1)-a(n)). - Jianing Song, Jul 06 2019
Numbers such that there is an m with t(n+m) = 3*t(m), where t(n) are the triangular numbers A000217. For instance, t(35) = 3*t(20) = 630, so 35 - 20 = 15 is in the sequence. - Floor van Lamoen, Oct 13 2005
a(n) = number of distinct matrix products in (A + B + C + D)^n where commutator [A,B] = 0 but neither A nor B commutes with C or D. - Paul D. Hanna and Max Alekseyev, Feb 01 2006
For n > 1, middle side (or long leg) of primitive Pythagorean triangles having an angle nearing Pi/3 with larger values of sides. [Complete triple (X, Y, Z), X < Y < Z, is given by X = A120892(n), Y = a(n), Z = A120893(n), with recurrence relations X(i+1) = 2*{X(i) - (-1)^i} + a(i); Z(i+1) = 2*{Z(i) + a(i)} - (-1)^i.] - Lekraj Beedassy, Jul 13 2006
From Dennis P. Walsh, Oct 04 2006: (Start)
Number of 2 X n simple rectangular mazes. A simple rectangular m X n maze is a graph G with vertex set {0, 1, ..., m} X {0, 1, ..., n} that satisfies the following two properties: (i) G consists of two orthogonal trees; (ii) one tree has a path that sequentially connects (0,0),(0,1), ..., (0,n), (1,n), ...,(m-1,n) and the other tree has a path that sequentially connects (1,0), (2,0), ..., (m,0), (m,1), ..., (m,n). For example, a(2) = 4 because there are four 2 X 2 simple rectangular mazes:
                                          
   |  |  |        |   |       |     |        |   |
   |   |        |   |       |  ||        |   |
(End)
[1, 4, 15, 56, 209, ...] is the Hankel transform of [1, 1, 5, 26, 139, 758, ...](see A005573). - Philippe Deléham, Apr 14 2007
The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4, 26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075, denominators=A001353. - Clark Kimberling, Aug 27 2008
From Gary W. Adamson, Jun 21 2009: (Start)
A001353 and A001835 = bisection of continued fraction [1, 2, 1, 2, 1, 2, ...], i.e., of [1, 3, 4, 11, 15, 41, ...].
For n>0, a(n) equals the determinant of an (n-1) X (n-1) tridiagonal matrix with ones in the super and subdiagonals and (4, 4, 4, ...) as the main diagonal. [Corrected by Johannes Boot, Sep 04 2011]
A001835 and A001353 = right and next to right borders of triangle A125077. (End)
a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 4's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011
2a(n) is the number of n-color compositions of 2n consisting of only even parts; see Guo in references. - Brian Hopkins, Jul 19 2011
Pisano period lengths: 1, 2, 6, 4, 3, 6, 8, 4, 18, 6, 10, 12, 12, 8, 6, 8, 18, 18, 5, 12, ... - R. J. Mathar, Aug 10 2012
From Michel Lagneau, Jul 08 2014: (Start)
a(n) is defined also by the recurrence a(1)=1; for n>1, a(n+1) = 2*a(n) + sqrt(3*a(n)^2 + 1) where a(n) is an integer for every n. This sequence is generalizable by the sequence b(n,m) of parameter m with the initial condition b(1,m) = 1, and for n > 1 b(n+1,m) = m*b(n,m) + sqrt((m^2 - 1)*b(n,m)^2 + 1) for m = 2, 3, 4, ... where b(n,m) is an integer for every n.
The first corresponding sequences are
   b(n,2)  = a(n) = A001353(n);
   b(n,3)  = A001109(n);
   b(n,4)  = A001090(n);
   b(n,5)  = A004189(n);
   b(n,6)  = A004191(n);
   b(n,7)  = A007655(n);
   b(n,8)  = A077412(n);
   b(n,9)  = A049660(n);
   b(n,10) = A075843(n);
   b(n,11) = A077421(n);
   ....................
We obtain a general sequence of polynomials {b(n,x)} = {1, 2*x, 4*x^2 - 1, 8*x^3 - 4*x, 16*x^4 - 12*x^2 + 1, 32*x^5 - 32*x^3 + 6*x, ...} with x = m where each b(n,x) is a Gegenbauer polynomial defined by the recurrence b(n,x)- 2*x*b(n-1,x) + b(n-2,x) = 0, the same relation as the Chebyshev recurrence, but with the initial conditions b(x,0) = 1 and b(x,1) = 2*x instead b(x,0) = 1 and b(x,1) = x for the Chebyshev polynomials. (End)
If a(n) denotes the n-th term of the above sequence and we construct a triangle whose sides are a(n) - 1, a(n) + 1 and sqrt(3a(n)^2 + 1), then, for every n the measure of one of the angles of the triangle so constructed will always be 120 degrees. This result of ours was published in Mathematics Spectrum (2012/2013), Vol. 45, No. 3, pp. 126-128. - K. S. Bhanu and Dr. M. N. Deshpande, Professor (Retd), Department of Statistics, Institute of Science, Nagpur (India).
For n >= 1, a(n) equals the number of 01-avoiding words of length n - 1 on alphabet {0, 1, 2, 3}. - Milan Janjic, Jan 25 2015
For n > 0, 10*a(n) is the number of vertices and roots on level n of the {4, 5} mosaic (see L. Németh Table 1 p. 6). - Michel Marcus, Oct 30 2015
(2 + sqrt(3))^n = A001075(n) + a(n)*sqrt(3), n >= 0; integers in the quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 16 2018
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 12 2019
The Cholesky decomposition A = C C* for tridiagonal A with A[i,i] = 4 and A[i+1,i] = A[i,i+1] = -1, as it arises in the discretized 2D Laplace operator (Poisson equation...), has nonzero elements C[i,i] = sqrt(a(i+1)/a(i)) = -1/C[i+1,i], i = 1, 2, 3, ... - M. F. Hasler, Mar 12 2021
The triples (a(n-1), 2a(n), a(n+1)), n=2,3,..., are exactly the triples (a,b,c) of positive integers a < b < c in arithmetic progression such that a*b+1, b*c+1, and c*a+1 are perfect squares. - Bernd Mulansky, Jul 10 2021
From Greg Dresden and Linyun Sheng, Jul 01 2025: (Start)
  a(n) is the number of ways to tile this strip of length n,
  .________________
  |  |  |  |  |  |  |\
  ||__||__||__|_\,
  where the last cell is a right triangle, with three types of tiles: 1 X 1 squares, 1 X 1 small right triangles, and large right triangles (with large side length 2) formed by joining two of those small right triangles along a short leg. As an example, here is one of the a(7)=2911 ways to tile the 1 X 7 strip with these kinds of tiles:
  .________________
  |\   /|\ | /|  | / \
  |\/_|\|/|__|/_\,
(End)

			For example, when n = 3:
  ****
  .***
  .***
can be packed with dominoes in 4 different ways: 3 in which the top row is tiled with two horizontal dominoes and 1 in which the top row has two vertical and one horizontal domino, as shown below, so a(2) = 4.
  ---- ---- ---- ||--
  .||| .--| .|-- .|||
  .||| .--| .|-- .|||
G.f. = x + 4*x^2 + 15*x^3 + 56*x^4 + 209*x^5 + 780*x^6 + 2911*x^7 + 10864*x^8 + ...

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Index entries for linear recurrences with constant coefficients, signature (4,-1)
Index entries for sequences related to Chebyshev polynomials.

Cf. A001075, A001542, A001571, A001834, A001835, A002531, A003500, A005246, A016064, A019679, A079935, A082840.
A bisection of A002530.
Cf. A125077.
A row of A116469.
Chebyshev sequence U(n, m): A000027 (m=1), this sequence (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
Cf. A323182.

a:=[0,1];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Feb 16 2018

a001353 n = a001353_list !! n
a001353_list =
   0 : 1 : zipWith (-) (map (4 *) $ tail a001353_list) a001353_list
-- Reinhard Zumkeller, Aug 14 2011

I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 06 2019

A001353 := proc(n) option remember; if n <= 1 then n else 4*A001353(n-1)-A001353(n-2); fi; end;
A001353:=z/(1-4*z+z**2); # Simon Plouffe in his 1992 dissertation.
seq( simplify(ChebyshevU(n-1, 2)), n=0..20); # G. C. Greubel, Dec 23 2019

a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 13 2005 *)
Table[GegenbauerC[n-1, 1, 2], {n, 0, 30}] (* Zerinvary Lajos, Jul 14 2009 *)
Table[-((I Sin[n ArcCos[2]])/Sqrt[3]), {n, 0, 30}] // FunctionExpand (* Eric W. Weisstein, Jul 16 2011 *)
Table[Sinh[n ArcCosh[2]]/Sqrt[3], {n, 0, 30}] // FunctionExpand (* Eric W. Weisstein, Jul 16 2011 *)
Table[ChebyshevU[n-1, 2], {n, 0, 30}] (* Eric W. Weisstein, Jul 16 2011 *)
a[0]:=0; a[1]:=1; a[n_]:= a[n]= 4a[n-1] - a[n-2]; Table[a[n], {n, 0, 30}] (* Alonso del Arte, Jul 19 2011 *)
LinearRecurrence[{4, -1}, {0, 1}, 30] (* Sture Sjöstedt, Dec 06 2011 *)
Round@Table[Fibonacci[2n, Sqrt[2]]/Sqrt[2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 15 2016 *)

M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=0,30,print1(([1,0,0]*M^i)[2],",")) \\ Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005

{a(n) = real( (2 + quadgen(12))^n / quadgen(12) )}; /* Michael Somos, Sep 19 2008 */

{a(n) = polchebyshev(n-1, 2, 2)}; /* Michael Somos, Sep 19 2008 */

concat(0, Vec(x/(1-4*x+x^2) + O(x^30))) \\ Altug Alkan, Oct 30 2015

a001353 = [0, 1]
for n in range(30): a001353.append(4*a001353[-1] - a001353[-2])
print(a001353)  # Gennady Eremin, Feb 05 2022

[lucas_number1(n,4,1) for n in range(30)] # Zerinvary Lajos, Apr 22 2009

[chebyshev_U(n-1,2) for n in (0..20)] # G. C. Greubel, Dec 23 2019

G.f.: x/(1-4*x+x^2).
a(n) = ((2 + sqrt(3))^n - (2 - sqrt(3))^n)/(2*sqrt(3)).
a(n) = sqrt((A001075(n)^2 - 1)/3).
a(n) = 2*a(n-1) + sqrt(3*a(n-1)^2 + 1). - Lekraj Beedassy, Feb 18 2002
Limit_{n->oo} a(n)/a(n-1) = 2 + sqrt(3). - Gregory V. Richardson, Oct 06 2002
Binomial transform of A002605.
E.g.f.: exp(2*x)*sinh(sqrt(3)*x)/sqrt(3).
a(n) = S(n-1, 4) = U(n-1, 2); S(-1, x) := 0, Chebyshev's polynomials of the second kind A049310.
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*4^(n - 2*k). - Paul Barry, Oct 25 2004
a(n) = Sum_{k=0..n-1} binomial(n+k,2*k+1)*2^k. - Paul Barry, Nov 30 2004
a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3), n>=3. - Lekraj Beedassy, Jul 13 2006
a(n) = -A106707(n). - R. J. Mathar, Jul 07 2006
M^n * [1,0] = [A001075(n), A001353(n)], where M = the 2 X 2 matrix [2,3; 1,2]; e.g., a(4) = 56 since M^4 * [1,0] = [97, 56] = [A001075(4), A001353(4)]. - Gary W. Adamson, Dec 27 2006
From Michael Somos, Sep 19 2008: (Start)
Sequence satisfies 1 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v.
a(n) = -a(-n) for all integer n. (End)
Rational recurrence: a(n) = (17*a(n-1)*a(n-2) - 4*(a(n-1)^2 + a(n-2)^2))/a(n-3) for n > 3. - Jaume Oliver Lafont, Dec 05 2009
If p[i] = Fibonacci(2i) and if A is the Hessenberg matrix of order n defined by A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j + 1), and A[i,j] = 0 otherwise, then, for n >= 1, a(n) = det A. - Milan Janjic, May 08 2010
From Eric W. Weisstein, Jul 16 2011: (Start)
a(n) = C_{n-1}^{(1)}(2), where C_n^{(m)}(x) is the Gegenbauer polynomial.
a(n) = -i*sin(n*arccos(2))/sqrt(3).
a(n) = sinh(n*arccosh(2))/sqrt(3). (End)
a(n) = b such that Integral_{x=0..Pi/2} (sin(n*x))/(2-cos(x)) dx = c + b*log(2). - Francesco Daddi, Aug 02 2011
a(n) = sqrt(A098301(n)) = sqrt([A055793 / 3]), base 3 analog of A031150. - M. F. Hasler, Jan 16 2012
a(n+1) = Sum_{k=0..n} A101950(n,k)*3^k. - Philippe Deléham, Feb 10 2012
1, 4, 15, 56, 209, ... = INVERT(INVERT(1, 2, 3, 4, 5, ...)). - David Callan, Oct 13 2012
From Peter Bala, Dec 23 2012: (Start)
Product_{n >= 1} (1 + 1/a(n)) = 1 + sqrt(3).
Product_{n >= 2} (1 - 1/a(n)) = 1/4*(1 + sqrt(3)). (End)
a(n+1) = (A001834(n) + A001835(n))/2. a(n+1) + a(n) = A001834(n). a(n+1) - a(n) = A001835(n). - Richard R. Forberg, Sep 04 2013
a(n) = -(-i)^(n+1)*Fibonacci(n, 4*i), i = sqrt(-1). - G. C. Greubel, Jun 06 2019
a(n)^2 - a(m)^2 = a(n+m) * a(n-m), a(n+2)*a(n-2) = 16*a(n+1)*a(n-1) - 15*a(n)^2, a(n+3)*a(n-2) = 15*a(n+2)*a(n-1) - 14*a(n+1)*a(n) for all integer n, m. - Michael Somos, Dec 12 2019
a(n) = 2^n*Sum_{k >= n} binomial(2*k,2*n-1)*(1/3)^(k+1). Cf. A102591. - Peter Bala, Nov 29 2021
a(n) = Sum_{k > 0} (-1)^((k-1)/2)*binomial(2*n, n+k)*(k|12), where (k|12) is the Kronecker symbol. - Greg Dresden, Oct 11 2022
Sum_{k=0..n} a(k) = (a(n+1) - a(n) - 1)/2. - Prabha Sivaramannair, Sep 22 2023
a(2n+1) = A001835(n+1) * A001834(n). - M. Farrokhi D. G., Oct 15 2023
Sum_{n>=1} arctan(1/(4*a(n)^2)) = Pi/12 (A019679) (Ohtskua, 2024). - Amiram Eldar, Aug 29 2024
From Peter Bala, May 21 2025: (Start)
Product_{n >= 1} (1 + 1/a(n))^2 = 2*(2 + sqrt(3)) (telescoping product: (1 + 1/a(2*n-1))^2 * (1 + 1/a(2*n-2))^2 = (4 + 2*A251963(n)/A005246(2*n)^2)/(4 + 2*A251963(n-1)/A005246(2*n-2)^2) ).
Product_{n >= 2} (1 - 1/a(n))^2 = (1/8)*(2 + sqrt(3)).
Product_{n >= 1} ((a(2*n) + 1)/(a(2*n) - 1))^2 = 3 (telescoping product: ((a(2*n) + 1)/(a(2*n) - 1))^2 = (3 - 2/A001835(n+1)^2)/(3 - 2/A001835(n)^2) ).
Product_{n >= 2} ((a(2*n-1) + 1)/(a(2*n-1) - 1))^2 = 4/3.
The o.g.f. A(x) satisfies A(x) + A(-x) + 8*A(x)*A(-x) = 0. The o.g.f. for A007655 equals -A(sqrt(x))*A(-sqrt(x)). (End)

A016945 a(n) = 6*n+3.

Original entry on oeis.org

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315, 321, 327

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(37).
Continued fraction expansion of tanh(1/3).
If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
Leaves of the Odd Collatz-Tree: a(n) has no odd predecessors in all '3x+1' trajectories where it occurs: A139391(2*k+1) <> a(n) for all k; A082286(n)=A006370(a(n)). - Reinhard Zumkeller, Apr 17 2008
Let random variable X have a uniform distribution on the interval [0,c] where c is a positive constant. Then, for positive integer n, the coefficient of determination between X and X^n is (6n+3)/(n+2)^2, that is, A016945(n)/A000290(n+2). Note that the result is independent of c. For the derivation of this result, see the link in the Links section below. - Dennis P. Walsh, Aug 20 2013
Positions of 3 in A020639. - Zak Seidov, Apr 29 2015
a(n+2) gives the sum of 6 consecutive terms of A004442 starting with A004442(n). - Wesley Ivan Hurt, Apr 08 2016
Numbers k such that Fibonacci(k) mod 4 = 2. - Bruno Berselli, Oct 17 2017
Also numbers k such that t^k == -1 (mod 7), where t is a member of A047389. - Bruno Berselli, Dec 28 2017

Friedrich L. Bauer, Der (ungerade) Collatz-Baum, Informatik Spektrum 31 (Springer, April 2008), pp. 379-384.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Dennis P. Walsh, The correlation for a power curve on nonnegative support.
Eric Weisstein's World of Mathematics, Collatz Problem.
Index entries for sequences related to 3x+1 (or Collatz) problem.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Third row of A092260.
Cf. A004442, A008588, A010503, A016921, A016933, A016957, A016969, A019679, A115754.
Subsequence of A061641; complement of A047263; bisection of A047241.
Cf. A000225. - Loren Pearson, Jul 02 2009
Cf. A020639. - Zak Seidov, Apr 29 2015
Odd numbers in A355200.

List([0..60], n-> 3*(1+2*n)); # G. C. Greubel, Sep 18 2019

a016945 = (+ 3) . (* 6)
a016945_list = [3, 9 ..]
-- Wesley Ivan Hurt, Sep 29 2013

[6*n+3 : n in [0..60]]; // Wesley Ivan Hurt, Sep 29 2013

seq(6*n+3, n=0..60); # Dennis P. Walsh, Aug 20 2013
A016945:=n->6*n+3; # Wesley Ivan Hurt, Sep 29 2013

Range[3, 350, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[6n+3, {n, 0, 60}] (* Wesley Ivan Hurt, Sep 29 2013 *)
LinearRecurrence[{2, -1}, {3, 9}, 55] (* Ray Chandler, Jul 17 2015 *)
CoefficientList[Series[3(1+x)/(1-x)^2, {x, 0, 60}], x] (* Robert G. Wilson v, Dec 14 2016 *)

makelist(6*n+3, n, 0, 60); /* Wesley Ivan Hurt, Sep 29 2013 */

{a(n) = 6*n + 3} \\ Wesley Ivan Hurt, Sep 29 2013

x='x+O('x^60); Vec(3*(1+x)/(1-x)^2) \\ Altug Alkan, Apr 08 2016

[3*(1+2*n) for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 3*(2*n + 1) = 3*A005408(n), odd multiples of 3.
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A103333(n+1). - Reinhard Zumkeller, Feb 24 2009
a(n) = 12*n - a(n-1) for n>0, a(0)=3. - Vincenzo Librandi, Nov 20 2010
G.f.: 3*(1+x)/(1-x)^2. - Mario C. Enriquez, Dec 14 2016
E.g.f.: 3*(1 + 2*x)*exp(x). - G. C. Greubel, Sep 18 2019
Sum_{n>=0} (-1)^n/a(n) = Pi/12 (A019679). - Amiram Eldar, Dec 10 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)/2 (A010503).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(3/2) (A115754). (End)
a(n) = (n+2)^2 - (n-1)^2. - Alexander Yutkin, Mar 15 2025

A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 12 2011

The Mathematica program includes a graph.
Guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected numbers b and c:
b.....c.......x
1.....2.......A197133
1.....3.......A197134
1.....4.......A197135
1.....5.......A197251
1.....6.......A197252
1.....7.......A197253
1.....8.......A197254
2.....1.......A105199, x=arctan(2)
2.....3.......A019679, x=Pi/12
2.....4.......A197255
2.....5.......A197256
2.....6.......A197257
2.....7.......A197258
2.....8.......A197259
3.....1.......A197260
3.....2.......A197261
3.....4.......A197262
3.....5.......A197263
3.....6.......A197264
3.....7.......A197265
3.....8.......A197266
4.....1.......A197267
4.....2.......A195693, x=arctan(1/(golden ratio))
4.....3.......A197268
1.....4*Pi....A197522
1.....3*Pi....A197571
1.....2*Pi....A197572
1.....3*Pi/2..A197573
1.....Pi......A197574
1.....Pi/2....A197575
1.....Pi/3....A197326
1.....Pi/4....A197327
1.....Pi/6....A197328
2.....Pi/3....A197329
2.....Pi/4....A197330
2.....Pi/6....A197331
3.....Pi/3....A197332
3.....Pi/6....A197375
3.....Pi/4....A197333
1.....1/2.....A197376
1.....1/3.....A197377
1.....2/3.....A197378
Pi....1.......A197576
Pi....2.......A197577
Pi....3.......A197578
2*Pi..1.......A197585
3*Pi..1.......A197586
4*Pi..1.......A197587
Pi/2..1.......A197579
Pi/2..2.......A197580
Pi/2..1/2.....A197581
Pi/3..Pi/4....A197379
Pi/3..Pi/6....A197380
Pi/4..Pi/3....A197381
Pi/4..Pi/6....A197382
Pi/6..Pi/3....A197383
Pi/6..Pi/4..........., x=1
Pi/3..1.......A197384
Pi/3..2.......A197385
Pi/3..3.......A197386
Pi/3..1/2.....A197387
Pi/3..1/3.....A197388
Pi/3..2/3.....A197389
Pi/4..1.......A197390
Pi/4..2.......A197391
Pi/4..3.......A197392
Pi/4..1/2.....A197393
Pi/4..1/3.....A197394
Pi/4..2/3.....A197411
Pi/4..1/4.....A197412
Pi/6..1.......A197413
Pi/6..2.......A197414
Pi/6..3.......A197415
Pi/6..1/2.....A197416
Pi/6..1/3.....A197417
Pi/6..2/3.....A197418
Cf. A197476 for a similar table for sin(b*x) = sin(c*x)^2.

			0.272971849236824950408616...

Index entries for transcendental numbers.

Cf. A002194, A197134, A197476 (cos), A333322.

b = 1; c = 2; f[x_] := Sin[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t] (* A197133 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
(* Second program: *)
RealDigits[ ArcSec[ Root[16 - 16 x^2 + x^6, 3]], 10, 100] // First (* Jean-François Alcover, Feb 19 2013 *)

asin(2*sin(asin(3*sqrt(3)/8)/3)/sqrt(3)) \\ Gleb Koloskov, Sep 15 2021

asin(polrootsreal(4*x^3-4*x+1)[2]) \\ Charles R Greathouse IV, Feb 12 2025

From Gleb Koloskov, Sep 15 2021: (Start)
Equals arcsin(2*sin(arcsin(3*sqrt(3)/8)/3)/sqrt(3))
     = arcsin(2*sin(arcsin(A333322)/3)/A002194). (End)

Edited and a(99) corrected by Georg Fischer, Jul 28 2021

A197476 Decimal expansion of least x>0 having cos(x) = cos(2*x)^2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 15 2011

The Mathematica program includes a graph.  Guide for least x>0 satisfying cos(b*x) = cos(c*x)^2, for selected b and c:
b.....c......x
....2.......A197476
....3.......A197477
....4.......A197478
....1.......A000796, Pi
....3.......A197479
....4.......A197480
....1.......A019669, Pi/2
....2.......A197482
....4.......A197483
....1.......A168229, arctan(sqrt(7))
....2.......A019669, Pi/2
....3.......A019679, Pi/12
....6.......A197485
....8.......A197486
....2.......A003881, Pi/4
....3.......A019670, Pi/3, tangency point
....4.......A197488
....8.......A197489
....4*Pi....A197334
....3*Pi....A197335
....2*Pi....A197490
....3*Pi/2..A197491
....Pi......A197492
....Pi/2....A197493
....Pi/3....A197494
....Pi/4....A197495
....2*Pi/3..A197506
....3*Pi....A197507
....3*Pi/2..A197508
....2*Pi....A197509
....Pi......A197510
....Pi/2....A197511
....Pi/3....A197512
....Pi/4....A197513
....Pi/6....A197514
Pi....1.......A197515
Pi....2.......A197516
Pi....1/2.....A197517
2*Pi..1.......A197518
2*Pi..2.......A197519
2*Pi..3.......A197520
Pi/2..Pi/3....A197521
Pi/2..Pi/6....3
Pi/3..1.......A197582
Pi/3..2.......A197583
Pi/3..1/3.....A197584
See A197133 for a guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected b and c.

			1.137743932905455557789449860055008349584...

Index entries for transcendental numbers.

Cf. A197133.

b = 1; c = 2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.1, 1.3}, WorkingPrecision -> 200]
RealDigits[t] (* A197476 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]
(* or *)
RealDigits[ ArcCos[ ((19 - 3*Sqrt[33])^(1/3) + (19 + 3*Sqrt[33])^(1/3) - 2)/6], 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

Edited by Georg Fischer, Jul 28 2021

A019683 Decimal expansion of Pi/16.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			Pi/16 = 0.19634954084936207740391521145496893026232308746094411381... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4.2, p. 494.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers.

Cf. A157332, A000796, A003881, A019669, A019670, A019675, A019679, A244978.

R:= RealField(100); Pi(R)/16; // G. C. Greubel, Aug 26 2019

RealDigits[N[Pi/16,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 100); Pi/16 \\ G. C. Greubel, Aug 26 2019

numerical_approx(pi/16, digits=100) # G. C. Greubel, Aug 26 2019

From  Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^3*sqrt(1 - x^8) dx.
Equals Integral_{x = 0..inf} x^2/(1 + x^2)^3 dx. (End)
From Amiram Eldar, Aug 04 2020: (Start)
Equals Sum_{k>=1} sin(k)^3 * cos(k)/k.
Equals Sum_{k>=1} sin(k)^3 * cos(k)^2/k.
Equals Sum_{k>=1} (-1)^(k+1) * sin((2*k-1)/4)/(2*k-1)^2. (End)

A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jun 24 2006

Side length of the largest equilateral triangle that can be inscribed in a unit square (as stated in MathWorld/Weisstein link).

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			1.03527618041008304939559535049619331339627560527972...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Eric Weisstein's World of Mathematics, Equilateral Triangle.
Index entries for algebraic numbers, degree 4.

Cf. A002193, A019679, A019691, A019884, A010464, A092242, A101263.

RealDigits[Sec[15 Degree],10,120][[1]] (* Harvey P. Dale, Jun 03 2015 *)

sqrt(6) - sqrt(2) \\ Charles R Greathouse IV, Aug 27 2017

Equals sec(Pi/12) = sec(A019679) = sqrt(6) - sqrt(2) = A010464 - A002193 = csc(5*Pi/12) = 1/sin(5*Pi/12) = 1/sin(10*A019691) = 1/A019884.
Equals Product_{k >= 1} 1/(1 - 1/(36*(2*k - 1)^2)). - Antonio Graciá Llorente, Mar 20 2024
From Amiram Eldar, Nov 24 2024: (Start)
Equals 2*A101263.
Equals Product_{k>=1} (1 - (-1)^k/A092242(k)). (End)
Smallest positive of the 4 real-valued roots of x^4-16*x^2+16=0. - R. J. Mathar, Aug 31 2025

A152584 Decimal expansion of (Pi^3)/24.

Original entry on oeis.org

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

Offset: 1

Eric Desbiaux, Dec 08 2008

Consider infinite sum made of areas of circles Pi*radius^2 with diameter 1/n.
The volume is (Pi/4)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ... + 1/n^2)
= (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ...)
= (Pi/4) * (Pi^2/6) = Pi^3/24.
Equals volume of a cone of height Pi^2/8 and radius 1.
Equals volume of a sphere (4*Pi*Pi^2/32)/3 with radius^3 = (Pi^2/32).

			1.291928195012492507311513127795891466759387023578...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A013661, A003881, A102753, A019679, A111003.

RealDigits[(Pi^3)/24, 10, 50][[1]] (* G. C. Greubel, Jun 09 2017 *)

PARI
```
(Pi^3)/24 \\ G. C. Greubel, Jun 09 2017
```

Equals Integral_{x=0..oo} arctan(x)^2/(x^2 + 1) dx. - Amiram Eldar, Aug 06 2020

A019691 Decimal expansion of Pi/24.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With a different offset, also decimal expansion of 5*Pi/12, 25*Pi/6 or 125*Pi/3. - Michel Marcus, Sep 09 2013

Volume of a quarter sphere of diameter 1. - Omar E. Pol, Aug 19 2019

			0.13089969389957471826927680763664595350821539164062940920728935801282...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. A000796, A019670, A019673, A019675, A019679.

RealDigits[N[Pi/24,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

Pi/24 \\ Charles R Greathouse IV, Sep 09 2013

Equals A019673/4 or A019675/3 or A019679/2. - Omar E. Pol, Aug 19 2019

Equals (1/10) * Sum_{k>=1} sin(k*Pi/6)/k. - Amiram Eldar, May 30 2021

A381689 Decimal expansion of the isoperimetric quotient of a truncated cuboctahedron (great rhombicuboctahedron).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 06 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.8390038051045342803688923433479361567507803475...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377343 (surface area), A377344 (volume).

Cf. A002193, A002194, A010549, A019679, A381684.

First[RealDigits[Pi/12*(11 + Sqrt[98])^2/(2 + Sqrt[2] + Sqrt[3])^3, 10, 100]]

Equals 36*Pi*A377344^2/(A377343^3).

Equals (Pi/12)*(11 + 7*sqrt(2))^2/((2 + sqrt(2) + sqrt(3))^3) = A019679*(11 + A010549)^2/((2 + A002193 + A002194)^3).

A381693 Decimal expansion of the isoperimetric quotient of a truncated icosahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 08 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.90317079253486933111536917369134449479629558352448...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377750 (surface area), A377751 (volume).

Cf. A019679, A002163, A002194, A010476, A381684.

First[RealDigits[Pi/12*(125 + 43*Sqrt[5])^2/(Sqrt[300] + Sqrt[25 + Sqrt[500]])^3, 10, 100]]

Equals 36*Pi*A377751^2/(A377750^3).

Equals (Pi/12)*(125 + 43*sqrt(5))^2/((10*sqrt(3) + sqrt(5*(5 + 2*sqrt(5))))^3) = A019679*(125 + 43*A002163)^2/((10*A002194 + sqrt(5*(5 + A010476)))^3).

cp's OEIS Frontend

A001353 a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

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A016945 a(n) = 6*n+3.

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A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.

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A197476 Decimal expansion of least x>0 having cos(x) = cos(2*x)^2.

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A019683 Decimal expansion of Pi/16.

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