cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A058331 a(n) = 2*n^2 + 1.

Original entry on oeis.org

1, 3, 9, 19, 33, 51, 73, 99, 129, 163, 201, 243, 289, 339, 393, 451, 513, 579, 649, 723, 801, 883, 969, 1059, 1153, 1251, 1353, 1459, 1569, 1683, 1801, 1923, 2049, 2179, 2313, 2451, 2593, 2739, 2889, 3043, 3201, 3363, 3529, 3699, 3873, 4051
Offset: 0

Views

Author

Erich Friedman, Dec 12 2000

Keywords

Comments

Maximal number of regions in the plane that can be formed with n hyperbolas.
Also the number of different 2 X 2 determinants with integer entries from 0 to n.
Number of lattice points in an n-dimensional ball of radius sqrt(2). - David W. Wilson, May 03 2001
Equals A112295(unsigned) * [1, 2, 3, ...]. - Gary W. Adamson, Oct 07 2007
Binomial transform of A166926. - Gary W. Adamson, May 03 2008
a(n) = longest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (2n^2 + 1, 2n^2 + 2, 4n^2 + 1).
{a(k): 0 <= k < 3} = divisors of 9. - Reinhard Zumkeller, Jun 17 2009
Number of ways to partition a 3*n X 2 grid into 3 connected equal-area regions. - R. H. Hardin, Oct 31 2009
Let A be the Hessenberg matrix of order n defined by: A[1, j] = 1, A[i, i] := 2, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 3, a(n - 1) = coeff(charpoly(A, x), x^(n - 2)). - Milan Janjic, Jan 26 2010
Except for the first term of [A002522] and [A058331] if X = [A058331], Y = [A087113], A = [A002522], we have, for all other terms, Pell's equation: [A058331]^2 - [A002522]*[A087113]^2 = 1; (X^2 - A*Y^2 = 1); e.g., 3^2 -2*2^2 = 1; 9^2 - 5*4^2 = 1; 129^2 - 65*16^2 = 1, and so on. - Vincenzo Librandi, Aug 07 2010
Niven (1961) gives this formula as an example of a formula that does not contain all odd integers, in contrast to 2n + 1 and 2n - 1. - Alonso del Arte, Dec 05 2012
Numbers m such that 2*m-2 is a square. - Vincenzo Librandi, Apr 10 2015
Number of n-tuples from the set {1,0,-1} where at most two elements are nonzero. - Michael Somos, Oct 19 2022
a(n) gives the x-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The y-value is given by 2*n (see Tattersall). - Stefano Spezia, Jul 23 2025

Examples

			a(1) = 3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different determinants.
G.f. = 1 + 3*x + 9*x^2 + 19*x^3 + 33*x^4 + 51*x^5 + 73*x^6 + ... - _Michael Somos_, Oct 19 2022

References

  • Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

Links

Crossrefs

Cf. A000124.
Second row of array A099597.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A112295.
Cf. A087113, A002552.
Cf. A005408, A016813, A086514, A000125, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.
Column 2 of array A188645.
Cf. A001105 and A247375. - Bruno Berselli, Sep 16 2014
Cf. A056106, A251599.
Cf. A000384, A000217, A166926.

Programs

  • Haskell
    a058331 = (+ 1) . a001105  -- Reinhard Zumkeller, Dec 13 2014
  • Magma
    [2*n^2 + 1 : n in [0..100]]; // Wesley Ivan Hurt, Feb 02 2017
  • Mathematica
    b[g_] := Length[Union[Map[Det, Flatten[ Table[{{i, j}, {k, l}}, {i, 0, g}, {j, 0, g}, {k, 0, g}, {l, 0, g}], 3]]]] Table[b[g], {g, 0, 20}]
2*Range[0, 49]^2 + 1 (* Alonso del Arte, Dec 05 2012 *)
  • PARI
    a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011

Formula

G.f.: (1 + 3x^2)/(1 - x)^3. - Paul Barry, Apr 06 2003
a(n) = M^n * [1 1 1], leftmost term, where M = the 3 X 3 matrix [1 1 1 / 0 1 4 / 0 0 1]. a(0) = 1, a(1) = 3; a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). E.g., a(4) = 33 since M^4 *[1 1 1] = [33 17 1]. - Gary W. Adamson, Nov 11 2004
a(n) = cosh(2*arccosh(n)). - Artur Jasinski, Feb 10 2010
a(n) = 4*n + a(n-1) - 2 for n > 0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(n) = (((n-1)^2 + n^2))/2 + (n^2 + (n+1)^2)/2. - J. M. Bergot, May 31 2012
a(n) = A251599(3*n) for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = sqrt(8*(A000217(n-1)^2 + A000217(n)^2) + 1). - J. M. Bergot, Sep 03 2015
E.g.f.: (2*x^2 + 2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A002378(n) + A002061(n). - Bruce J. Nicholson, Aug 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(2))*coth(Pi/sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(2))*csch(Pi/sqrt(2)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(2))*sinh(Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(2))*csch(Pi/sqrt(2)). (End)
From Leo Tavares, May 23 2022: (Start)
a(n) = A000384(n+1) - 3*n.
a(n) = 3*A000217(n) + A000217(n-2). (End)
a(n) = a(-n) for all n in Z and A037235(n) = Sum_{k=0..n-1} a(k). - Michael Somos, Oct 19 2022

Extensions

Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001

A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).

Original entry on oeis.org

1, 2, 1, 0, 4, 1, 0, 0, 6, 1, 0, 0, 0, 8, 1, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 14, 1, 0, 0, 0, 0, 0, 0, 0, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 1
Offset: 0

Views

Author

Gary W. Adamson, Oct 07 2007

Keywords

Comments

A134082 * [1,2,3,...] = A084849: (1, 4, 11, 22, 37, ...).
Binomial transform of A134082 = A134083.
A112295 replaces subdiagonal with (-1,-3,-5, ...).

Examples

			First few rows of the triangle:
  1;
  2,  1;
  0,  4,  1;
  0,  0,  6,  1;
  0,  0,  0,  8,  1;
  0,  0,  0,  0, 10,  1;
  ...

Links

Crossrefs

Cf. A005408, A084849, A112295, A134083.

Programs

  • Magma
    A134082:= func< n,k | k eq n select 1 else k eq n-1 select 2*n else 0 >;
[A134082(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 17 2021
  • Mathematica
    T[n_, k_]:= If[k==n, 1, If[k==n-1, 2*n, 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)
  • Sage
    def A134082(n,k): return 1 if k==n else 2*n if k==n-1 else 0
flatten([[A134082(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 17 2021

Formula

Triangle read by rows, (n-1) zeros followed by (2n, 1). As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (2,4,6,8,...) in the subdiagonal.
From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] (1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e., I_o[2*(u*x)^(1/2)] = Sum_{j>=0} u^j/j! * x^j/j!. - Tom Copeland, Dec 07 2007
Row polynomial e.g.f.: exp(x*y)(1+2x). - Tom Copeland, Dec 03 2013
Sum_{k=0..n} T(n,k) = 2*n+1 = A005408(n). - G. C. Greubel, Feb 17 2021

Extensions

More terms added by G. C. Greubel, Feb 17 2021

A112292 An invertible triangle of ratios of double factorials.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 15, 5, 1, 105, 105, 35, 7, 1, 945, 945, 315, 63, 9, 1, 10395, 10395, 3465, 693, 99, 11, 1, 135135, 135135, 45045, 9009, 1287, 143, 13, 1, 2027025, 2027025, 675675, 135135, 19305, 2145, 195, 15, 1, 34459425, 34459425, 11486475, 2297295, 328185, 36465, 3315, 255, 17, 1
Offset: 0

Views

Author

Paul Barry, Sep 01 2005

Keywords

Comments

As a square array read by antidiagonals, column k has e.g.f. (1/(1-2x)^(1/2))*(1/(1-2x))^k. - Paul Barry, Sep 04 2005
Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) is this sequence and T(n, k, 3) = A136214. - Peter Luschny, Jun 01 2009, revised Jun 18 2019

Examples

			Triangle begins
      1;
      1,     1;
      3,     3,    1;
     15,    15,    5,  1;
    105,   105,   35,  7,  1;
    945,   945,  315, 63,  9,  1;
  10395, 10395, 3465,693, 99, 11, 1;
Inverse is A112295, which begins
   1;
  -1,  1;
   0, -3,  1;
   0,  0, -5,  1;
   0,  0,  0, -7,  1;
   0,  0,  0,  0, -9,  1;
Similar results arise for higher factorials.

Crossrefs

Columns include A001147, A051577, A051579.
Row sums are A112293.
Diagonal sums are A112294.
Cf. A094587 (p=1), this sequence (p=2), A136214 (p=3).

Programs

  • Mathematica
    T[n_, k_] := If[k <= n, (2n-1)!!/(2k-1)!!, 0];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

Formula

T(n, k)=if(k<=n, (2n-1)!!/(2k-1)!!, 0);
T(n, k)=if(k<=n, n!*C(2n, n)2^(k-n)/(k!*C(2k, k)), 0);
T(n, k)=if(k<=n, 2^(n-k)(n-1/2)!/(k-1/2)!, 0);
T(n, k)=if(k<=n, (n+1)!*C(n)2^(k-n)/((k+1)!*C(k)), 0).

A134081 Triangle T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1), read by rows.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 12, 8, 1, 5, 22, 26, 11, 1, 6, 35, 60, 45, 14, 1, 7, 51, 115, 125, 69, 17, 1, 8, 70, 196, 280, 224, 98, 20, 1, 9, 92, 308, 546, 574, 364, 132, 23, 1, 10, 117, 456, 966, 1260, 1050, 552, 171, 26, 1
Offset: 0

Views

Author

Gary W. Adamson, Oct 07 2007

Keywords

Examples

			First few rows of the triangle are:
  1;
  2,  1;
  3,  5,   1;
  4, 12,   8,   1;
  5, 22,  26,  11,  1;
  6, 35,  60,  45, 14,  1;
  7, 51, 115, 125, 69, 17, 1;
  ...

Links

Crossrefs

Cf. A002064, A007318, A112295.
Columns: A000027, A000326, A002413, A051740, A051879.

Programs

  • Magma
    A134081:= func< n,k | Binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1) >;
[A134081(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021
  • Mathematica
    T[n_, k_]:= Binomial[n, k]*((2*k+1)*(n-k) +k+1)/(k+1);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)
  • Sage
    def A134081(n,k): return binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1)
flatten([[A134081(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

Formula

Binomial transform of A112295(unsigned).
From G. C. Greubel, Feb 17 2021: (Start)
T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1).
Sum_{k=0..n} T(n, k) = 2^n *n + 1 = A002064(n). (End)
Showing 1-4 of 4 results.

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