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.

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A130299 A130296 * A051340.

Original entry on oeis.org

1, 3, 2, 5, 3, 3, 7, 4, 4, 4, 9, 5, 5, 5, 5, 11, 6, 6, 6, 6, 6, 13, 7, 7, 7, 7, 7, 7, 15, 8, 8, 8, 8, 8, 8, 8, 17, 9, 9, 9, 9, 9, 9, 9, 9, 19, 10, 10, 10, 10, 10, 10, 10, 10, 10
Offset: 1

Views

Author

Gary W. Adamson, May 20 2007

Keywords

Comments

Row sums = A018387: (1, 5, 11, 19, 29, 41, 55, ...).
A130298 = A051340 * A130296.

Examples

			First few rows of the triangle:
   1;
   3, 2;
   5, 3, 3;
   7, 4, 4, 4;
   9, 5, 5, 5, 5;
  11, 6, 6, 6, 6, 6;
  13, 7, 7, 7, 7, 7, 7;
  ...

Crossrefs

Cf. A051340, A130296, A130298, A028387.

Formula

A130296 * A051340 as infinite lower triangular matrices.

A130307 A051731 * A130296.

Original entry on oeis.org

1, 3, 1, 4, 1, 1, 7, 2, 1, 1, 6, 1, 1, 1, 1, 12, 3, 2, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 15, 3, 2, 2, 1, 1, 1, 1, 13, 2, 2, 1, 1, 1, 1, 1, 1, 18, 3, 2, 2, 2, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Gary W. Adamson, May 20 2007

Keywords

Comments

Row sums = A129235, (1, 4, 6, 11, 10, 20, 14, ...).
Left column = sigma(n), A000203.

Examples

			First few rows of the triangle:
   1;
   3, 1;
   4, 1, 1;
   7, 2, 1, 1;
   6, 1, 1, 1, 1;
  12, 3, 2, 1, 1, 1;
   8, 1, 1, 1, 1, 1, 1;
  ...

Crossrefs

Cf. A130296, A051731, A129235, A000203.

Formula

Inverse Moebius transform of A130296.

A131231 3*A130296 - 2*A128174.

Original entry on oeis.org

1, 6, 1, 7, 3, 1, 12, 1, 3, 1, 13, 3, 1, 3, 1, 18, 1, 3, 1, 3, 1, 19, 3, 1, 3, 1, 3, 1, 24, 1, 3, 1, 3, 1, 3, 1, 25, 3, 1, 3, 1, 3, 1, 3, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 20 2007

Keywords

Comments

Left column = A047225, numbers congruent to {0,1} mod 6: (1, 6, 7, 12, 13, 18, 19, ...).
Row sums = A131229, numbers congruent to {1,7} mod 10: (1, 7, 11, 17, ...).

Examples

			First few rows of the triangle:
   1;
   6, 1;
   7, 3, 1;
  12, 1, 3, 1;
  13, 3, 1, 3, 1;
  ...

Crossrefs

Cf. A131229, A047225, A130296, A128174.

Formula

3*A130296 - 2*A128174 as infinite lower triangular matrices.

A131230 Triangle read by rows: 2*A130296 - A128174.

Original entry on oeis.org

1, 4, 1, 5, 2, 1, 8, 1, 2, 1, 9, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 13, 2, 1, 2, 1, 2, 1, 16, 1, 2, 1, 2, 1, 2, 1, 17, 2, 1, 2, 1, 2, 1, 2, 1, 20, 1, 2, 1, 2, 1, 2, 1, 2, 1, 21, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 25, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 20 2007

Keywords

Comments

Left column = A042948, numbers congruent to {1,0} mod 4: (1, 4, 5, 8, 9, 12, ...).
Row sums = A047383, numbers congruent to {1,5} mod 7: (1, 5, 8, 12, 15, ...).

Examples

			First few rows of the triangle:
   1;
   4, 1;
   5, 2, 1;
   8, 1, 2, 1;
   9, 2, 1, 2, 1;
  12, 1, 2, 1, 2, 1;
  ...

Crossrefs

Cf. A047383, A042948, A128174, A131231, A130296.

Extensions

Incorrect formula removed and more terms from Georg Fischer, Jun 08 2023

A131922 Triangle read by rows: 2*A002024 - A130296 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 4, 7, 7, 7, 5, 9, 9, 9, 9, 6, 11, 11, 11, 11, 11, 7, 13, 13, 13, 13, 13, 13, 8, 15, 15, 15, 15, 15, 15, 15, 9, 17, 17, 17, 17, 17, 17, 17, 17, 10, 19, 19, 19, 19, 19, 19, 19, 19, 19, 11, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 12, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23
Offset: 0

Views

Author

Gary W. Adamson, Jul 29 2007

Keywords

Comments

Row sums = A001844: (1, 5, 13, 25, 41, 61, 85, ...).

Examples

			First few rows of the triangle:
  1;
  2, 3;
  3, 5, 5;
  4, 7, 7, 7;
  5, 9, 9, 9, 9;
  ...

Crossrefs

Cf. A001844, A002024, A130296.

Extensions

Edited and more terms from Georg Fischer, Jun 08 2023

A034856 a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.

Original entry on oeis.org

1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, 151, 169, 188, 208, 229, 251, 274, 298, 323, 349, 376, 404, 433, 463, 494, 526, 559, 593, 628, 664, 701, 739, 778, 818, 859, 901, 944, 988, 1033, 1079, 1126, 1174, 1223, 1273, 1324, 1376, 1429, 1483
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of 1's in the n X n lower Hessenberg (0,1)-matrix (i.e., the matrix having 1's on or below the superdiagonal and 0's above the superdiagonal).
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Number of binary operations which have to be added to Moisil's algebras to obtain algebraic counterparts of n-valued Łukasiewicz logics. See the Wójcicki and Malinowski book, page 31. - Artur Jasinski, Feb 25 2010
Also (n + 1)!(-1)^(n + 1) times the determinant of the n X n matrix given by m(i,j) = i/(i+1) if i=j and otherwise 1. For example, (5+1)! * ((-1)^(5+1)) * Det[{{1/2, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1}, {1, 1, 3/4, 1, 1}, {1, 1, 1, 4/5, 1}, {1, 1, 1, 1, 5/6}}] = 19 = a(5), and (6+1)! * ((-1)^(6+1)) * Det[{{1/2, 1, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1, 1}, {1, 1, 3/4, 1, 1, 1}, {1, 1, 1, 4/5, 1, 1}, {1, 1, 1, 1, 5/6, 1}, {1, 1, 1, 1, 1, 6/7}}] = 26 = a(6). - John M. Campbell, May 20 2011
2*a(n-1) = n*(n+1) - 4, n>=0, with a(-1) = -2 and a(0) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 17 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
a(n) is not divisible by 3, 5, 7, or 11. - Vladimir Shevelev, Feb 03 2014
With a(0) = 1 and a(1) = 2, a(n-1) is the number of distinct values of 1 +- 2 +- 3 +- ... +- n, for n > 0. - Derek Orr, Mar 11 2015
Also, numbers m such that 8*m+17 is a square. - Bruno Berselli, Sep 16 2015
Omar E. Pol's formula from Apr 23 2008 can be interpreted as the position of an element located on the third diagonal of an triangular array (read by rows) provided n > 1. - Enrique Pérez Herrero, Aug 29 2016
a(n) is the sum of the numerator and denominator of the fraction that is the sum of 2/(n-1) + 2/n; all fractions are reduced and n > 2. - J. M. Bergot, Jun 14 2017
a(n) is also the number of maximal irredundant sets in the (n+2)-path complement graph for n > 1. - Eric W. Weisstein, Apr 12 2018
From Klaus Purath, Dec 07 2020: (Start)
a(n) is not divisible by primes listed in A038890. The prime factors are given in A038889 and the prime terms of the sequence are listed in A124199.
Each odd prime factor p divides exactly 2 out of any p consecutive terms with the exception of 17, which appears only once in such an interval of terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -3 (mod p), see examples.
If A is a sequence satisfying the recurrence t(n) = 5*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 4 or A(0) = -1, A(1) = n-1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i for i>0. (End)
Mark each point on a 4^n grid with the number of points that are visible from the point; for n > 1, a(n) is the number of distinct values in the grid. - Torlach Rush, Mar 23 2021
The sequence gives the number of "ON" cells in the cellular automaton on a quadrant of a square grid after the n-th stage, where the "ON" cells lie only on the external perimeter and the perimeter of inscribed squares having the cell (1,1) as a unique common vertex. See Spezia link. - Stefano Spezia, May 28 2025

Examples

			From _Bruno Berselli_, Mar 09 2015: (Start)
By the definition (first formula):
----------------------------------------------------------------------
  1       4         8           13            19              26
----------------------------------------------------------------------
                                                              X
                                              X              X X
                                X            X X            X X X
                    X          X X          X X X          X X X X
          X        X X        X X X        X X X X        X X X X X
  X      X X      X X X      X X X X      X X X X X      X X X X X X
          X        X X        X X X        X X X X        X X X X X
----------------------------------------------------------------------
(End)
From _Klaus Purath_, Dec 07 2020: (Start)
Assuming a(i) is divisible by p with 0 < i < p and a(k) is the next term divisible by p, then from i + k == -3 (mod p) follows that k = min(p*m - i - 3) != i for any integer m.
(1) 17|a(7) => k = min(17*m - 10) != 7 => m = 2, k = 24 == 7 (mod 17). Thus every a(17*m + 7) is divisible by 17.
(2) a(9) = 53 => k = min(53*m - 12) != 9 => m = 1, k = 41. Thus every a(53*m + 9) and a(53*m + 41) is divisible by 53.
(3) 101|a(273) => 229 == 71 (mod 101) => k = min(101*m - 74) != 71 => m = 1, k = 27. Thus every a(101*m + 27) and a(101*m + 71) is divisible by 101. (End)
From _Omar E. Pol_, Aug 08 2021: (Start)
Illustration of initial terms:                             _ _
.                                           _ _           |_|_|_
.                              _ _         |_|_|_         |_|_|_|_
.                   _ _       |_|_|_       |_|_|_|_       |_|_|_|_|_
.          _ _     |_|_|_     |_|_|_|_     |_|_|_|_|_     |_|_|_|_|_|_
.   _     |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.  |_|    |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.
.   1       4         8          13            19              26
------------------------------------------------------------------------ (End)

References

  • A. S. Karpenko, Łukasiewicz's Logics and Prime Numbers, 2006 (English translation).
  • G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972.
  • Wójcicki and Malinowski, eds., Łukasiewicz Sentential Calculi, Wrocław: Ossolineum, 1977.

Links

Crossrefs

Subsequence of A165157.
Triangular numbers (A000217) minus two.
Third diagonal of triangle in A059317.
Cf. A000096, A000124, A000217, A000290, A005408, A005843, A008277, A024206, A027379.
Cf. A028347, A038889, A038890, A049600, A091823, A101881, A113452, A113453, A113454.
Cf. A113455, A124199, A130296, A131818, A134225, A161680, A168244, A253909, A267633.

Programs

  • Haskell
    a034856 = subtract 1 . a000096 -- Reinhard Zumkeller, Feb 20 2015
  • Magma
    [Binomial(n + 1, 2) + n - 1: n in [1..60]]; // Vincenzo Librandi, May 21 2011
  • Maple
    a := n -> hypergeom([-2, n-1], [1], -1);
seq(simplify(a(n)), n=1..53); # Peter Luschny, Aug 02 2014
  • Mathematica
    f[n_] := n (n + 3)/2 - 1; Array[f, 55] (* or *) k = 2; NestList[(k++; # + k) &, 1, 55] (* Robert G. Wilson v, Jun 11 2010 *)
Table[Binomial[n + 1, 2] + n - 1, {n, 53}] (* or *)
Rest@ CoefficientList[Series[x (1 + x - x^2)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Aug 29 2016 *)
  • Maxima
    A034856(n) := block(
        n-1+(n+1)*n/2
)$ /* R. J. Mathar, Mar 19 2012 */
  • PARI
    A034856(n)=(n+3)*n\2-1 \\ M. F. Hasler, Jan 21 2015
  • Python
    def A034856(n): return n*(n+3)//2 -1 # G. C. Greubel, Jun 15 2025

Formula

G.f.: A(x) = x*(1 + x - x^2)/(1 - x)^3.
a(n) = A049600(3, n-2).
a(n) = binomial(n+2, 2) - 2. - Paul Barry, Feb 27 2003
With offset 5, this is binomial(n, 0) - 2*binomial(n, 1) + binomial(n, 2), the binomial transform of (1, -2, 1, 0, 0, 0, ...). - Paul Barry, Jul 01 2003
Row sums of triangle A131818. - Gary W. Adamson, Jul 27 2007
Binomial transform of (1, 3, 1, 0, 0, 0, ...). Also equals A130296 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007
Row sums of triangle A134225. - Gary W. Adamson, Oct 14 2007
a(n) = A000217(n+1) - 2. - Omar E. Pol, Apr 23 2008
From Jaroslav Krizek, Sep 05 2009: (Start)
a(n) = a(n-1) + n + 1 for n >= 1.
a(n) = n*(n-1)/2 + 2*n - 1.
a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). (End)
a(n) = Hyper2F1([-2, n-1], [1], -1). - Peter Luschny, Aug 02 2014
a(n) = floor[1/(-1 + Sum_{m >= n+1} 1/S2(m,n+1))], where S2 is A008277. - Richard R. Forberg, Jan 17 2015
a(n) = A101881(2*(n-1)). - Reinhard Zumkeller, Feb 20 2015
a(n) = A253909(n+3) - A000217(n+3). - David Neil McGrath, May 23 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - David Neil McGrath, May 23 2015
For n > 1, a(n) = 4*binomial(n-1,1) + binomial(n-2,2), comprising the third column of A267633. - Tom Copeland, Jan 25 2016
From Klaus Purath, Dec 07 2020: (Start)
a(n) = A024206(n) + A024206(n+1).
a(2*n-1) = -A168244(n+1).
a(2*n) = A091823(n). (End)
Sum_{n>=1} 1/a(n) = 3/2 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - Amiram Eldar, Jan 06 2021
a(n) + a(n+1) = A028347(n+2). - R. J. Mathar, Mar 13 2021
a(n) = A000290(n) - A161680(n-1). - Omar E. Pol, Mar 26 2021
E.g.f.: 1 + exp(x)*(x^2 + 4*x - 2)/2. - Stefano Spezia, Jun 05 2021
a(n) = A024916(n) - A244049(n). - Omar E. Pol, Aug 01 2021
a(n) = A000290(n) - A000217(n-2). - Omar E. Pol, Aug 05 2021

Extensions

More terms from Zerinvary Lajos, May 12 2006

A193091 Augmentation of the triangular array A158405. See Comments.

Original entry on oeis.org

1, 1, 3, 1, 6, 14, 1, 9, 37, 79, 1, 12, 69, 242, 494, 1, 15, 110, 516, 1658, 3294, 1, 18, 160, 928, 3870, 11764, 22952, 1, 21, 219, 1505, 7589, 29307, 85741, 165127, 1, 24, 287, 2274, 13355, 61332, 224357, 638250, 1217270, 1, 27, 364, 3262, 21789, 115003
Offset: 0

Views

Author

Clark Kimberling, Jul 30 2011

Keywords

Comments

Suppose that P is an infinite triangular array of numbers:
p(0,0)
p(1,0)...p(1,1)
p(2,0)...p(2,1)...p(2,2)
p(3,0)...p(3,1)...p(3,2)...p(3,3)...
...
Let w(0,0)=1, w(1,0)=p(1,0), w(1,1)=p(1,1), and define
W(n)=(w(n,0), w(n,1), w(n,2),...w(n,n-1), w(n,n)) recursively by W(n)=W(n-1)*PP(n), where PP(n) is the n X (n+1) matrix given by
...
row 0 ... p(n,0) ... p(n,1) ...... p(n,n-1) ... p(n,n)
row 1 ... 0 ..... p(n-1,0) ..... p(n-1,n-2) .. p(n-1,n-1)
row 2 ... 0 ..... 0 ............ p(n-2,n-3) .. p(n-2,n-2)
...
row n-1 . 0 ..... 0 ............. p(2,1) ..... p(2,2)
row n ... 0 ..... 0 ............. p(1,0) ..... p(1,1)
...
The augmentation of P is here introduced as the triangular array whose n-th row is W(n), for n>=0. The array P may be represented as a sequence of polynomials; viz., row n is then the vector of coefficients: p(n,0), p(n,1),...,p(n,n), from p(n,0)*x^n+p(n,1)*x^(n-1)+...+p(n,n). For example, (C(n,k)) is represented by ((x+1)^n); using this choice of P (that is, Pascal's triangle), the augmentation of P is calculated one row at a time, either by the above matrix products or by polynomial substitutions in the following manner:
...
row 0 of W: 1, by decree
row 1 of W: 1 augments to 1,1
...polynomial version: 1 -> x+1
row 2 of W: 1,1 augments to 1,3,2
...polynomial version: x+1 -> (x^2+2x+1)+(x+1)=x^2+3x+2
row 3 to W: 1,3,2 augments to 1,6,11,6
...polynomial version:
x^2+3x+2 -> (x+1)^3+3(x+1)^2+2(x+1)=(x+1)(x+2)(x+3)
...
Examples of augmented triangular arrays:
(p(n,k)=1) augments to A009766, Catalan triangle.
Catalan triangle augments to A193560.
Pascal triangle augments to A094638, Stirling triangle.
A002260=((k+1)) augments to A023531.
A154325 augments to A033878.
A158405 augments to A193091.
((k!)) augments to A193092.
A094727 augments to A193093.
A130296 augments to A193094.
A004736 augments to A193561.
...
Regarding the specific augmentation W=A193091: w(n,n)=A003169.
From Peter Bala, Aug 02 2012: (Start)
This is the table of g(n,k) in the notation of Carlitz (p. 124). The triangle enumerates two-line arrays of positive integers
............a_1 a_2 ... a_n..........
............b_1 b_2 ... b_n..........
such that
1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
2) max(a_i, b_i) <= i for 1 <= i <= n
3) max(a_n, b_n) = k.
See A071948 and A211788 for other two-line array enumerations.
(End)

Examples

			The triangle P, at A158405, is given by rows
  1
  1...3
  1...3...5
  1...3...5...7
  1...3...5...7...9...
The augmentation of P is the array W starts with w(0,0)=1, by definition of W.
Successive polynomials (rows of W) arise from P as shown here:
  ...
  1->x+3, so that W has (row 1)=(1,3);
  ...
  x+3->(x^2+3x+5)+3*(x+3), so that W has (row 2)=(1,6,14);
  ...
  x^2+6x+14->(x^3+3x^2+5x+7)+6(x^2+3x+5)+14(x+3), so that (row 3)=(1,9,37,79).
  ...
First 7 rows of W:
  1
  1    3
  1    6    14
  1    9    37    79
  1   12    69   242    494
  1   15   110   516   1658    3294
  1   18   160   928   3870   11764   22952

Links

Crossrefs

Cf. A003169, A193092, A193093, A193094.
Cf. A071948, A211788.

Programs

  • Mathematica
    p[n_, k_] := 2 k + 1
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A158405 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193091 *)
Flatten[Table[v[n], {n, 0, 9}]]

Formula

From Peter Bala, Aug 02 2012: (Start)
T(n,k) = (n-k+1)/n*Sum_{i=0..k} C(n+1,n-k+i+1)*C(2*n+i+1,i) for 0 <= k <= n.
Recurrence equation: T(n,k) = Sum_{i=0..k} (2*k-2*i+1)*T(n-1,i).
(End)

A131821 Triangle read by rows: row n consists of n followed by (n-2) ones then n.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Jul 19 2007

Keywords

Examples

			First few rows of the triangle are:
  1;
  2, 2;
  3, 1, 3;
  4, 1, 1, 4;
  5, 1, 1, 1, 5;
  6, 1, 1, 1, 1, 6;
  7, 1, 1, 1, 1, 1, 7;
  ...

Crossrefs

Row sums: A016777.
Cf. A130296, A051340, A016777, A131830.

Programs

  • Maple
    A131821 := proc(n,c) if c=1 or c=n then n ; else 1 ; fi ; end: for n from 1 to 16 do for c from 1 to n do printf("%d,",A131821(n,c)) ; od: od: # R. J. Mathar, May 30 2008
  • Mathematica
    T[n_, k_] := If[k == 1 || k == n, n, 1];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 05 2023 *)
  • Maxima
    T(n,k) := if k = 1 or k = n then n else 1$
create_list(T(n, k), n, 1, 12, k, 1, n); /* Franck Maminirina Ramaharo, Dec 19 2018 */
  • Python
    from math import isqrt
def A131821(n): return m+(k>r) if 0<=(k:=n<<1)-(r:=(m:=isqrt(k))*(m+1))<=2 or n<=2 else 1 # Chai Wah Wu, Nov 07 2024

Formula

T(n,k) = A130296(n,k) + A051340(k-1,n-k) - 1.
T(n,1) = T(n,n) = n for n >= 1, and T(n,k) = 1 for 2 <= k <= n - 1, n >= 3.
From Franck Maminirina Ramaharo, Dec 19 2018: (Start)
T(n,k) = A130296(n,k) + A130296(n,n-k+1) - 1.
G.f.: y*(x - 3*x^2*y^2 + (x^2 + x^3)*y^3)/((1 - y)^2*(1 - x*y)^2).
E.g.f.: (1 - x^2 +(x^2 + (x - x^2)*y)*exp(y) - (1 - (x - x^2)*y)*exp(x*y))/(1 - x). (End)

Extensions

More terms from R. J. Mathar, May 30 2008

A004201 Accept one, reject one, accept two, reject two, ...

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 36, 43, 44, 45, 46, 47, 48, 49, 57, 58, 59, 60, 61, 62, 63, 64, 73, 74, 75, 76, 77, 78, 79, 80, 81, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 133, 134, 135
Offset: 1

Views

Author

Alexander Stasinski

Keywords

Comments

a(n) are the numbers satisfying m - 0.5 < sqrt(a(n)) <= m for some positive integer m. - Floor van Lamoen, Jul 24 2001
Lower s(n)-Wythoff sequence (as defined in A184117) associated to s(n) = A002024(n) = floor(1/2+sqrt(2n)), with complement (upper s(n)-Wythoff sequence) in A004202.

Links

Crossrefs

Cf. A002024, A004202, A007606.

Programs

  • Haskell
    a004201 n = a004201_list !! (n-1)
a004201_list = f 1 [1..] where
   f k xs = us ++ f (k + 1) (drop (k) vs) where (us, vs) = splitAt k xs
-- Reinhard Zumkeller, Jun 20 2015, Feb 12 2011
  • Mathematica
    f[x_]:=Module[{c=1-x+x^2},Range[c,c+x-1]]; Flatten[Array[f,20]] (* Harvey P. Dale, Jul 31 2012 *)
  • PARI
    A004201(n)=n+(n=(sqrtint(8*n-7)+1)\2)*(n-1)\2  \\ M. F. Hasler, Feb 13 2011
  • Python
    from math import comb, isqrt
def A004201(n): return n+comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2) # Chai Wah Wu, Nov 09 2024

Formula

a(n) = A061885(n-1)+1. - Franklin T. Adams-Watters, Jul 05 2009
a(n+1) - a(n) = A130296(n+1). - Reinhard Zumkeller, Jul 16 2008
a(A000217(n)) = n^2. - Reinhard Zumkeller, Feb 12 2011
a(n) = A004202(n)-A002024(n). - M. F. Hasler, Feb 13 2011
a(n) = n+A000217(A003056(n-1)) = n+A000217(A002024(n)-1). - M. F. Hasler, Feb 13 2011
a(n) = n + t(t+1)/2, where t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
a(n) = (2*n - r + r^2)/2, where r = round(sqrt(2*n)). - Wesley Ivan Hurt, Sep 20 2021

A130295 Erroneous duplicate of A125026.

Original entry on oeis.org

1, 3, 2, 5, 7, 3, 7, 15, 13, 4, 9, 26, 34, 21, 5, 11, 40, 70, 65, 31, 6, 13, 57, 125, 155, 111, 43, 7, 15, 77, 203, 315, 301, 175, 57, 8, 17, 100, 308, 574, 686, 532, 260, 73, 9, 19, 126, 444, 966, 1386, 1344, 876369, 91, 10
Offset: 1

Views

Author

Gary W. Adamson, May 20 2007

Keywords

Comments

This sequence was initially defined as "A051340 * A007318". However, the matrix product A051340 * A007318 is not well defined, because all elements of A051340 are strictly positive integers, as are all elements of the lower left of A007318. Therefore the matrix A051340 must be truncated to its lower left (setting A[i,j]=0 if j>i), which actually equals A130296. Then the product yields this sequence, which is identical to A125026.
Row sums = A099035 (not A083706 as stated initially): (1, 5, 15, 39, 95, 223, 511, ...).

Examples

			First few rows of the triangle A125026:
   1;
   3,  2;
   5,  7,   3;
   7, 15,  13,   4;
   9, 26,  34,  21,   5;
  11, 40,  70,  65,  31,  6;
  13, 57, 125, 155, 111, 43, 7;
  ...

Crossrefs

Cf. A051340, A099035.

Formula

(A051340) * A007318 as infinite lower triangular matrices. [Here (A051340) is that matrix with the upper right triangle set to zero, which is actually A130296. - M. F. Hasler, Aug 15 2015]

Extensions

Restored and edited by M. F. Hasler, Aug 15 2015
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