A029578 -id:A029578 - cp's OEIS frontend

A048759 Longest perimeter of a Pythagorean triangle with n as length of one of the three sides.

12, 12, 30, 24, 56, 40, 90, 60, 132, 84, 182, 112, 240, 144, 306, 180, 380, 220, 462, 264, 552, 312, 650, 364, 756, 420, 870, 480, 992, 544, 1122, 612, 1260, 684, 1406, 760, 1560, 840, 1722, 924, 1892, 1012, 2070, 1104, 2256, 1200, 2450, 1300, 2652

Offset: 3

Henry Bottomley, Jun 15 2000

Stefano Spezia, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A010814, A029578, A055522, A055523, A055524.

[(3*n^2+4*n-n^2*(-1)^n)/4: n in [3..60]]; // Vincenzo Librandi, Jul 19 2015

A048759[n_] := (3 - (-1)^n)*n^2 / 4 + n; Array[A048759, 100, 3] (* or *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {12, 12, 30, 24, 56, 40}, 100] (* Paolo Xausa, Feb 29 2024 *)

Vec(-2*x^3*(2*x^5+x^4-6*x^3-3*x^2+6*x+6)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 13 2014

a(n) = n*A029578(n+2) = n+A055523(n)+A055524(n).
a(2*k) = 2*k*(k+1), a(2*k+1) = 2*(2*k+1)*(k+1).
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). - Colin Barker, Sep 13 2014
G.f.: -2*x^3*(2*x^5+x^4-6*x^3-3*x^2+6*x+6) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 13 2014
a(n) = (3*n^2+4*n-n^2*(-1)^n)/4. - Luce ETIENNE, Jul 18 2015
E.g.f.: x*((4 + x)*cosh(x) + (3 + 2*x)*sinh(x) - 4*(1 + x))/2. - Stefano Spezia, May 24 2021

A056136 Largest positive integer whose harmonic mean with another positive integer is n.

Original entry on oeis.org

1, 2, 6, 6, 15, 12, 28, 20, 45, 30, 66, 42, 91, 56, 120, 72, 153, 90, 190, 110, 231, 132, 276, 156, 325, 182, 378, 210, 435, 240, 496, 272, 561, 306, 630, 342, 703, 380, 780, 420, 861, 462, 946, 506, 1035, 552, 1128, 600, 1225, 650, 1326, 702, 1431, 756, 1540

Offset: 1

Henry Bottomley, Jun 13 2000

nonn

Cf. A008619 for the smallest integer whose harmonic mean with another integer is n (i.e. harmonic mean of A008619 and a(n) is n).

a(n) =1/(2/n-1/floor[n/2+1]) =n*A008619(n)/(2*A008619(n)-n) =(n/2)*A029578(n+2)

A082404 Triangle in which n-th row gives trajectory of n under the map x -> x/2 if x is even, x -> x-1 if x is odd, stopping when we reach 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 4, 2, 1, 6, 3, 2, 1, 7, 6, 3, 2, 1, 8, 4, 2, 1, 9, 8, 4, 2, 1, 10, 5, 4, 2, 1, 11, 10, 5, 4, 2, 1, 12, 6, 3, 2, 1, 13, 12, 6, 3, 2, 1, 14, 7, 6, 3, 2, 1, 15, 14, 7, 6, 3, 2, 1, 16, 8, 4, 2, 1, 17, 16, 8, 4, 2, 1

Offset: 1

Cino Hilliard, Apr 14 2003

If you write down 0 when dividing by 2, 1 when subtracting 1, the trajectory gives the binary expansion of n.

The length of the n-th row of the triangle is A056792(n). - Nathaniel Johnston, Apr 21 2011

			Triangle begins:
  1;
  2, 1;
  3, 2, 1;
  4, 2, 1,
  5, 4, 2, 1;
  6, 3, 2, 1;
  7, 6, 3, 2, 1;
  8, 4, 2, 1;
  9, 8, 4, 2, 1;
  ...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A056792, A080825.

A082404 := proc(n,k) option remember: if(k = 1)then return n:elif(A082404(n,k-1) mod 2 = 0)then return A082404(n,k-1)/2: else return A082404(n,k-1)-1: fi: end:
for n from 1 to 20 do k:=1: while A082404(n,k)>=1 do printf("%d, ",A082404(n,k)); k:=k+1: od:printf("\n");od: # Nathaniel Johnston, Apr 21 2011

T(n, 1) = n, T(n, 2) = A029578(n).

More terms and changed offset from Nathaniel Johnston, Apr 21 2011

A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

Original entry on oeis.org

1, -2, 1, -1, -2, 1, 2, -2, -2, 1, 1, 4, -3, -2, 1, -2, 3, 6, -4, -2, 1, -1, -6, 6, 8, -5, -2, 1, 2, -4, -12, 10, 10, -6, -2, 1, 1, 8, -10, -20, 15, 12, -7, -2, 1, -2, 5, 20, -20, -30, 21, 14, -8, -2, 1, -1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 03 2006

			Triangular sequence begins as:
   1;
  -2,   1;
  -1,  -2,   1;
   2,  -2,  -2,   1;
   1,   4,  -3,  -2,   1;
  -2,   3,   6,  -4,  -2,   1;
  -1,  -6,   6,   8,  -5,  -2,  1;
   2,  -4, -12,  10,  10,  -6, -2,  1;
   1,   8, -10, -20,  15,  12, -7, -2,  1;
  -2,   5,  20, -20, -30,  21, 14, -8, -2,  1;
  -1, -10,  15,  40, -35, -42, 28, 16, -9, -2, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A005809, A045721, A066170.
Row reversal of: A374439.
Columns are related to: A000034 (k=0), A029578 (k=1), A131259 (k=2).
Diagonals are related to: A113679 (k=n-1), A022958 (k=n-2), A005843 (k=n-3), A000217 (k=n-4), -A002378 (k=n-5).
Sums include: (-1)^floor((n+1)/2)*A016116 (signed diagonal), A057079 (row), A119910 (signed row), (-1)^n*A130706 (diagonal).

function T(n,k) // T = A124038
  if k lt 0 or k gt n then return 0;
  elif k ge n-2 then return k-n + (-1)^(n+k);
  else return T(n-1,k-1) - T(n-2,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 22 2025

(* First program *)
t[n_, m_, d_]:= If[n==m && n>1 && m>1, x, If[n==m-1 || n==m+1, -1, If[n==m== 1, x-2, 0]]];
M[d_]:= Table[t[n,m,d], {n,d}, {m,d}];
Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d,10}][[d]], x], {d,10}]]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k>n-2, k-n+(-1)^(n-k), T[n-1, k- 1] -T[n-2,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 22 2025 *)

@CachedFunction
def A124038(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A124038(n-1,k) if n==1 else 0
    return A124038(n-1,k-1) - A124038(n-2,k) - h
for n in (0..9): [A124038(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

from sage.combinat.q_analogues import q_stirling_number2
def A124038(n,k): return (1 + ((n-k)%2))*q_stirling_number2(n+1, n-k+1, -1)
print(flatten([[A124038(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 22 2025

From G. C. Greubel, Jan 22 2025: (Start)
T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
T(n, k) = (-1)^floor((n-k+1)/2)*(1 + (n-k mod 2))*qStirling2(n+1, n-k+1,-1).
T(2*n, n) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)* A045721((n-1)/2) ). (End)

Edited by G. C. Greubel, Jan 22 2025

A174595 a(n) = 5n^2/8 - n + 1/2 + (-1)^n(-3*n^2/8 + n - 1/2).

Original entry on oeis.org

0, 0, 1, 4, 4, 16, 9, 36, 16, 64, 25, 100, 36, 144, 49, 196, 64, 256, 81, 324, 100, 400, 121, 484, 144, 576, 169, 676, 196, 784, 225, 900, 256, 1024, 289, 1156, 324, 1296, 361, 1444, 400, 1600, 441, 1764, 484, 1936, 529, 2116, 576, 2304, 625, 2500, 676, 2704, 729, 2916, 784, 3136, 841, 3364, 900

Offset: 0

Paul Curtz, Nov 29 2010

Based on A174571.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

[5*n^2/8-n+1/2+(-1)^n*(-3*n^2/8+n-1/2): n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

LinearRecurrence[{0,3,0,-3,0,1},{0,0,1,4,4,16},70] (* Harvey P. Dale, Jun 26 2012 *)
CoefficientList[Series[1/8 E^-x (-4 - 5 x - 3 x^2 + E^(2 x) (4 - 3 x + 5 x^2)), {x, 0, 50}], x]*Table[k!, {k, 0, 50}] (* Stefano Spezia, Nov 02 2018 *)

vector(50, n, n--; (5*n^2 -8*n + 4 - (-1)^n*(3*n^2 - 8*n +4))/8) \\ G. C. Greubel, Nov 02 2018

a(n) = A029578(n)^2.
Interleaving of A000290 and 4*A000290.
G.f.: -x^2*(4*x+1)*(x^2+1) / ( (x-1)^3*(1+x)^3 ).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
E.g.f.: (1/8)*exp(-x)*(- 4 - 5*x - 3*x^2 +exp(2*x)*(4 - 3*x + 5*x^2)). - Stefano Spezia, Nov 02 2018

A260316 n/3 if 3 divides n, else n-1.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 2, 6, 7, 3, 9, 10, 4, 12, 13, 5, 15, 16, 6, 18, 19, 7, 21, 22, 8, 24, 25, 9, 27, 28, 10, 30, 31, 11, 33, 34, 12, 36, 37, 13, 39, 40, 14, 42, 43, 15, 45, 46, 16, 48, 49, 17, 51, 52, 18, 54, 55, 19, 57, 58, 20, 60, 61, 21, 63, 64, 22, 66, 67, 23

Offset: 0

Peter Kagey, Jul 22 2015

			a(5) = 5 - 1 = 4 because 5 is not divisible by 3.
a(12) = 12/3 = 4 because 12 is divisible by 3.

Peter Kagey, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

A029578 is an analogous case where the divisor is 2 instead of 3.

Table[If[Mod[n, 3] == 0, n/3, n - 1], {n, 0, 69}] (* or *)
CoefficientList[Series[x^2*(2 x^3 + 3 x^2 + x + 1)/((x - 1)^2*(x^2 + x + 1)^2), {x, 0, 69}], x] (* Michael De Vlieger, Jul 23 2015 *)
LinearRecurrence[{0,0,2,0,0,-1},{0,0,1,1,3,4},80] (* Harvey P. Dale, May 20 2018 *)

concat([0,0], Vec(x^2*(2*x^3+3*x^2+x+1) / ((x-1)^2*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, Jul 23 2015

Ruby
```
def a(n);(n%3==0)?n/3:n-1 end
```

a(3k) = k; a(3k + 1) = 3k; a(3k + 2) = 3k + 1.
a(n) = 2*a(n-3) - a(n-6) for n>5. - Colin Barker, Jul 23 2015
G.f.: x^2*(2*x^3+3*x^2+x+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Jul 23 2015

A300153 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the number of parts inscribed in a rose or rhodonea curve with polar coordinates r = cos(t * (k/n)).

Original entry on oeis.org

1, 4, 4, 2, 1, 3, 8, 12, 12, 8, 3, 4, 1, 4, 5, 12, 20, 24, 24, 20, 12, 4, 2, 9, 1, 10, 3, 7, 16, 28, 4, 40, 40, 4, 28, 16, 5, 8, 12, 12, 1, 12, 14, 8, 9, 20, 36, 48, 56, 60, 60, 56, 48, 36, 20, 6, 3, 2, 4, 20, 1, 21, 4, 3, 5, 11, 24, 44, 60, 72, 80, 84, 84, 80

Offset: 1

Rémy Sigrist, Feb 26 2018

For any real p > 0, the rose or rhodonea curve with polar coordinates r = cos(t * p):
- is dense in the unit disk when p is irrational,
- is closed when p is rational, say p = u/v in reduced form; in that case, the number of parts inscribed in the curve is T(v, u),
- see also the illustration in Links section.

			Array T(n, k) begins:
  n\k|    1    2    3    4    5    6    7    8    9
  ---+---------------------------------------------
    1|    1    4    3    8    5   12    7   16    9
    2|    4    1   12    4   20    3   28    8   36
    3|    2   12    1   24   10    4   14   48    3
    4|    8    4   24    1   40   12   56    4   72
    5|    3   20    9   40    1   60   21   80   27
    6|   12    2    4   12   60    1   84   24   12
    7|    4   28   12   56   20   84    1  112   36
    8|   16    8   48    4   80   24  112    1  144
    9|    5   36    2   72   25   12   35  144    1
   10|   20    3   60   20    4    9  140   40  180
   11|    6   44   18   88   30  132   42  176   54
...
The following diagram shows the curve for T(2, 1) and the corresponding 4 parts:
                         |
               ########     ########
           #####      #######      #####
        ###          ###   ###          ###
      ###           ##   |   ##           ###
     ##            ##         ##            ##
    ##             #  Part #2  #             ##
   ##              ##         ##              ##
   #                ###  |  ###                #
  -#- - - Part #3  - -#######- -  Part #1 - - -#-
   #                ###  |  ###                #
   ##              ##         ##              ##
    ##             #  Part #4  #             ##
     ##            ##         ##            ##
      ###           ##   |   ##           ###
        ###          ###   ###          ###
           #####      #######      #####
               ########     ########
                         |

Rémy Sigrist, Illustration of the first terms
Eric Weisstein's World of Mathematics, Rose
Wikipedia, Rose (mathematics)

Cf. A022998, A029578.

T(1, k) = A022998(k).
T(n, k) = T(n/gcd(n, k), k/gcd(n, k)).
Empirically, when gcd(n, k) = 1, we have the following formulas depending on the parity of n and of k:
             | k is odd                       | k is even
   ----------+--------------------------------+--------------------
   n is odd  | T(n, k) =     k * A029578(n+1) | T(n, k) = 2 * k * n
   n is even | T(n, k) = 2 * k * A029578(n+1) | N/A

A071054 a(2n)=3n+1, a(2n+1)=2n+2.

Original entry on oeis.org

1, 3, 4, 5, 7, 7, 10, 9, 13, 11, 16, 13, 19, 15, 22, 17, 25, 19, 28, 21, 31, 23, 34, 25, 37, 27, 40, 29, 43, 31, 46, 33, 49, 35, 52, 37, 55, 39, 58, 41, 61, 43, 64, 45, 67, 47, 70, 49, 73, 51, 76, 53, 79, 55, 82, 57, 85, 59, 88, 61, 91, 63, 94, 65, 97, 67, 100, 69

Offset: 0

Hans Havermann, May 26 2002

nonn

Number of ON cells at n-th generation of 1-D CA defined by Rule 158, starting with a single ON cell at generation 0. Equivalently, number of 1's in n-th row of triangle in A071037. - N. J. A. Sloane, Aug 10 2014

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, 2016.
N. J. A. Sloane, Illustration of first 20 generations of Rule 158
Index entries for sequences related to cellular automata

Cf. A029578.

CoefficientList[Series[(-x^3 + 2 x^2 + 3 x + 1)/(1 - x^2)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Aug 11 2014 *)
ArrayPlot[CellularAutomaton[158, {{1}, 0}, 20]] (* N. J. A. Sloane, Aug 11 2014 *)

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

a(n) = (5/4)*n + 3/2 + (n/4 - 1/2)*(-1)^n. - Robert Israel, Aug 11 2014

Simpler definition from N. J. A. Sloane, Aug 11 2014

A236203 Interleave A005563(n), A028347(n).

Original entry on oeis.org

0, 0, 3, 5, 8, 12, 15, 21, 24, 32, 35, 45, 48, 60, 63, 77, 80, 96, 99, 117, 120, 140, 143, 165, 168, 192, 195, 221, 224, 252, 255, 285, 288, 320, 323, 357, 360, 396, 399, 437, 440, 480, 483, 525, 528, 572, 575, 621, 624, 672, 675, 725, 728, 780, 783, 837, 840, 896

Offset: 2

Paul Curtz, Jan 20 2014

A175628 gives the numerators of interleaved Lyman and Balmer series, i.e., A005563(n)/A000290(n+1) and A061037(n+2)/A061038(n+2).
Difference table of a(n):
   -1,  -3,  0,  0,  3,   5,   8,  12,  15,  21,  24, ...
   -2,   3,  0,  3,  2,   3,   4,   3,   6,   3,   8, ...
    5,  -3,  3, -1,  1,   1,  -1,   3,  -3,   5,  -5, ...
   -8,   6, -4,  2,  0,  -2,   4,  -6,   8, -10,  12, ...
   14, -10,  6, -2, -2,   6, -10,  14, -18,  22, -26, ...
  -24,  16, -8,  0,  8, -16,  24, -32,  40, -48,  56, ... .
a(n+2) gives the numerators of 0/1, 0/16, 3/4, 5/36, 8/9, 12/64, 15/16, 21/100, 24/25, 32/144, ... . The denominators are A097362(n+1)^2. (Compare A097362 to A029578.)
Note the particular distribution of a(-n). Example:
a(n-9) = 12,15, 5,8, 0,3, -3,0, -4,-1, -3,0, 0,3, 5,8, 12,15, ... .
a(2n) + a(2n+1) = a(-2n-1) + a(-2n-2) = -4,0,8,20,36,56,80,... = 4*A000096(n-1).
a(2n) + a(2n-1) = a(-2n) + a(-2n-1) = -5,-3,3,13,... = A001105(n) - A010716(n).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A005843, A008590, A016825, A109613, A147658.

List([2..60], n-> (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 ); # G. C. Greubel, Dec 04 2019

[(2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8: n in [2..60]]; // Vincenzo Librandi, Jul 27 2014

seq( (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8, n=2..60); # G. C. Greubel, Dec 04 2019

CoefficientList[Series[x^2(3x^2-2x-3)/((x-1)^3(x+1)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Jul 27 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{0,0,3,5,8},60] (* Harvey P. Dale, Aug 30 2018 *)

concat([0,0], Vec(x^4*(3*x^2-2*x-3)/((x-1)^3*(x+1)^2) + O(x^60))) \\ Colin Barker, Jan 26 2014

[(2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 for n in (2..60)] # G. C. Greubel, Dec 04 2019

a(n+2) = (period 8: repeat 1, 16, 1, 1, 1, 4, 1, 1)*A175628(n+1).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n+4) - a(n-4) = 0, 8, 8, ... = A168397.
From Colin Barker, Jan 26 2014: (Start)
a(n) = (n^2 -4)/4 for n even, a(n) = (n^2 +2*n -15)/4 for n odd.
G.f.: x^4*(3 + 2*x - 3*x^2)/ ((1-x)^3*(1+x)^2). (End)
a(n) = (2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8. - Luce ETIENNE, Jul 26 2014
Sum_{n>=4} (-1)^n/a(n) = 11/48. - Amiram Eldar, Aug 21 2022

More terms from Colin Barker, Jan 26 2014

A287638 Odd primes with no other odd primes as prefixes in binary.

Original entry on oeis.org

3, 5, 17, 19, 37, 67, 73, 131, 257, 521, 523, 577, 1033, 1039, 1061, 1063, 1069, 1153, 1163, 2053, 2069, 2081, 2083, 2089, 2113, 2129, 2131, 2137, 2141, 2143, 2333, 4099, 4111, 4129, 4153, 4177, 4229, 4231, 4241, 4243, 4261, 4271, 4273, 4637, 4639, 4643, 4649, 4651, 4657, 4663

Offset: 1

Dan Brumleve, May 28 2017

The sum of the reciprocals converges by the Kraft-McMillan inequality.

Odd primes p such that iterating the map A029578 on p reaches 2 without going through a prime. - Ya-Ping Lu, Oct 20 2021

			4663 is an odd prime with proper binary prefixes 2331, 1165, 582, 291, 145, 72, 36, 18, 9, 4, 2, 1, and none of these are odd primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Dan Brumleve, Does the sum of reciprocals of all prime-prefix-free numbers converge?, Math StackExchange, May 20 2017.
Wikipedia, Kraft-McMillan inequality

Odd prime elements of A287117.

Cf. A029578.

Select[Prime@ Range[2, 800], Function[w, NoneTrue[Array[FromDigits[w[[1 ;; #]], 2] &, Length[w] - 1], And[PrimeQ[#], # != 2] &]]@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Oct 20 2021 *)

is(n)=if(!isprime(n) || n<3, return(0)); while(n>3, if(isprime(n>>=1), return(n==2))); 1 \\ Charles R Greathouse IV, Jun 12 2017

sub isp {
  my $x = shift;
  return 0 if $x < 2;
  for my $d (2 .. $x - 1) {
    return 0 if $x % $d == 0;
  }
  return 1;
}
sub rots {
  my $x = shift;
  my @x;
  while ($x > 5) {
    $x = int($x / 2);
    push @x, $x;
  }
  @x
}
for my $i (1 .. $ARGV[0] // 8000) {
  my @np = grep isp($_), rots($i);
  push @z, $i if isp($i) && $i % 2 && @np == 0;
}
print join(", ", @z) . "\n";

cp's OEIS Frontend

A048759 Longest perimeter of a Pythagorean triangle with n as length of one of the three sides.

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Magma

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A056136 Largest positive integer whose harmonic mean with another positive integer is n.

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A082404 Triangle in which n-th row gives trajectory of n under the map x -> x/2 if x is even, x -> x-1 if x is odd, stopping when we reach 1.

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Maple

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A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

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SageMath

SageMath

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A174595 a(n) = 5*n^2/8 - n + 1/2 + (-1)^n*(-3*n^2/8 + n - 1/2).

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A260316 n/3 if 3 divides n, else n-1.

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Ruby

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A300153 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the number of parts inscribed in a rose or rhodonea curve with polar coordinates r = cos(t * (k/n)).

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A071054 a(2n)=3n+1, a(2n+1)=2n+2.

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A174595 a(n) = 5n^2/8 - n + 1/2 + (-1)^n(-3*n^2/8 + n - 1/2).