A316841 -id:A316841 - cp's OEIS frontend

A087811 Numbers k such that ceiling(sqrt(k)) divides k.

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784, 812, 841

Offset: 1

Reinhard Zumkeller, Oct 16 2003

Essentially the same as the quarter-squares A002620.
Nonsquare terms of this sequence are given by A002378. - Max Alekseyev, Nov 27 2006
This also gives the number of ways to make change for "c" cents using only pennies, nickels and dimes. You must first set n=floor(c/5), to account for the 5-repetitive nature of the task. - Adam Sasson, Feb 09 2011
These are the segment boundaries of Oppermann's conjecture (1882): n^2-n < p < n^2 < p < n^2+n. - Fred Daniel Kline, Apr 07 2011
a(n) is the number of triples (w,x,y) having all terms in {0..n} and w=2*x+y. - Clark Kimberling, Jun 04 2012
a(n+1) is also the number of points with integer coordinates inside a rectangle isosceles triangle with hypotenuse [0,n] (see A115065 for an equilateral triangle). - Michel Marcus, Aug 05 2013
a(n) = degree of generating polynomials of Galois numbers in (n+1)-dimensional vector space, defined as total number of subspaces in (n+1) space over GF(n) (see Mathematica procedure), when n is a power of a prime. - Artur Jasinski, Aug 31 2016, corrected by Robert Israel, Sep 23 2016
Also number of pairs (x,y) with 0 < x <= y <= n, x + y > n. - Ralf Steiner, Jan 05 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Ya-Ping Lu, Illustration of the Terms in A087811 on the Square Spiral
Wikipedia, Oppermann's conjecture.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002378, A002620, A003059, A024206, A110835, A316841, A059260.

Subsequence of A006446.

a087811 n = (n + n `mod` 2) * (n + 2 - n `mod` 2) `div` 4
-- Reinhard Zumkeller, Oct 27 2012

[ n: n in [1..841] | n mod Ceiling(Sqrt(n)) eq 0 ]; // Bruno Berselli, Feb 09 2011

f:= gfun:-rectoproc({a(n)=n+a(n-2),a(1)=1,a(2)=2},a(n),remember):
map(f, [$1..100]); # Robert Israel, Aug 31 2016

a[1] := 1; a[2] := 2; a[n_] := n + a[n - 2]; Table[a[n], {n, 57}] (* Alonso del Arte *)
GaloisNumber[n_, q_] :=
Sum[QBinomial[n, m, q], {m, 0, n}]; aa = {}; Do[
sub = Table[GaloisNumber[m, n], {n, 0, 200}];
pp = InterpolatingPolynomial[sub, x]; pol = pp /. x -> n + 1;
coef = CoefficientList[pol, n];
AppendTo[aa, Length[coef] - 1], {m, 2, 25}]; aa (* Artur Jasinski, Aug 31 2016 *)
Select[Range[900],Divisible[#,Ceiling[Sqrt[#]]]&] (* or *) LinearRecurrence[ {2,0,-2,1},{1,2,4,6},60] (* Harvey P. Dale, Nov 06 2016 *)

a(n)=(n+n%2)*(n+2-n%2)/4 \\ Charles R Greathouse IV, Apr 03 2012

j=0;for(k=1,850,s=sqrtint(4*k+1);if(s>j,j=s;print1(k,", "))) \\ Hugo Pfoertner, Sep 17 2018

def A087811(n): return n*(n+2)+(n&1)>>2 # Chai Wah Wu, Jul 27 2022

a(n) = (n + n mod 2)*(n + 2 - n mod 2)/4.
Numbers of the form m^2 or m^2 - m. - Don Reble, Oct 17 2003
a(1) = 1, a(2) = 2, a(n) = n + a(n - 2). - Alonso del Arte, Jun 18 2005
From Bruno Berselli, Feb 09 2011: (Start)
G.f.: x/((1+x)*(1-x)^3).
a(n) = (2*n*(n+2)-(-1)^n+1)/8. (End)
G.f.: G(0)/(2*(1-x^2)*(1-x)), where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = (C(n+2,2) - floor((n+2)/2))/2. - Mircea Merca, Nov 23 2013
a(n) = ((-1)^n*(-1 + (-1)^n*(1 + 2*n*(2 + n))))/8. - Fred Daniel Kline, Jan 06 2015
a(n) = Product_{k=1...n-1} (1 + 2 / (k + k mod 2)), n >= 1. - Fred Daniel Kline, Oct 30 2016
E.g.f.: (1/4)*(x*(3 + x)*cosh(x) + (1 + 3*x + x^2)*sinh(x)). - Stefano Spezia, Jan 05 2020
a(n) = (n*(n+2)+(n mod 2))/4. - Chai Wah Wu, Jul 27 2022
Sum_{n>=1} 1/a(n) = Pi^2/6 + 1. - Amiram Eldar, Sep 17 2022
a(n) = A024206(n) + 1. - Ya-Ping Lu, Dec 29 2023

A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

6, 12, 12, 24, 48, 30, 60, 54, 24, 84, 48, 36, 60, 120, 108, 66, 42, 96, 84, 126, 60, 108, 192, 90, 150, 84, 168, 120, 36, 204, 240, 210, 210, 60, 120, 216, 132, 300, 96, 336, 72, 192, 144, 240, 480, 294, 84, 252, 360, 432, 114, 156, 180, 210, 420, 120, 210, 420

Offset: 1

Henry Bottomley, May 26 2000

nonn

This is the ordering of triangles used for A316841.

Ray Chandler, Table of n, a(n) for n = 1..10000
A. Dendane, Heron's Formula For Area of a Triangle-Geometry Calculator

The sides are given by A055592, A055593, A055594.
Cf. A046131, A070786, A316841, A316853.
Range of values: A188158.

max = 42; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055595 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 4]](* Jean-François Alcover, Jun 12 2012 *)

a(n) = sqrt(s(n)*(s(n)-A055592(n))*(s(n)-A055593(n))*(s(n)-A055594(n))) where s(n) = (A055592(n)+A055593(n)+A055594(n))/2 i.e. half the perimeter of the triangle

A316842 Three-column table read by rows giving primitive integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i, gcd(i,j,k) = 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

			Table begins:
[1,1,1],
[2,2,1],
[3,2,2],
[3,3,1],
[3,3,2],
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,3],
[5,3,3],
[5,4,2],
...

Lars Blomberg, Table of n, a(n) for n = 1..9000

There are A123323(k) rows that begin with k.
The three columns are A316846, A316847, A316848.
A316850 is a compressed version.
See A316841 for all triples (including imprimitive triples).
See A316852 and A317181 & A317183 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

A331251 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives shortest side i. The other sides are in A331252 and A331253.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 3, 1, 3, 3, 1, 2, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 1, 3, 4, 2, 4, 1, 3, 2, 3, 1, 3, 4, 2, 4, 1, 2, 4, 1, 3, 3, 2, 3, 1, 2, 4, 1, 4, 4, 5, 2, 3, 4, 2, 1, 3, 3, 1, 2, 5, 4, 2, 1, 3, 4, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 5, 3, 4, 1, 5

Offset: 1

Hugo Pfoertner, Jan 19 2020

nonn

			List of first triangles:
   n
   | 16*A^2
   |    | i .... (this sequence)
   |    | | j .. (A331252)
   |    | | | k  (A331253)
   |    | | | |
   1    3 1 1 1
   2   15 1 2 2
   3   35 1 3 3
   4   48 2 2 2
   5   63 1 4 4
   6   63 2 2 3
   7   99 1 5 5
   8  128 2 3 3
   9  135 2 3 4
  10  143 1 6 6
  11  195 1 7 7

Cf. A316841, A316843, A316844, A316845, A316853, A317182, A331011.

Cf. A331252 (middle side j), A331253 (longest side k).

A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

Original entry on oeis.org

6, 24, 30, 54, 24, 84, 36, 60, 66, 42, 96, 84, 126, 90, 150, 84, 120, 36, 204, 210, 210, 60, 216, 132, 96, 336, 72, 144, 240, 294, 84, 252, 360, 114, 156, 180, 210, 120, 210, 420, 168, 270, 264, 168, 384, 240, 468, 126, 180, 336, 336, 504, 264, 330, 486, 216

Offset: 0

Eric W. Weisstein

nonn

This is the ordering of triangles used for A316841.

Eric Weisstein's World of Mathematics, Heronian Triangle.

The sides are given by A046128, A046129, A046130.
Cf. A055595, A072294, A316841, A316853.
Range of values: A383413.

sideMax = 60; r[c_] := Reap[Do[ p = (a + b + c)/2; red = Reduce[ area > 1 && a < b < c && area^2 == p*(p - a)*(p - b)*(p - c), area, Integers]; If[red =!= False, sol = {a, b, c, area} /. {ToRules[red]}; Sow[sol]], {b, 1, c - 1}, {a, c - b, b - 1}]]; triangles = Flatten[ Reap[ Do[rc = r[c]; If[rc[[2]] =!= {}, Sow[rc[[2, 1]]]], {c, 5, sideMax}]][[2, 1]] , 2]; Sort[ triangles, Which[#1[[3]] < #2[[3]], True, #1[[3]] > #2[[3]], False, #1[[2]] < #2[[2]], True,  #1[[2]] > #2[[2]], False, #1[[1]] <= #2[[1]], True, True, False] &][[All, 4]] (* Jean-François Alcover, Oct 29 2012 *)

A317182 Numbers k such that k = 16*area(T)^2 for an integer triangle, T.

Original entry on oeis.org

3, 15, 35, 48, 63, 99, 128, 135, 143, 195, 231, 240, 243, 255, 275, 320, 323, 351, 384, 399, 455, 483, 495, 560, 575, 576, 663, 675, 735, 768, 783, 819, 855, 896, 899, 935, 975, 1008, 1023, 1071, 1155, 1215, 1235, 1280, 1295, 1311, 1344, 1443, 1463, 1536, 1539

Offset: 1

N. J. A. Sloane, Jul 25 2018

nonn

The possible lengths of the chord connecting the cusps of the lens-like intersection area between two circles with integer radii and integer distance d between their centers are of the form sqrt(a(n))/d. - Hugo Pfoertner, Sep 05 2020

Lars Blomberg, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circle-Circle Intersection

Cf. A316841.

Sorted and uniqued values of A135622 or A316853.

More terms from Lars Blomberg, Apr 25 2019

Name edited by Peter Munn, Jul 30 2025

A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

Original entry on oeis.org

3, 15, 48, 35, 63, 128, 63, 135, 243, 240, 320, 99, 231, 275, 495, 384, 576, 768, 143, 351, 455, 819, 975, 560, 896, 1008, 1344, 195, 495, 675, 1215, 735, 1575, 1875, 768, 1280, 1536, 2048, 2304, 255, 663, 935, 1683, 1071, 2295, 2499, 2975, 1008, 1728

Offset: 1

Franz Vrabec, Feb 29 2008

			A070080(4)=1, A070081(4)=3, A070082(4)=3, so a(4)=(1+3+3)*(-1+3+3)*(1-3+3)*(1+3-3)=35.

See the formula section for the relationships with A070080, A070081, A070082, A070086.
Cf. A317182 (range of values), A331011 (nonunique values), A331250 (counts triangles by area).
Cf. A316853 (with terms ordered as for A316841), and using this order for other sets of triangles: A046131, A055595, A070786.

a(n)=(u+v+w)*(-u+v+w)*(u-v+w)*(u+v-w), where u=A070080(n), v=A070081(n), w=A070082(n).

A070086(n) = round(sqrt(a(n))/4).

A316852 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives perimeters of these triangles in the same order.

Original entry on oeis.org

3, 5, 7, 7, 8, 9, 10, 9, 11, 11, 11, 12, 13, 11, 12, 13, 14, 13, 13, 14, 15, 16, 13, 17, 15, 15, 16, 17, 15, 16, 17, 18, 19, 15, 16, 17, 18, 19, 20, 17, 18, 17, 19, 17, 18, 19, 20, 21, 22, 17, 19, 21, 23, 19, 19, 20, 19, 20, 21, 22, 23, 19, 20, 21, 22, 23, 24

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

The only missing numbers are 1,2,4,6.

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316841, A316842, A316851.

a(n) = A316846(n)+A316847(n)+A316848(n).

Terms a(54) and beyond from Lars Blomberg, Jul 27 2018

A331696 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331695 and A331697.

Original entry on oeis.org

2, 4, 1, 2, 6, 3, 2, 8, 1, 8, 2, 8, 1, 2, 10, 5, 2, 10, 5, 10, 5, 2, 12, 1, 4, 3, 4, 1, 12, 3, 4, 3, 12, 1, 2, 14, 7, 2, 14, 7, 14, 7, 2, 14, 7, 14, 7, 14, 7, 2, 16, 1, 16, 4, 16, 1, 16, 2, 16, 2, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 2, 18, 9, 2, 18, 3, 6, 9, 2

Offset: 1

Hugo Pfoertner, Jan 25 2020

			See A331695.

Cf. A316841.
Sides of triangle: A316843, A316844, A316845.
Cf. A331695, A331697.

A331697 Squared numerator of y-coordinate of 3rd point of the n-th triangle with integer sides in a sorted list corresponding to A331695. The y-coordinate is given by sqrt(a(n))/A331696(n).

Original entry on oeis.org

3, 15, 3, 7, 35, 32, 27, 135, 5, 63, 15, 495, 12, 11, 231, 144, 39, 99, 96, 819, 336, 75, 455, 7, 39, 56, 175, 16, 143, 35, 135, 128, 2975, 27, 15, 675, 384, 51, 495, 320, 2295, 864, 95, 195, 192, 1683, 720, 4275, 1440, 147, 1071, 9, 935, 135, 3591, 20, 663, 27

Offset: 1

Hugo Pfoertner, Jan 25 2020

nonn

			a(1) = 3 because the 3rd point of the smallest triangle with integer sides (1,1,1) is at (x,y)=(1/2,sqrt(3)/2), A331695(1)=1, A331696(1)=2.
See A331695 for an extended list.

Cf. A316841.
Sides of triangle: A316843, A316844, A316845.
Cf. A331695, A331696.

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A087811 Numbers k such that ceiling(sqrt(k)) divides k.

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A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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A316842 Three-column table read by rows giving primitive integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i, gcd(i,j,k) = 1.

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A331251 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives shortest side i. The other sides are in A331252 and A331253.

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A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

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A317182 Numbers k such that k = 16*area(T)^2 for an integer triangle, T.

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A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

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A316852 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives perimeters of these triangles in the same order.

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A331696 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331695 and A331697.

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A331697 Squared numerator of y-coordinate of 3rd point of the n-th triangle with integer sides in a sorted list corresponding to A331695. The y-coordinate is given by sqrt(a(n))/A331696(n).

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