A002623 -id:A002623 - cp's OEIS frontend

A247587 Number of obtuse triangles with integer sides at most n.

0, 0, 1, 2, 4, 8, 13, 20, 30, 42, 57, 74, 95, 120, 149, 182, 219, 261, 309, 362, 420, 485, 556, 632, 715, 806, 906, 1012, 1125, 1247, 1377, 1517, 1666, 1824, 1993, 2170, 2358, 2555, 2765, 2986

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(4) = 2 because there are 2 obtuse triangles with integer sides less than or equal to 4: (2,2,3); (2,3,4).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A002623, A224921, A247586, A247588, A247589.

tr_o:=proc(n) local a, b, c, t, d; t:=0:
  for a to n do
  for b from a to n do
  for c from b to min(a+b-1, n) do
  d:=a^2+b^2-c^2:
  if d<0 then t:=t+1 fi
  od od od;
  [n, t]; end;

a(n)=sum(a=2,n-1,sum(b=a,n-1,max(0,min(n,a+b-1)-sqrtint(a^2+b^2)))) \\ Charles R Greathouse IV, Sep 20 2014

obtuse(n)=sum(a=2,n-1, max(0, sqrtint(n^2-1-a^2)-max(a,n-a+1)+1))
s=0; vector(100,n, s+=obtuse(n)) \\ Charles R Greathouse IV, Sep 20 2014

A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 4, 1, 1, 0, 4, 2, 1, 10, 4, 5, 3, 1, 0, 10, 8, 7, 4, 1, 20, 10, 14, 13, 10, 5, 1, 0, 20, 20, 22, 20, 14, 6, 1, 35, 20, 30, 34, 35, 30, 19, 7, 1, 0, 35, 40, 50, 56, 55, 44, 25, 8, 1, 56, 35, 55, 70, 84, 91, 85, 63, 32, 9, 1, 0, 56, 70, 95, 120, 140, 146, 129, 88, 40, 10, 1

Offset: 0

Henry Bottomley, Nov 20 2000

			Rows are (1,0,4,0,10,0,20,...), (1,1,4,4,10,10,20,...), (1,2,5,8,14,20,30,...), (1,3,7,13,22,34,50,...), (1,4,10,20,35,56,84,...) etc.

Rows are A000292 with zeros, A058187 (A000292 with terms duplicated), A006918, A002623, A000292, A000330, A005900, A001845, A008412.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(4, 1)=4, T(0, 2n)=T(4, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^4.

A078126 Negative determinant of n X n matrix M_{i,j}=1 if i=j or i+j=1 (mod 2).

Original entry on oeis.org

-1, -1, 0, 1, 3, 5, 8, 11, 15, 19, 24, 29, 35, 41, 48, 55, 63, 71, 80, 89, 99, 109, 120, 131, 143, 155, 168, 181, 195, 209, 224, 239, 255, 271, 288, 305, 323, 341, 360, 379, 399, 419, 440, 461, 483, 505, 528, 551, 575, 599, 624, 649, 675, 701, 728, 755, 783, 811

Offset: 0

Michael Somos, Nov 18 2002

Apparently, also 6(n+3) times the Dedekind sum s(2,n+3). - Ralf Stephan, Sep 16 2013

			G.f. = -1 - x + x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 15*x^8 + 19*x^9 + ...

Moira Chas and Anthony Phillips, Self-intersection numbers of curves in the doubly-punctured plane , arXiv:1001.4568 [math.GT], 2010. [_Jonathan Vos Post_, Jan 27 2010]
Eric Weisstein's World of Mathematics, Dedekind Sum
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002620, A002623, A004526, A024206.

A078126:=n->floor((n + 2)*(n - 2)/4); seq(A078126(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2014

Table[Floor[(n + 2)(n - 2)/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2014 *)
LinearRecurrence[{2,0,-2,1},{-1,-1,0,1},60] (* Harvey P. Dale, Sep 10 2015 *)
a[n_]:=-Det[Table[If[i==j ||Mod[i+j,2]==1,1,0],{i,n},{j,n}]]; Join[{-1},Array[a,57]] (* Stefano Spezia, Aug 06 2024 *)

a(n)=-matdet(matrix(n,n,i,j,i==j||((i+j)%2))) /* Ralf Stephan, Sep 16 2013 */

a(n)=sumdedekind(2,n+3)*6*(n+3) /* Ralf Stephan, Sep 16 2013 */

G.f.: (-1 + x + 2*x^2 - x^3) / ((1 - x^2) * (1 - x)^2).
a(n) = A002620(n) - 1.
a(n) = A002623(n-2) - A002623(n-3) - 1.
a(n) = A024206(n-1) for all n in Z.
a(n) = floor( (n+2)(n-2)/4 ). - Wesley Ivan Hurt, Jun 16 2013
A004526(n) = a(n) - a(n-1) for all n in Z. - Michael Somos, Aug 22 2016
a(n) = Sum_{i=1..n+2} floor((n-i+1)/2). - Wesley Ivan Hurt, Sep 12 2017
E.g.f.: ((x^2 + x - 4)*cosh(x) + (x^2 + x - 5)*sinh(x))/4. - Stefano Spezia, Aug 06 2024

A-number twister corrected in cross-refs by R. J. Mathar, Feb 11 2010

A095941 Number of subsets of {1,2,...,n} such that every number in the set is no larger than the sum of the other numbers in the set.

Original entry on oeis.org

0, 0, 1, 4, 13, 35, 85, 194, 425, 904, 1885, 3878, 7904, 16008, 32282, 64913, 130280, 261145, 523036, 1047017, 2095222, 4191927, 8385695, 16773663, 33550117, 67103645, 134211440, 268427907, 536861880, 1073731053, 2147470842, 4294952115, 8589916646, 17179848025

Offset: 1

Michael Rieck and W. Edwin Clark, Jul 13 2004

These are the lengths of the sides of a (possibly degenerate) polygon.

Might be called "coalition sets": no member of the set can outnumber all of the others, so a coalition is needed in order to get a majority. - Jaap Spies, Jul 14 2004

Alois P. Heinz, Table of n, a(n) for n = 1..3321 (first 500 terms from T. D. Noe)

See A095944 for formula. Cf. A002623.

b:= proc(n, s) option remember; `if`(s<1, 2^n,
     `if`(n*(n+1)/2 add(b(j-1, j), j=1..n):
seq(a(n), n=1..37);  # Alois P. Heinz, Feb 13 2021

max = 30; -Accumulate[ Accumulate[q = PartitionsQ[ Range[max]]] + 1] + Accumulate[q] + 2^Range[max] - 1 (* Jean-François Alcover, Aug 01 2013, after A095944 *)

Extended by Alexander D. Healy using data from A095944, Nov 18 2005

A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, -1, -2, 1, 1, 1, 2, -2, -3, 1, 1, 0, 2, 4, -3, -4, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, -2, -6, 7, 11, -5, -6, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, 3, 9, -13, -24, 16, 22, -7, -8, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, -3, -12, 22, 46, -40, -62, 29, 37, -9, -10, 1, 1, 1, 4, -12

Offset: 0

Wolfdieter Lang, Apr 04 2007

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k) = a(2*p+1,2*k) = A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1) = a(2*p+2,2*k+1) = A059260(p+k+1,2*k+1)*(-1)^(p+k), k >= 0.

			The triangle a(n,m) begins:
  n\m  0   1   2   3   4   5   6   7   8   9  10
   0:  1
   1:  1   1
   2:  0   1   1
   3:  0  -1   1   1
   4:  1  -1  -2   1   1
   5:  1   2  -2  -3   1   1
   6:  0   2   4  -3  -4   1   1
   7:  0  -2   4   7  -4  -5   1   1
   8:  1  -2  -6   7  11  -5  -6   1   1
   9:  1   3  -6 -13  11  16  -6  -7   1   1
  10:  0   3   9 -13 -24  16  22  -7  -8   1   1
... reformatted by _Wolfdieter Lang_, Oct 16 2012
Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).
S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).
From _Wolfdieter Lang_, Oct 16 2012: (Start)
S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).
S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)

Wolfdieter Lang, First 15 rows
Per Alexandersson, Luis Angel González-Serrano, and Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables, arXiv:1912.12725 [math.CO], 2019.
Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, and Mario Alberto Moctezuma-Salazar, Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables z_j = x_j + x_j^(-1), The Elec. J. of Combinatorics (2021) Vol. 28, No. 1, #P1.56.

Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).
Cf. A128495 for S(2; n, x) coefficient table.
The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.
For m >= 1 the column sequences (without leading zeros) are of the form a(m, 2*k) = a(m, 2*k+1) = ((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.

S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.
a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
From Wolfdieter Lang, Oct 16 2012: (Start)
a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.
a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)

A247586 Number of acute triangles with integer sides less than or equal to n.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 36, 49, 64, 81, 102, 127, 154, 185, 219, 258, 301, 349, 401, 457, 520, 587, 660, 740, 824, 914, 1010, 1114, 1225, 1342, 1468, 1600, 1740, 1887, 2041, 2206, 2378, 2561, 2750, 2948

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(2) = 3 because there are 3 acute triangles with integer sides less than or equal to 2: (1,1,1); (1,2,2); (2,2,2).

Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A002623, A224921, A247587, A247588, A247589.

tr_a:=proc(n) local a,b,c,t,d;t:=0:
  for a to n do
  for b from a to n do
  for c from b to min(a+b-1,n) do
  d:=a^2+b^2-c^2:
  if d>0 then t:=t+1 fi
  od od od;
  [n,t]; end;

a[n_] := Module[{a, b, c, d, t = 0}, Do[d = a^2 + b^2 - c^2; If[d>0, t++], {a, n}, {b, a, n}, {c, b, Min[a+b-1, n]}]; t]; Array[a, 40] (* Jean-François Alcover, Jun 19 2019, from Maple *)

import itertools
def A247586(n):
    I = itertools.combinations_with_replacement(range(1,n+1),3)
    F = filter(lambda c: c[0]**2 + c[1]**2 > c[2]**2, I)
    return len(list(F))
print([A247586(n) for n in range(41)]) # Peter Luschny, Sep 22 2014

A248851 a(n) = ( 2n(2n^2 + 9n + 14) + (-1)^n - 1 )/16.

Original entry on oeis.org

0, 3, 10, 22, 41, 68, 105, 153, 214, 289, 380, 488, 615, 762, 931, 1123, 1340, 1583, 1854, 2154, 2485, 2848, 3245, 3677, 4146, 4653, 5200, 5788, 6419, 7094, 7815, 8583, 9400, 10267, 11186, 12158, 13185, 14268, 15409, 16609, 17870, 19193, 20580, 22032

Offset: 0

Luce ETIENNE, Mar 03 2015

Consider a grid of small triangles of side 1 forming a regular polygon with side n*(n+2); a(n) is the number of equilateral triangles of side length >= 1 in this figure which are oriented with the sides of the figure.
This sequence gives the number of triangles of all sizes in a (n^2+2*n)-iamond with a 4*n-gon configuration.
Equals (1/2)*Sum_{j=0..n-1} (n-j)*(n+1-j) + (-1 + (1/8)*Sum_{j=0..(2*n+1-(-1)^n)/4} (2*n+3+(-1)^n-4*j)*(2*n+3-(-1)^n-4*j)) numbers of triangles in a direction and in the opposite direction.

			From third comment: a(0)=0, a(1)=1+2, a(2)=4+6, a(3)=10+12, a(4)=20+21, a(5)=35+33.

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Illustration for a(1), a(2), a(3), a(4), a(5)
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A117143, A000292, A002623, A045947, A005563, A008586.

[(4*n^3+18*n^2+28*n-(1-(-1)^n)) div 16: n in [0..50]]; // Vincenzo Librandi, Mar 21 2015

CoefficientList[Series[x (x^3 - 2 x^2 + x + 3) / ((x - 1)^4(x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 21 2015 *)
LinearRecurrence[{3,-2,-2,3,-1},{0,3,10,22,41},50] (* Harvey P. Dale, Jan 17 2023 *)

concat(0, Vec(x*(x^3-2*x^2+x+3)/((x-1)^4*(x+1)) + O(x^100))) \\ Colin Barker, Mar 03 2015

G.f.: x*(x^3-2*x^2+x+3) / ((x-1)^4*(x+1)). - Colin Barker, Mar 03 2015

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - Colin Barker, Mar 03 2015

Typo in formula fixed by Vincenzo Librandi, Mar 21 2015

Name rewritten using the closed form by Bruno Berselli, Apr 19 2015

A007518 a(n) = floor(n(n+2)(2*n-1)/8).

Original entry on oeis.org

0, 3, 9, 21, 39, 66, 102, 150, 210, 285, 375, 483, 609, 756, 924, 1116, 1332, 1575, 1845, 2145, 2475, 2838, 3234, 3666, 4134, 4641, 5187, 5775, 6405, 7080, 7800, 8568, 9384, 10251, 11169, 12141, 13167, 14250, 15390, 16590, 17850, 19173, 20559, 22011, 23529, 25116, 26772, 28500, 30300

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

From a problem on p. 151 of J. Rec. Math., 7 (1975).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3, -2, -2, 3, -1).

Column 4 of triangle A094953.

List([1..50],n->Int(n*(n+2)*(2*n-1)/8)); # Muniru A Asiru, Mar 22 2018

[Floor(n*(n+2)(2*n-1)/8): n in [1..50]]; // G. C. Greubel, Mar 21 2018

[seq(floor(n*(n+2)*(2*n-1)/8),n=1..50)]; # Muniru A Asiru, Mar 22 2018

Table[Floor[(n(n+2)(2n-1))/8],{n,50}] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,3,9,21,39},40] (* Harvey P. Dale, Oct 06 2014 *)

vector(50, n, n*(n+2)*(2*n-1)\8) \\ Michel Marcus, Oct 12 2014

a(n) = 3*A002623(n) for n>0. - M. F. Hasler, Sep 15 2009
G.f.: 3*x/((x+1)*(x-1)^4). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]
a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5) with a(1)=0, a(2)=3, a(3)=9, a(4)=21, a(5)=39. - Harvey P. Dale, Oct 06 2014

Offset corrected by Harvey P. Dale, Oct 06 2014

Terms a(40) onward added by G. C. Greubel, Mar 21 2018

A052267 Number of 2 X n matrices over GF(3) under row and column permutations.

Original entry on oeis.org

1, 6, 27, 92, 267, 678, 1561, 3312, 6582, 12372, 22194, 38232, 63594, 102564, 160974, 246576, 369567, 543114, 784069, 1113684, 1558557, 2151578, 2933151, 3952416, 5268796, 6953544, 9091668, 11783856, 15148836, 19325736, 24476940, 30790944, 38485773, 47812398

Offset: 0

Vladeta Jovovic, Feb 04 2000

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1).

Cf. A002623, A002727.

Vec((3*x^2+1) / ((1-x^2)^3*(1-x)^6) + O(x^40)) \\ Colin Barker, Jan 16 2017

G.f.: (3*x^2+1) /((1-x^2)^3*(1-x)^6).

a(n) = ((315*(475+37*(-1)^n) + 6*(54959+945*(-1)^n)*n + (298618+630*(-1)^n)*n^2 + 150528*n^3 + 46788*n^4 + 9156*n^5 + 1092*n^6 + 72*n^7 + 2*n^8)) / 161280. - Colin Barker, Jan 16 2017

A111746 Number of squares in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.

Original entry on oeis.org

0, 5, 17, 42, 82, 143, 227, 340, 484, 665, 885, 1150, 1462, 1827, 2247, 2728, 3272, 3885, 4569, 5330, 6170, 7095, 8107, 9212, 10412, 11713, 13117, 14630, 16254, 17995, 19855, 21840, 23952, 26197, 28577, 31098, 33762, 36575, 39539, 42660, 45940

Offset: 0

Floor van Lamoen, Nov 19 2005

Cf. A000330, A000447, A002623.

[n*(4*n^2+12*n+11)/6+1/4-(-1)^n/4: n in [0..60]]; // Vincenzo Librandi, May 12 2015

seq(n*(4*n^2-1)/6 - 1/4 + 1/4*(-1)^n,n=1..50);

Table[n (4 n^2 + 12 n + 11)/6 + 1/4 - (-1)^n/4, {n, 0, 60}] (* Vincenzo Librandi, May 12 2015 *)

a(n) = n*(4*n^2 + 12*n + 11)/6 + 1/4 - (-1)^n/4 = floor(A000447(n+1)/2).
a(n) = 4*A002623(n-1) + A000330(n), with A002623(-1)=0. - Luce ETIENNE, May 12 2015
G.f.: x*(5 + 2*x + x^2)/((1-x)^4*(1+x)). - Vincenzo Librandi, May 12 2015

Closed formula adapted to the offset by Bruno Berselli, May 12 2015

cp's OEIS Frontend

A247587 Number of obtuse triangles with integer sides at most n.

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A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

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A078126 Negative determinant of n X n matrix M_{i,j}=1 if i=j or i+j=1 (mod 2).

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A095941 Number of subsets of {1,2,...,n} such that every number in the set is no larger than the sum of the other numbers in the set.

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A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

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A247586 Number of acute triangles with integer sides less than or equal to n.

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A248851 a(n) = ( 2*n*(2*n^2 + 9*n + 14) + (-1)^n - 1 )/16.

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A007518 a(n) = floor(n*(n+2)*(2*n-1)/8).

A248851 a(n) = ( 2n(2n^2 + 9n + 14) + (-1)^n - 1 )/16.

A007518 a(n) = floor(n(n+2)(2*n-1)/8).