A211795 -id:A211795 - cp's OEIS frontend

A212069 Number of (w,x,y,z) with all terms in {1,...,n} and 3*w = x+y+z.

0, 1, 2, 9, 22, 41, 72, 115, 170, 243, 334, 443, 576, 733, 914, 1125, 1366, 1637, 1944, 2287, 2666, 3087, 3550, 4055, 4608, 5209, 5858, 6561, 7318, 8129, 9000, 9931, 10922, 11979, 13102, 14291, 15552, 16885, 18290, 19773, 21334, 22973

Offset: 0

Clark Kimberling, May 01 2012

w is the average of {x,y,z}, as well as {w,x,y,z}.
For a guide to related sequences, see A211795.
a(n) is also the number of (w,x,y,z) with all terms in {0,1,...,n-1} and 3*w = x+y+z. - Clark Kimberling, May 16 2012

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A049347, A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w == x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212087 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -3, 2, -3, 3, -1},{0, 1, 2, 9, 22, 41},42] (* Ray Chandler, Aug 02 2015 *)

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
From R. J. Mathar, Jun 25 2012: (Start)
G.f.: x*(1 - x + 6*x^2 - x^3 + x^4)/((1 + x + x^2)*(1 - x)^4).
a(n) = (n^3 + 2*A049347(n-1))/3. (End)
E.g.f.: (3*exp(x)*x*(1 + 3*x + x^2) + 4*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Aug 06 2025

A212135 Number of (w,x,y,z) with all terms in {1,...,n} and median

Original entry on oeis.org

0, 0, 4, 24, 84, 220, 480, 924, 1624, 2664, 4140, 6160, 8844, 12324, 16744, 22260, 29040, 37264, 47124, 58824, 72580, 88620, 107184, 128524, 152904, 180600, 211900, 247104, 286524, 330484, 379320, 433380, 493024, 558624, 630564, 709240, 795060, 888444

Offset: 0

Clark Kimberling, May 05 2012

Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>mean.
For a guide to related sequences, see A211795.
Also, a(n+1) is the length of the long leg of the unique primitive Pythagorean triple whose inradius is A000217(n). - Miguel-Ángel Pérez García-Ortega, Jul 13 2025

Colin Barker, Table of n, a(n) for n = 0..1001
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[
Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 > (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Flatten[Map[{t[#]} &, Range[0, 20]]]  (* A212135 *)
%/4 (* A002817 *)

concat(vector(2), Vec(4*x*(1 + x + x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Dec 02 2017

a(n) + A212134(n) = n^4.
a(n) = n*(n - 1)*(n^2 - n + 2)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 4*x*(1 + x + x^2) / (1 - x)^5. - Colin Barker, Dec 02 2017

A212523 Number of (w,x,y,z) with all terms in {1,...,n} and w+x

Original entry on oeis.org

0, 0, 5, 31, 106, 270, 575, 1085, 1876, 3036, 4665, 6875, 9790, 13546, 18291, 24185, 31400, 40120, 50541, 62871, 77330, 94150, 113575, 135861, 161276, 190100, 222625, 259155, 300006, 345506, 395995, 451825, 513360, 580976, 655061

Offset: 0

Clark Kimberling, May 21 2012

For a guide to related sequences, see A211795.

If the initial 0 is omitted, the sequence {b(n): n>=0} = {0, 5, 31, 106, 270, 575, 1085, 1876, 3036, 4665, 6875, 9790, ...} is given by b(n) = n*(n+1)*(3*n^2+7*n+5)/6. - N. J. A. Sloane, Jul 25 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A211795.

I:=[0, 0, 5, 31, 106]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2) +10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jun 09 2012

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x < y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212523 *)
LinearRecurrence[{5,-10, 10, -5, 1},{0,0,5,31,106},60] (* Vincenzo Librandi, Jun 09 2012 *)

a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5).
a(n) = Sum_{i=0..n-1} A048395(i). - J. M. Bergot, Jun 08 2012
G.f.: -x^2*(x+5)*(1+x)/(x-1)^5 . - R. J. Mathar, Sep 23 2016

A212683 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y| = w + |y-z|.

Original entry on oeis.org

0, 0, 2, 8, 22, 46, 84, 138, 212, 308, 430, 580, 762, 978, 1232, 1526, 1864, 2248, 2682, 3168, 3710, 4310, 4972, 5698, 6492, 7356, 8294, 9308, 10402, 11578, 12840, 14190, 15632, 17168, 18802, 20536, 22374, 24318, 26372, 28538, 30820

Offset: 0

Clark Kimberling, May 24 2012

For a guide to related sequences, see A211795.

Also the number of (w,x,y) with all terms in {0,...,n-1} and |w-x| < |x-y|, see A212959. - Clark Kimberling, Jun 02 2012

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A019298, A211795, A212684.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[x - y] == w + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212683 *)
%/2  (* A019298 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 8, 22}, 40]

a(n) = 2*A019298(n-1) for n>=1.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
G.f.: (2*x^2 + 2*x^3 + 2*x^4)/(1 - 3*x + 2*x^2 + 2*x^3 - 3*x^4 + x^5).
a(n) + A212684(n) = n^3. - Clark Kimberling, Jun 02 2012 [corrected by Jason Yuen, Aug 19 2025]
a(n) = (2*n^3 - 3*n^2 + 2*n - (n mod 2))/4. - Ayoub Saber Rguez, Sep 02 2021

A212760 Number of (w,x,y,z) with all terms in {0,...,n}, w even, and x = y + z.

Original entry on oeis.org

1, 3, 12, 20, 45, 63, 112, 144, 225, 275, 396, 468, 637, 735, 960, 1088, 1377, 1539, 1900, 2100, 2541, 2783, 3312, 3600, 4225, 4563, 5292, 5684, 6525, 6975, 7936, 8448, 9537, 10115, 11340, 11988, 13357, 14079, 15600, 16400, 18081, 18963, 20812, 21780, 23805

Offset: 0

Clark Kimberling, May 29 2012

A signed version is A122576.
For a guide to related sequences, see A211795.
Partial sums of the positive elements of A129194. - Omar E. Pol, Dec 28 2013

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

Cf. A211795.

Cf. A001318, A260706.

a212760 = a260706 . fromInteger . a001318 . (+ 1)
-- Reinhard Zumkeller, Nov 17 2015

[(n+1)*(n+2)*(2*n+3+(-1)^n)/8 : n in [0..50]]; // Wesley Ivan Hurt, Jul 22 2014

A212760:=n->(n+1)*(n+2)*(2*n+3+(-1)^n)/8: seq(A212760(n), n=0..50); # Wesley Ivan Hurt, Jul 22 2014

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && x == y + z, s++],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 50]]  (* A212760 *)
Table[(n + 1) (n + 2) (2 n + 3 + (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Jul 22 2014 *)
CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 + x)^3 (1 - x)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 23 2014 *)

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).
G.f.: ( 1+2*x+6*x^2+2*x^3+x^4 ) / ( (1+x)^3*(1-x)^4 ).
a(n) = (n+1)*(n+2)*(2*n+3+(-1)^n)/8. - Wesley Ivan Hurt, Jul 22 2014
a(n) = A260706(A001318(n+1)). - Reinhard Zumkeller, Nov 17 2015
a(n) = Sum_{i=1..n+1} numerator(i^2/2). - Wesley Ivan Hurt, Feb 26 2017

A211920 Number of (w,x,y,z) with all terms in {1,...,n} and 2wx<3yz.

Original entry on oeis.org

0, 1, 11, 50, 160, 387, 792, 1480, 2511, 4001, 6104, 8943, 12623, 17396, 23374, 30765, 39803, 50707, 63656, 79050, 96985, 117795, 141871, 169444, 200754, 236327, 276397, 321322, 371547, 427471, 489342, 557906, 633285, 716040, 806754

Offset: 0

Clark Kimberling, Apr 28 2012

nonn

a(n)+A211923(n)=n^4.

See A211795 for a guide to related sequences.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w*x < 3 y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211920 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212087 Number of (w,x,y,z) with all terms in {1,...,n} and w^2+x^2=y^2+z^2.

Original entry on oeis.org

0, 1, 6, 15, 28, 45, 66, 95, 132, 173, 210, 267, 320, 385, 458, 523, 600, 693, 786, 899, 1000, 1109, 1226, 1367, 1492, 1629, 1778, 1931, 2084, 2269, 2426, 2615, 2812, 3013, 3222, 3427, 3624, 3857, 4094, 4335, 4564, 4841, 5082, 5379, 5656, 5913

Offset: 0

Clark Kimberling, May 02 2012

nonn

For a guide to related sequences, see A211795.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 + x^2 == y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 60]] (* A212087 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212151 Number of 2 X 2 matrices M of positive integers such that permanent(M) < n.

Original entry on oeis.org

0, 0, 0, 1, 5, 13, 27, 47, 75, 112, 156, 214, 278, 358, 444, 552, 660, 796, 930, 1099, 1259, 1457, 1649, 1885, 2101, 2377, 2623, 2933, 3221, 3569, 3879, 4279, 4623, 5056, 5452, 5926, 6334, 6878, 7328, 7892, 8404, 9018, 9540, 10228, 10788, 11504, 12142, 12898

Offset: 0

Clark Kimberling, May 07 2012

For a guide to related sequences, see A211795.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000005, A211795, A212240.

Cf. A055507.

N:= 100: # to get a(0)..a(N)
g:= z*(1-z)^(-1)*add(z^i/(1-z^i),i=1..N-2)^2:
S:=series(g,z,N+1):
seq(coeff(S,z,n),n=0..N); # Robert Israel, Nov 16 2017

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x + y*z < n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212151 *)
(* Peter J. C. Moses, Apr 13 2012 *)

from sympy import divisor_count
def A212151(n): return  sum((sum(divisor_count(i+1)*divisor_count(j-i) for i in range(j>>1))<<1)+(divisor_count(j+1>>1)**2 if j&1 else 0) for j in range(1,n-1)) # Chai Wah Wu, Jul 26 2024

a(n) + A212240(n) = n^4.
a(n) = Sum_{k=1..n-1} Sum_{i=1..n-1} d(k) * floor((n-k-1)/i), where d(k) is the number of divisors of k (A000005). - Wesley Ivan Hurt, Nov 16 2017
G.f.: (x/(1-x))*(Sum_{i>=1} x^i/(1-x^i))^2. - Robert Israel, Nov 16 2017
from Ridouane Oudra, Oct 10 2023: (Start)
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} tau(i*j)*floor((n-1)/(i+j)) ;
a(n) = Sum_{i=1..n-1} Sum_{j=1..i-1} tau(j)*tau(i-j) ;
a(n+2) = Sum_{i=1..n} A055507(i). (End)

A212246 Number of (w,x,y,z) with all terms in {1,...,n} and w <= x > y <= z.

Original entry on oeis.org

0, 0, 4, 21, 65, 155, 315, 574, 966, 1530, 2310, 3355, 4719, 6461, 8645, 11340, 14620, 18564, 23256, 28785, 35245, 42735, 51359, 61226, 72450, 85150, 99450, 115479, 133371, 153265, 175305, 199640, 226424, 255816, 287980, 323085

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w <= x > y <= z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212246 *)

a(n) = n*(n-1)*(n+1)*(5*n+6)/24 \\ Charles R Greathouse IV, Feb 16 2015

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
G.f.: x^2*(4+x)/(1-x)^5. - Bruno Berselli, May 31 2012
a(n) = (n-1)*n*(n+1)*(5*n+6)/24. - Bruno Berselli, May 31 2012

Offset corrected by Charles R Greathouse IV, Feb 16 2015

A212509 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=3z.

Original entry on oeis.org

0, 1, 12, 63, 180, 437, 891, 1628, 2736, 4392, 6600, 9646, 13608, 18669, 24990, 32955, 42432, 54033, 67797, 84010, 102900, 125118, 150282, 179444, 212544, 249977, 292032, 339687, 392196, 451165, 516375, 588336, 667392, 754908, 849660, 953922, 1067256

Offset: 0

Clark Kimberling, May 20 2012

For a guide to related sequences, see A211795.

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

Cf. A211795, A212508.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y <= 3 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]]   (* A212509 *)

concat(0, Vec(x*(1 +12*x +61*x^2 +154*x^3 +288*x^4 +421*x^5 +505*x^6 +510*x^7 +487*x^8 +387*x^9 +246*x^10 +120*x^11 +42*x^12 +6*x^13) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^60))) \\ Colin Barker, Dec 17 2015

a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)+2*a(n-7)-2*a(n-9)+4*a(n-11)+a(n-12)-2*a(n-13)-2*a(n-14)+a(n-16).

G.f.: x*(1 +12*x +61*x^2 +154*x^3 +288*x^4 +421*x^5 +505*x^6 +510*x^7 +487*x^8 +387*x^9 +246*x^10 +120*x^11 +42*x^12 +6*x^13) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - Colin Barker, Dec 17 2015

cp's OEIS Frontend

A212069 Number of (w,x,y,z) with all terms in {1,...,n} and 3*w = x+y+z.

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A212135 Number of (w,x,y,z) with all terms in {1,...,n} and median

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PARI

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A212523 Number of (w,x,y,z) with all terms in {1,...,n} and w+x

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Magma

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A212683 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y| = w + |y-z|.

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A212760 Number of (w,x,y,z) with all terms in {0,...,n}, w even, and x = y + z.

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A211920 Number of (w,x,y,z) with all terms in {1,...,n} and 2*w*x<3*y*z.

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A212087 Number of (w,x,y,z) with all terms in {1,...,n} and w^2+x^2=y^2+z^2.

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A212151 Number of 2 X 2 matrices M of positive integers such that permanent(M) < n.

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A211920 Number of (w,x,y,z) with all terms in {1,...,n} and 2wx<3yz.