A211422 -id:A211422 - cp's OEIS frontend

A211520 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 4y = 2x.

0, 0, 0, 1, 2, 3, 5, 7, 10, 12, 16, 19, 24, 27, 33, 37, 44, 48, 56, 61, 70, 75, 85, 91, 102, 108, 120, 127, 140, 147, 161, 169, 184, 192, 208, 217, 234, 243, 261, 271, 290, 300, 320, 331, 352, 363, 385, 397, 420, 432, 456, 469, 494, 507, 533, 547, 574

Offset: 0

Clark Kimberling, Apr 14 2012

For a guide to related sequences, see A211422.

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

Cf. A178804, A211422, A282513.

a211520 n = a211520_list !! n
a211520_list = 0 : 0 : 0 : scanl1 (+) a178804_list
-- Reinhard Zumkeller, Nov 15 2014

seq(floor((n-1)^2/4)-floor((n-1)/4)*floor((n+1)/4), n=0..60); # Ridouane Oudra, Nov 21 2024

t[n_] := t[n] = Flatten[Table[w - 2 x + 4 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* this sequence *)
FindLinearRecurrence[t]
LinearRecurrence[{1,1,-1,1,-1,-1,1},{0,0,0,1,2,3,5},57] (* Ray Chandler, Aug 02 2015 *)

{ my(x='x+O('x^66)); concat([0,0,0],Vec( x^3*(1+x+x^3) / ( (1-x)^3*(1+x)^2*(1+x^2) ) ) ) } \\ Joerg Arndt, Apr 02 2017

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).
a(n) - a(n-1) = A178804(n-2). - Reinhard Zumkeller, Nov 15 2014
a(n) = (6*n^2-10*n+3+(2*n-7)*(-1)^n-4*(-1)^((2*n-3-(-1)^n)/4))/32. - Luce ETIENNE, Dec 31 2015
a(n) = Sum_{k=1..floor(n/2)} floor((n-k)/2). - Wesley Ivan Hurt, Apr 01 2017
G.f.: x^3 * (1+x+x^3) / ( (1-x)^3*(1+x)^2*(1+x^2) ). - Joerg Arndt, Apr 02 2017
a(n)+a(n-1) = A282513(n-2). - R. J. Mathar, Jun 23 2021
a(n) = floor((n-1)^2/4) - floor((n-1)/4)*floor((n+1)/4). - Ridouane Oudra, Nov 21 2024

A211521 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 2x = 4y.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 9, 11, 16, 18, 25, 28, 36, 39, 49, 53, 64, 68, 81, 86, 100, 105, 121, 127, 144, 150, 169, 176, 196, 203, 225, 233, 256, 264, 289, 298, 324, 333, 361, 371, 400, 410, 441, 452, 484, 495, 529, 541, 576, 588, 625, 638, 676, 689, 729, 743

Offset: 0

Clark Kimberling, Apr 14 2012

For a guide to related sequences, see A211422.

Also, number of ordered pairs (w,x) with both terms in {1,...,n} and w+2x divisible by 4. - Pontus von Brömssen, Jan 19 2020

Colin Barker, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Cf. A211422.

a:=[0]; for n in [1..55] do m:=0; for i, j in [1..n] do if (i+2*j) mod 4 eq 0  then m:=m+1; end if; end for; Append(~a, m); end for; a; // Marius A. Burtea, Jan 19 2020

R:=PowerSeriesRing(Integers(), 57); [0,0] cat Coefficients(R!( x^3*(1 + x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)))); // Marius A. Burtea, Jan 19 2020

t[n_] := t[n] = Flatten[Table[w + 2 x - 4 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211521 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,1,-1,1,-1,-1,1},{0,0,1,2,4,5,9},56] (* Ray Chandler, Aug 02 2015 *)

concat(vector(2), Vec(x^2*(1 + x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)) + O(x^40))) \\ Colin Barker, Dec 02 2017

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).
a(n) = (2*n^2-n+1+(n-1)*(-1)^n+(-1)^((2*n+1-(-1)^n)/4)-(-1)^((6*n+1-(-1)^n)/4))/8. - Luce ETIENNE, Dec 31 2015
G.f.: x^2*(1 + x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)). - Colin Barker, Dec 02 2017

Offset corrected by Pontus von Brömssen, Jan 19 2020

A211523 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w+2x=5y.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 10, 13, 17, 20, 24, 29, 34, 40, 45, 51, 58, 65, 73, 80, 88, 97, 106, 116, 125, 135, 146, 157, 169, 180, 192, 205, 218, 232, 245, 259, 274, 289, 305, 320, 336, 353, 370, 388, 405, 423, 442, 461, 481, 500, 520, 541, 562, 584, 605, 627

Offset: 0

Clark Kimberling, Apr 14 2012

For a guide to related sequences, see A211422.

Also, number of ordered pairs (w,x) with both terms in {1,...,n} and w+2x divisible by 5. - Pontus von Brömssen, Jan 17 2020

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

Cf. A211422.

a:=[]; for n in [0..57] do m:=0; for i,j in [1..n] do if (i+2*j) mod 5 eq 0  then m:=m+1; end if; end for; Append(~a,m); end for; a; // Marius A. Burtea, Jan 17 2020

R:=PowerSeriesRing(Integers(), 57);[0,0] cat Coefficients(R!( x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4) ))); // Marius A. Burtea, Jan 17 2020

t[n_] := t[n] = Flatten[Table[w + 2 x - 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]   (* A211523 *)
FindLinearRecurrence[t]
LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,1,2,4,5,7},57] (* Ray Chandler, Aug 02 2015 *)

concat(vector(2), Vec(x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 02 2017

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).

G.f.: x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 02 2017

A211534 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w = 3x + 3y.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 3, 3, 6, 6, 6, 10, 10, 10, 15, 15, 15, 21, 21, 21, 28, 28, 28, 36, 36, 36, 45, 45, 45, 55, 55, 55, 66, 66, 66, 78, 78, 78, 91, 91, 91, 105, 105, 105, 120, 120, 120, 136, 136, 136, 153, 153, 153, 171, 171, 171, 190, 190, 190, 210

Offset: 0

Clark Kimberling, Apr 15 2012

This sequence consists of six 0's followed by triply repeated triangular numbers.

For a guide to related sequences, see A211422.

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

Cf. A211422, A008805 (w = 2x + 2y and doubly repeated triangular numbers).

[Floor(n/3)*(Floor(n/3)-1)/2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 05 2015

[n le 7 select Floor(n/7) else Self(n-1)+2*Self(n-3)-2*Self(n-4)-Self(n-6)+ Self(n-7): n in [1..70]]; // Vincenzo Librandi, Apr 05 2015

A211534:=n->floor(n/3)*(floor(n/3)-1)/2: seq(A211534(n), n=0..100); # Wesley Ivan Hurt, Apr 05 2015

t[n_] := t[n] = Flatten[Table[-w + 3 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211534 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {0, 0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Apr 05 2015 *)

concat([0,0,0,0,0,0], Vec(-x^6/((x-1)^3*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, Feb 17 2015

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).
a(n) = floor(n/3)*( floor(n/3) - 1 )/2. - Luce ETIENNE, Jul 08 2014
G.f.: -x^6 / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Feb 17 2015
a(n) = Sum_{i=0..n-3} i*0^(i mod 3)/3. - Wesley Ivan Hurt, Apr 05 2015

A211635 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2

Original entry on oeis.org

0, 1, 7, 22, 51, 97, 166, 263, 391, 554, 756, 1004, 1303, 1653, 2061, 2530, 3068, 3677, 4362, 5126, 5973, 6912, 7942, 9071, 10304, 11640, 13087, 14649, 16333, 18142, 20078, 22150, 24357, 26707, 29202, 31845, 34649, 37610, 40739, 44031

Offset: 0

Clark Kimberling, Apr 18 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w^2 < x^2 + y^2, s = s + 1],
        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]]  (* A211635 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211638 Number of ordered triples (w, x, y) with all terms in {1, ..., n} and w^2 + x^2 + y^2 < n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 4, 4, 4, 7, 7, 10, 11, 11, 17, 17, 17, 20, 23, 26, 26, 32, 35, 35, 38, 38, 44, 48, 48, 54, 60, 60, 60, 66, 69, 75, 78, 78, 87, 87, 87, 96, 102, 105, 108, 114, 120, 120, 121, 127, 133, 139, 139, 145, 157, 157, 163, 169, 169, 178, 178, 184

Offset: 0

Clark Kimberling, Apr 18 2012

nonn

For a guide to related sequences, see A211422.

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A211422.

t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w^2 + x^2 + y^2 < n, s = s + 1],
        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 80]]     (* A211638 *)
(* Peter J. C. Moses, Apr 13 2012 *)

first(n) = {n = max(n, 2); n-=2; my(res = vector(n), v = vector(n)); forvec(x = vector(3, i, [1,sqrtint(n)]), c = sum(i = 1, 3, x[i]^2); if(c <= n, v[c]++)); for(i = 2, #v, v[i]+=v[i-1]); concat([0,0],v)} \\ David A. Corneth, Jun 16 2023

a(n) + A063691(n) = A211639(n). - R. J. Mathar, Jun 16 2023

a(n) = A211639(n-1). - R. J. Mathar, Jun 16 2023

A211649 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2+x^2+y^2=2n.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 1, 6, 0, 3, 0, 3, 3, 6, 0, 6, 0, 3, 3, 9, 0, 6, 3, 6, 1, 6, 0, 12, 6, 0, 0, 12, 0, 12, 3, 6, 3, 12, 3, 6, 0, 3, 6, 15, 3, 12, 0, 12, 3, 12, 0, 6, 6, 6, 4, 18, 0, 12, 6, 9, 6, 12, 0, 18, 0, 0, 6, 21, 3, 12, 6, 6, 3, 21, 0, 15, 9, 12, 0, 12, 0, 12, 9, 15, 6, 12, 3, 18

Offset: 0

Clark Kimberling, Apr 18 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w^2 + x^2 + y^2 == 2 n, s = s + 1],
        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 400]]    (* A211649 *)
-1 + Flatten[Position[%, 0]]  (* A182195 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A063468 Number of Pythagorean triples in the range [1..n], i.e., the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 2, 2, 2, 4, 4, 4, 6, 6, 8, 8, 10, 10, 10, 12, 12, 12, 12, 12, 16, 18, 18, 18, 20, 22, 22, 22, 22, 24, 26, 26, 28, 28, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 36, 40, 42, 44, 46, 46, 48, 48, 48, 50, 50, 52, 54, 54, 54, 54, 62, 62, 62, 64, 64, 66, 66, 66, 68, 70

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 27 2001

nonn

			For n = 5 the Pythagorean triples are (3, 4, 5) and (4, 3, 5), so a (5) = 2.
For n = 10 the Pythagorean triples are (3, 4, 5), (4, 3, 5), (6, 8, 10) and (8, 6, 10), so a(10) = 4.
For n = 17 the Pythagorean triples are (3, 4, 5), (4, 5, 3), (5, 12, 13), (12, 5, 13), (6, 8, 10), (8, 6, 10), (8, 15, 17), (15, 8, 17), (9, 12, 15) and (12, 9, 15), so a(17) = 10.

Marius A. Burtea, Table of n, a(n) for n = 1..1000

Cf. A062775, A211422.

a(n) = 2*partial sums of A046080(n).

[#[: x in [1..n], y in [1..n]| IsSquare(x^2+y^2) and Floor(Sqrt(x^2+y^2)) le n]:n in [1..74]]; // Marius A. Burtea, Jan 22 2020

nq[n_] := SquaresR[2, n^2]/4 - 1; Accumulate@ Array[nq, 80] (* Giovanni Resta, Jan 23 2020 *)

Corrected and extended by Vladeta Jovovic, Jul 28 2001

A182074 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+x*y=n.

Original entry on oeis.org

12, 16, 24, 56, 40, 48, 56, 64, 148, 80, 72, 96, 120, 96, 104, 304, 120, 128, 168, 144, 184, 176, 120, 208, 524, 160, 184, 256, 232, 192, 264, 224, 264, 304, 184, 760, 344, 176, 280, 400, 328, 256, 344, 352, 392, 416, 200, 384, 1156, 288, 376, 496

Offset: 1

Clark Kimberling, Apr 12 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[w^2 + x*y - n, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 1, 80}]  (* A182074 *)
t/4                          (* A028387 *)

A182112 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and (w+n)^2=x+y+w.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 20, 24, 28, 32, 36, 40, 45, 50, 55, 60, 65, 70, 75, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336

Offset: 0

Clark Kimberling, Apr 12 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[(n + w)^2 - x - y - w, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A182112 *)

cp's OEIS Frontend

A211520 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 4y = 2x.

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A211521 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 2x = 4y.

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A211523 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w+2x=5y.

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A211534 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w = 3x + 3y.

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A211635 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2

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A211638 Number of ordered triples (w, x, y) with all terms in {1, ..., n} and w^2 + x^2 + y^2 < n.

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A211649 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2+x^2+y^2=2n.

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