A211422 -id:A211422 - cp's OEIS frontend

A211424 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+3xy=0.

1, 5, 9, 21, 25, 29, 49, 53, 57, 69, 73, 77, 105, 109, 113, 125, 137, 141, 161, 165, 169, 181, 185, 189, 217, 229, 233, 261, 265, 269, 297, 301, 313, 325, 329, 333, 369, 373, 377, 389, 393, 397, 417, 421, 425, 445, 449, 453, 497, 517, 529, 541, 545

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Brandon Crofts, Table of n, a(n) for n = 0..20000
Brandon Crofts, Faster Mathematica code for A211424

Cf. A211422.

t[n_] := t[n] = Flatten[Table[w^2 + 3 x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211424 *)
(t - 1)/4                    (* integers  *)

A211425 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+4xy=0.

Original entry on oeis.org

1, 5, 13, 17, 33, 37, 45, 49, 65, 77, 85, 89, 113, 117, 125, 129, 153, 157, 173, 177, 193, 197, 205, 209, 241, 261, 269, 281, 297, 301, 317, 321, 345, 349, 357, 361, 401, 405, 413, 417, 441, 445, 453, 457, 473, 485, 493, 497, 537, 565, 589, 593, 609

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[w^2 + 4 x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211425 *)
(t - 1)/4                    (* integers *)

A211426 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w^2+xy=0.

Original entry on oeis.org

1, 5, 17, 21, 33, 37, 49, 53, 73, 93, 105, 109, 121, 125, 137, 141, 161, 165, 201, 205, 217, 221, 233, 237, 257, 285, 297, 317, 329, 333, 345, 349, 393, 397, 409, 413, 449, 453, 465, 469, 489, 493, 505, 509, 521, 541, 553, 557, 577, 613, 673, 677

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[2 w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211426 *)
(t - 1)/4                    (* integers *)

A211427 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 3w^2+xy=0.

Original entry on oeis.org

1, 5, 9, 21, 33, 37, 49, 53, 65, 77, 81, 85, 113, 117, 121, 133, 153, 157, 169, 173, 185, 197, 201, 205, 233, 253, 257, 301, 313, 317, 329, 333, 353, 365, 369, 373, 401, 405, 409, 421, 433, 437, 449, 453, 465, 477, 481, 485, 537, 573, 593, 605, 617

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[3 w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 90}]  (* A211427 *)
(t - 1)/4                    (* integers *)

A211428 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w^2+3x*y=0.

Original entry on oeis.org

1, 5, 9, 21, 25, 29, 49, 53, 65, 77, 81, 85, 105, 109, 113, 125, 137, 141, 161, 165, 169, 181, 185, 189, 225, 237, 241, 269, 273, 277, 305, 309, 329, 341, 345, 349, 377, 381, 385, 397, 409, 413, 433, 437, 441, 453, 457, 461, 497, 517, 545, 557, 561

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[2 w^2 + 3 x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211428 *)
(t - 1)/4                    (* integers  *)

A211429 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^3 + x*y = 0.

Original entry on oeis.org

1, 9, 13, 17, 29, 33, 37, 41, 57, 69, 73, 77, 81, 85, 89, 93, 105, 109, 121, 125, 129, 133, 137, 141, 153, 165, 169, 193, 197, 201, 205, 209, 229, 233, 237, 241, 253, 257, 261, 265, 277, 281, 285, 289, 293, 297, 301, 305, 317, 329, 341, 345, 349, 353

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A211422.

t[n_] := t[n] = Flatten[Table[w^3 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211429 *)
(t - 1)/4                    (* integers *)

A211430 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+x+y=0.

Original entry on oeis.org

1, 7, 15, 25, 35, 49, 63, 77, 93, 111, 129, 147, 165, 187, 209, 231, 253, 275, 299, 325, 351, 377, 403, 429, 455, 485, 515, 545, 575, 605, 635, 665, 697, 731, 765, 799, 833, 867, 901, 935, 969, 1007, 1045, 1083, 1121, 1159, 1197, 1235, 1273, 1311

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[w^2 + x + y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211430 *)
(t - 1)/2                    (* integers *)

A211431 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^3+(x+y)^2=0.

Original entry on oeis.org

1, 7, 13, 19, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 131, 145, 159, 173, 187, 201, 215, 229, 243, 257, 271, 285, 299, 313, 327, 341, 355, 369, 385, 403, 421, 439, 457, 475, 493, 511, 529, 547, 565, 583, 601, 619, 637, 655, 673, 691, 709, 727, 745

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211422.

t[n_] := t[n] = Flatten[Table[w^3 + (x + y)^2, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}] (* A211431 *)
(t - 1)/2                   (* integers *)

A211433 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+4y=0.

Original entry on oeis.org

1, 1, 7, 11, 23, 27, 45, 53, 77, 85, 115, 127, 163, 175, 217, 233, 281, 297, 351, 371, 431, 451, 517, 541, 613, 637, 715, 743, 827, 855, 945, 977, 1073, 1105, 1207, 1243, 1351, 1387, 1501, 1541, 1661, 1701, 1827, 1871, 2003, 2047, 2185, 2233

Offset: 0

Clark Kimberling, Apr 10 2012

For a guide to related sequences, see A211422.

Also, a(n) is the number of ordered pairs (w,x) with both terms in {-n,...,0,...,n} and w+2x divisible by 4. If (w,x) is such a pair it is easy to see that (-w,x), (-w,-x), and (w,-x) also are such pairs. The number of pairs with both w and x positive is given by A211521(n). If w=0, x must be even, and if x=0, w must be divisible by 4. This means that a(n) = 4*A211521(n) + 2*floor(n/2) + 2*floor(n/4) + 1. Since the sequences A211521(n), floor(n/2), floor(n/4), and the constant sequence all satisfy the recurrence conjectured in the formula section, a(n) must also satisfy the recurrence, so this proves the conjecture. - Pontus von Brömssen, Jan 19 2020

Pontus von Brömssen, Table of n, a(n) for n = 0..1024
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Cf. A211422, A211521.

a:=[]; for n in [0..50] do m:=0; for i, j in [-n..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(), 50); Coefficients(R!( (1 + 5*x^2 + 4*x^3 + 5*x^4 + x^6) / ((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, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211433 *)
(t - 1)/2 (* integers *)

Conjectures from Colin Barker, May 15 2017: (Start)
G.f.: (1 + 5*x^2 + 4*x^3 + 5*x^4 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n>6.
(End)

A211434 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+5y=0.

Original entry on oeis.org

1, 1, 5, 9, 17, 25, 33, 45, 57, 73, 89, 105, 125, 145, 169, 193, 217, 245, 273, 305, 337, 369, 405, 441, 481, 521, 561, 605, 649, 697, 745, 793, 845, 897, 953, 1009, 1065, 1125, 1185, 1249, 1313, 1377, 1445, 1513, 1585, 1657, 1729, 1805, 1881

Offset: 0

Clark Kimberling, Apr 11 2012

For a guide to related sequences, see A211422.

Also, a(n) is the number of ordered pairs (w,x) with both terms in {-n,...,0,...,n} and w+2x divisible by 5. If (w,x) is such a pair it is easy to see that (-x,w), (-w,-x), and (x,-w) also are such pairs. If both w and x are nonzero these four pairs lie one in each quadrant. If one of w or x is zero, the other must be a multiple of 5. This means that a(n) equals 4*A211523(n) (the nonzero pairs) plus 4*floor(n/5) + 1 (pairs with w or x equal to zero). Since the sequences A211523(n), floor(n/5), and the constant sequence all satisfy the recurrence conjectured in the formula section, a(n) must also satisfy the recurrence, so this proves the conjecture. - Pontus von Brömssen, Jan 17 2020

Pontus von Brömssen, Table of n, a(n) for n = 0..1024
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A211422, A211523.

a:=[]; for n in [0..50] do m:=0; for i, j in [-n..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 19 2020

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1 - x + 4*x^2 + 4*x^4 - x^5 + x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)))); // Marius A. Burtea, Jan 19 2020

t[n_] := t[n] = Flatten[Table[w + 2 x + 5 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211434 *)
(t - 1)/4                    (* A011858 *)

Conjectures from Colin Barker, May 15 2017: (Start)
G.f.: (1 - x + 4*x^2 + 4*x^4 - x^5 + x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>6.
(End)
a(n) = (4*n*(n+1) + c(n))/5, where c(n) is 5 if n is 0 or 4 (mod 5), -3 if n is 1 or 3 (mod 5), and 1 if n is 2 (mod 5). - Pontus von Brömssen, Jan 17 2020

cp's OEIS Frontend

A211424 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+3*x*y=0.

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A211425 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+4*x*y=0.

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A211426 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2*w^2+x*y=0.

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A211427 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 3*w^2+x*y=0.

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A211428 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2*w^2+3*x*y=0.

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A211429 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^3 + x*y = 0.

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A211430 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+x+y=0.

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A211431 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^3+(x+y)^2=0.

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Mathematica

A211433 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+4y=0.

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Formula

A211434 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+5y=0.

A211424 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+3xy=0.

A211425 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2+4xy=0.

A211426 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w^2+xy=0.

A211427 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 3w^2+xy=0.

A211428 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w^2+3x*y=0.