A211423 -id:A211423 - cp's OEIS frontend

A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

Suppose that S={-n,...,0,...,n} and that f(w,x,y,n) is a function, where w,x,y are in S. The number of ordered triples (w,x,y) satisfying f(w,x,y,n)=0, regarded as a function of n, is a sequence t of nonnegative integers. Sequences such as t/4 may also be integer sequences for all except certain initial values of n. In the following guide, such sequences are indicated in the related sequences column and may be included in the corresponding Mathematica programs.
...
sequence... f(w,x,y,n) ..... related sequences
A211415 ... w^2+x*y-1 ...... t+2, t/4, (t/4-1)/4
A211422 ... w^2+x*y ........ (t-1)/8, A120486
A211423 ... w^2+2x*y ....... (t-1)/4
A211424 ... w^2+3x*y ....... (t-1)/4
A211425 ... w^2+4x*y ....... (t-1)/4
A211426 ... 2w^2+x*y ....... (t-1)/4
A211427 ... 3w^2+x*y ....... (t-1)/4
A211428 ... 2w^2+3x*y ...... (t-1)/4
A211429 ... w^3+x*y ........ (t-1)/4
A211430 ... w^2+x+y ........ (t-1)/2
A211431 ... w^3+(x+y)^2 .... (t-1)/2
A211432 ... w^2-x^2-y^2 .... (t-1)/8
A003215 ... w+x+y .......... (t-1)/2, A045943
A202253 ... w+2x+3y ........ (t-1)/2, A143978
A211433 ... w+2x+4y ........ (t-1)/2
A211434 ... w+2x+5y ........ (t-1)/4
A211435 ... w+4x+5y ........ (t-1)/2
A211436 ... 2w+3x+4y ....... (t-1)/2
A211435 ... 2w+3x+5y ....... (t-1)/2
A211438 ... w+2x+2y ....... (t-1)/2, A118277
A001844 ... w+x+2y ......... (t-1)/4, A000217
A211439 ... w+3x+3y ........ (t-1)/2
A211440 ... 2w+3x+3y ....... (t-1)/2
A028896 ... w+x+y-1 ........ t/6, A000217
A211441 ... w+x+y-2 ........ t/3, A028387
A182074 ... w^2+x*y-n ...... t/4, A028387
A000384 ... w+x+y-n
A000217 ... w+x+y-2n
A211437 ... w*x*y-n ........ t/4, A007425
A211480 ... w+2x+3y-1
A211481 ... w+2x+3y-n
A211482 ... w*x+w*y+x*y-w*x*y
A211483 ... (n+w)^2-x-y
A182112 ... (n+w)^2-x-y-w
...
For the following sequences, S={1,...,n}, rather than
{-n,...,0,...n}. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A132188 ... w^2-x*y
A211506 ... w^2-x*y-n
A211507 ... w^2-x*y+n
A211508 ... w^2+x*y-n
A211509 ... w^2+x*y-2n
A211510 ... w^2-x*y+2n
A211511 ... w^2-2x*y ....... t/2
A211512 ... w^2-3x*y ....... t/2
A211513 ... 2w^2-x*y ....... t/2
A211514 ... 3w^2-x*y ....... t/2
A211515 ... w^3-x*y
A211516 ... w^2-x-y
A211517 ... w^3-(x+y)^2
A063468 ... w^2-x^2-y^2 .... t/2
A000217 ... w+x-y
A001399 ... w-2x-3y
A211519 ... w-2x+3y
A008810 ... w+2x-3y
A001399 ... w-2x-3y
A008642 ... w-2x-4y
A211520 ... w-2x+4y
A211521 ... w+2x-4y
A000115 ... w-2x-5y
A211522 ... w-2x+5y
A211523 ... w+2x-5y
A211524 ... w-3x-5y
A211533 ... w-3x+5y
A211523 ... w+3x-5y
A211535 ... w-4x-5y
A211536 ... w-4x+5y
A008812 ... w+4x-5y
A055998 ... w+x+y-2n
A074148 ... 2w+x+y-2n
A211538 ... 2w+2x+y-2n
A211539 ... 2w+2x-y-2n
A211540 ... 2w-3x-4y
A211541 ... 2w-3x+4y
A211542 ... 2w+3x-4y
A211543 ... 2w-3x-5y
A211544 ... 2w-3x+5y
A008812 ... 2w+3x-5y
A008805 ... w-2x-2y (repeated triangular numbers)
A001318 ... w-2x+2y
A000982 ... w+x-2y
A211534 ... w-3x-3y
A211546 ... w-3x+3y (triply repeated triangular numbers)
A211547 ... 2w-3x-3y (triply repeated squares)
A082667 ... 2w-3x+3y
A055998 ... w-x-y+2
A001399 ... w-2x-3y+1
A108579 ... w-2x-3y+n
...
Next, S={-n,...-1,1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated inequality. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A211545 ... w+x+y>0; recurrence degree: 4
A211612 ... w+x+y>=0
A211613 ... w+x+y>1
A211614 ... w+x+y>2
A211615 ... |w+x+y|<=1
A211616 ... |w+x+y|<=2
A211617 ... 2w+x+y>0; recurrence degree: 5
A211618 ... 2w+x+y>1
A211619 ... 2w+x+y>2
A211620 ... |2w+x+y|<=1
A211621 ... w+2x+3y>0
A211622 ... w+2x+3y>1
A211623 ... |w+2x+3y|<=1
A211624 ... w+2x+2y>0; recurrence degree: 6
A211625 ... w+3x+3y>0; recurrence degree: 8
A211626 ... w+4x+4y>0; recurrence degree: 10
A211627 ... w+5x+5y>0; recurrence degree: 12
A211628 ... 3w+x+y>0; recurrence degree: 6
A211629 ... 4w+x+y>0; recurrence degree: 7
A211630 ... 5w+x+y>0; recurrence degree: 8
A211631 ... w^2>x^2+y^2; all terms divisible by 8
A211632 ... 2w^2>x^2+y^2; all terms divisible by 8
A211633 ... w^2>2x^2+2y^2; all terms divisible by 8
...
Next, S={1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated relation.
A211634 ... w^2<=x^2+y^2
A211635 ... w^2A211790
A211636 ... w^2>=x^2+y^2
A211637 ... w^2>x^2+y^2
A211638 ... w^2+x^2+y^2A211639 ... w^2+x^2+y^2<=n
A211640 ... w^2+x^2+y^2>n
A211641 ... w^2+x^2+y^2>=n
A211642 ... w^2+x^2+y^2<2n
A211643 ... w^2+x^2+y^2<=2n
A211644 ... w^2+x^2+y^2>2n
A211645 ... w^2+x^2+y^2>=2n
A211646 ... w^2+x^2+y^2<3n
A211647 ... w^2+x^2+y^2<=3n
A063691 ... w^2+x^2+y^2=n
A211649 ... w^2+x^2+y^2=2n
A211648 ... w^2+x^2+y^2=3n
A211650 ... w^3A211790
A211651 ... w^3>x^3+y^3; see Comments at A211790
A211652 ... w^4A211790
A211653 ... w^4>x^4+y^4; see Comments at A211790

			a(1) counts these 9 triples: (-1,-1,1), (-1, 1,-1), (0, -1, 0), (0, 0, -1), (0,0,0), (0,0,1), (0,1,0), (1,-1,1), (1,1,-1).

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

Cf. A120486.

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}] (* A211422 *)
(t - 1)/8                   (* A120486 *)

A334524 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + 5xy = 0.

Original entry on oeis.org

1, 5, 9, 13, 17, 29, 33, 37, 41, 45, 65, 69, 73, 77, 81, 101, 105, 109, 113, 117, 153, 157, 161, 165, 169, 181, 185, 189, 193, 197, 233, 237, 241, 245, 249, 261, 273, 277, 281, 285, 321, 325, 329, 333, 337, 373, 377, 381, 385, 397, 417, 421, 425, 429, 433, 445

Offset: 0

Brandon Crofts, Jun 15 2020

nonn

If n is squarefree and not divisible by 5, a(n) = a(n-1)+4. - Robert Israel, Jun 29 2020

Brandon Crofts, Table of n, a(n) for n = 0..20000 (first 10001 terms from Robert Israel)
Brandon Crofts, Mathematica code for 334524

Cf. A211423.

df:= proc(n) local t,s,m0,m;
   if n mod 5 = 0 then
     m:= n/5;
     t:= 4*nops(select(s -> s < n and s > m, numtheory:-divisors(5*m^2)))
   else t:= 0
   fi;
   m0:= mul(`if`(s[1]=5, s[1]^ceil((s[2]-1)/2),
       s[1]^ceil(s[2]/2)),s=ifactors(n)[2]);
   t + 4 + 8*floor(n/m0/5);
end proc:
df(0):= 1:
ListTools:-PartialSums(map(df,[$0..100])); # Robert Israel, Jun 29 2020

A334525 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + 7xy = 0.

Original entry on oeis.org

1, 5, 9, 13, 17, 21, 25, 37, 41, 45, 49, 53, 57, 61, 81, 85, 89, 93, 97, 101, 105, 125, 129, 133, 137, 141, 145, 149, 185, 189, 193, 197, 201, 205, 209, 229, 233, 237, 241, 245, 249, 253, 297, 301, 305, 309, 313, 317, 321, 333, 337, 341, 345, 349, 353, 357, 393, 397

Offset: 0

Brandon Crofts, Jun 15 2020

nonn

If n is squarefree and not divisible by 7, a(n) = a(n-1)+4. - Robert Israel, Jul 01 2020

Brandon Crofts, Table of n, a(n) for n = 0..20000
Brandon Crofts, Mathematica code for A334525

Cf. A211423.

df:= proc(n) local t, s, m0, m;
   if n mod 7 = 0 then
     m:= n/7;
     t:= 4*nops(select(s -> s < n and s > m, numtheory:-divisors(7*m^2)))
   else t:= 0
   fi;
   m0:= mul(`if`(s[1]=7, s[1]^ceil((s[2]-1)/2),
       s[1]^ceil(s[2]/2)), s=ifactors(n)[2]);
   t + 4 + 8*floor(n/m0/7);
end proc:
df(0):= 1:
ListTools:-PartialSums(map(df, [$0..100])); # Robert Israel, Jul 01 2020

A334526 Number of triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + 11xy = 0.

Original entry on oeis.org

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 249, 253, 257, 261, 265, 269, 273, 277, 281, 285, 289, 309, 313, 317

Offset: 0

Brandon Crofts, Jul 10 2020

Brandon Crofts, Mathematica code for A334526

Cf. A211423.

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cp's OEIS Frontend

A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

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A334524 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + 5xy = 0.

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A334525 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + 7xy = 0.

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Maple

A334526 Number of triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + 11xy = 0.

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