A082667 -id:A082667

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A211422

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

+10
106

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

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OFFSET

0,2

COMMENTS

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^2<x^2+y^2; see Comments at A211790

A211636 ... w^2>=x^2+y^2

A211637 ... w^2>x^2+y^2

A211638 ... w^2+x^2+y^2<n

A211639 ... 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^3<x^3+y^3; see Comments at A211790

A211651 ... w^3>x^3+y^3; see Comments at A211790

A211652 ... w^4<x^4+y^4; see Comments at A211790

A211653 ... w^4>x^4+y^4; see Comments at A211790

LINKS

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

EXAMPLE

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).

MATHEMATICA

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 *)

CROSSREFS

Cf. A120486.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Apr 10 2012

STATUS

approved

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