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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.
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
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
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.
Sequence in context: A004768 A226323 A211432 * A035198 A271186 A253705
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 10 2012
STATUS
approved