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.

%I A211422 #52 Jan 27 2020 03:11:17
%S A211422 1,9,17,25,41,49,57,65,81,105,113,121,137,145,153,161,193,201,225,233,
%T A211422 249,257,265,273,289,329,337,361,377,385,393,401,433,441,449,457,505,
%U A211422 513,521,529,545,553,561,569,585,609,617,625,657,713,753,761
%N A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.
%C A211422 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.
%C A211422 ...
%C A211422 sequence... f(w,x,y,n) ..... related sequences
%C A211422 A211415 ... w^2+x*y-1 ...... t+2, t/4, (t/4-1)/4
%C A211422 A211422 ... w^2+x*y ........ (t-1)/8, A120486
%C A211422 A211423 ... w^2+2x*y ....... (t-1)/4
%C A211422 A211424 ... w^2+3x*y ....... (t-1)/4
%C A211422 A211425 ... w^2+4x*y ....... (t-1)/4
%C A211422 A211426 ... 2w^2+x*y ....... (t-1)/4
%C A211422 A211427 ... 3w^2+x*y ....... (t-1)/4
%C A211422 A211428 ... 2w^2+3x*y ...... (t-1)/4
%C A211422 A211429 ... w^3+x*y ........ (t-1)/4
%C A211422 A211430 ... w^2+x+y ........ (t-1)/2
%C A211422 A211431 ... w^3+(x+y)^2 .... (t-1)/2
%C A211422 A211432 ... w^2-x^2-y^2 .... (t-1)/8
%C A211422 A003215 ... w+x+y .......... (t-1)/2, A045943
%C A211422 A202253 ... w+2x+3y ........ (t-1)/2, A143978
%C A211422 A211433 ... w+2x+4y ........ (t-1)/2
%C A211422 A211434 ... w+2x+5y ........ (t-1)/4
%C A211422 A211435 ... w+4x+5y ........ (t-1)/2
%C A211422 A211436 ... 2w+3x+4y ....... (t-1)/2
%C A211422 A211435 ... 2w+3x+5y ....... (t-1)/2
%C A211422 A211438 ... w+2x+2y ....... (t-1)/2, A118277
%C A211422 A001844 ... w+x+2y ......... (t-1)/4, A000217
%C A211422 A211439 ... w+3x+3y ........ (t-1)/2
%C A211422 A211440 ... 2w+3x+3y ....... (t-1)/2
%C A211422 A028896 ... w+x+y-1 ........ t/6, A000217
%C A211422 A211441 ... w+x+y-2 ........ t/3, A028387
%C A211422 A182074 ... w^2+x*y-n ...... t/4, A028387
%C A211422 A000384 ... w+x+y-n
%C A211422 A000217 ... w+x+y-2n
%C A211422 A211437 ... w*x*y-n ........ t/4, A007425
%C A211422 A211480 ... w+2x+3y-1
%C A211422 A211481 ... w+2x+3y-n
%C A211422 A211482 ... w*x+w*y+x*y-w*x*y
%C A211422 A211483 ... (n+w)^2-x-y
%C A211422 A182112 ... (n+w)^2-x-y-w
%C A211422 ...
%C A211422 For the following sequences, S={1,...,n}, rather than
%C A211422 {-n,...,0,...n}. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
%C A211422 A132188 ... w^2-x*y
%C A211422 A211506 ... w^2-x*y-n
%C A211422 A211507 ... w^2-x*y+n
%C A211422 A211508 ... w^2+x*y-n
%C A211422 A211509 ... w^2+x*y-2n
%C A211422 A211510 ... w^2-x*y+2n
%C A211422 A211511 ... w^2-2x*y ....... t/2
%C A211422 A211512 ... w^2-3x*y ....... t/2
%C A211422 A211513 ... 2w^2-x*y ....... t/2
%C A211422 A211514 ... 3w^2-x*y ....... t/2
%C A211422 A211515 ... w^3-x*y
%C A211422 A211516 ... w^2-x-y
%C A211422 A211517 ... w^3-(x+y)^2
%C A211422 A063468 ... w^2-x^2-y^2 .... t/2
%C A211422 A000217 ... w+x-y
%C A211422 A001399 ... w-2x-3y
%C A211422 A211519 ... w-2x+3y
%C A211422 A008810 ... w+2x-3y
%C A211422 A001399 ... w-2x-3y
%C A211422 A008642 ... w-2x-4y
%C A211422 A211520 ... w-2x+4y
%C A211422 A211521 ... w+2x-4y
%C A211422 A000115 ... w-2x-5y
%C A211422 A211522 ... w-2x+5y
%C A211422 A211523 ... w+2x-5y
%C A211422 A211524 ... w-3x-5y
%C A211422 A211533 ... w-3x+5y
%C A211422 A211523 ... w+3x-5y
%C A211422 A211535 ... w-4x-5y
%C A211422 A211536 ... w-4x+5y
%C A211422 A008812 ... w+4x-5y
%C A211422 A055998 ... w+x+y-2n
%C A211422 A074148 ... 2w+x+y-2n
%C A211422 A211538 ... 2w+2x+y-2n
%C A211422 A211539 ... 2w+2x-y-2n
%C A211422 A211540 ... 2w-3x-4y
%C A211422 A211541 ... 2w-3x+4y
%C A211422 A211542 ... 2w+3x-4y
%C A211422 A211543 ... 2w-3x-5y
%C A211422 A211544 ... 2w-3x+5y
%C A211422 A008812 ... 2w+3x-5y
%C A211422 A008805 ... w-2x-2y (repeated triangular numbers)
%C A211422 A001318 ... w-2x+2y
%C A211422 A000982 ... w+x-2y
%C A211422 A211534 ... w-3x-3y
%C A211422 A211546 ... w-3x+3y (triply repeated triangular numbers)
%C A211422 A211547 ... 2w-3x-3y (triply repeated squares)
%C A211422 A082667 ... 2w-3x+3y
%C A211422 A055998 ... w-x-y+2
%C A211422 A001399 ... w-2x-3y+1
%C A211422 A108579 ... w-2x-3y+n
%C A211422 ...
%C A211422 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.
%C A211422 A211545 ... w+x+y>0; recurrence degree: 4
%C A211422 A211612 ... w+x+y>=0
%C A211422 A211613 ... w+x+y>1
%C A211422 A211614 ... w+x+y>2
%C A211422 A211615 ... |w+x+y|<=1
%C A211422 A211616 ... |w+x+y|<=2
%C A211422 A211617 ... 2w+x+y>0; recurrence degree: 5
%C A211422 A211618 ... 2w+x+y>1
%C A211422 A211619 ... 2w+x+y>2
%C A211422 A211620 ... |2w+x+y|<=1
%C A211422 A211621 ... w+2x+3y>0
%C A211422 A211622 ... w+2x+3y>1
%C A211422 A211623 ... |w+2x+3y|<=1
%C A211422 A211624 ... w+2x+2y>0; recurrence degree: 6
%C A211422 A211625 ... w+3x+3y>0; recurrence degree: 8
%C A211422 A211626 ... w+4x+4y>0; recurrence degree: 10
%C A211422 A211627 ... w+5x+5y>0; recurrence degree: 12
%C A211422 A211628 ... 3w+x+y>0; recurrence degree: 6
%C A211422 A211629 ... 4w+x+y>0; recurrence degree: 7
%C A211422 A211630 ... 5w+x+y>0; recurrence degree: 8
%C A211422 A211631 ... w^2>x^2+y^2; all terms divisible by 8
%C A211422 A211632 ... 2w^2>x^2+y^2; all terms divisible by 8
%C A211422 A211633 ... w^2>2x^2+2y^2; all terms divisible by 8
%C A211422 ...
%C A211422 Next, S={1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated relation.
%C A211422 A211634 ... w^2<=x^2+y^2
%C A211422 A211635 ... w^2<x^2+y^2; see Comments at A211790
%C A211422 A211636 ... w^2>=x^2+y^2
%C A211422 A211637 ... w^2>x^2+y^2
%C A211422 A211638 ... w^2+x^2+y^2<n
%C A211422 A211639 ... w^2+x^2+y^2<=n
%C A211422 A211640 ... w^2+x^2+y^2>n
%C A211422 A211641 ... w^2+x^2+y^2>=n
%C A211422 A211642 ... w^2+x^2+y^2<2n
%C A211422 A211643 ... w^2+x^2+y^2<=2n
%C A211422 A211644 ... w^2+x^2+y^2>2n
%C A211422 A211645 ... w^2+x^2+y^2>=2n
%C A211422 A211646 ... w^2+x^2+y^2<3n
%C A211422 A211647 ... w^2+x^2+y^2<=3n
%C A211422 A063691 ... w^2+x^2+y^2=n
%C A211422 A211649 ... w^2+x^2+y^2=2n
%C A211422 A211648 ... w^2+x^2+y^2=3n
%C A211422 A211650 ... w^3<x^3+y^3; see Comments at A211790
%C A211422 A211651 ... w^3>x^3+y^3; see Comments at A211790
%C A211422 A211652 ... w^4<x^4+y^4; see Comments at A211790
%C A211422 A211653 ... w^4>x^4+y^4; see Comments at A211790
%H A211422 Chai Wah Wu, <a href="/A211422/b211422.txt">Table of n, a(n) for n = 0..10000</a>
%e A211422 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).
%t A211422 t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
%t A211422 c[n_] := Count[t[n], 0]
%t A211422 t = Table[c[n], {n, 0, 70}] (* A211422 *)
%t A211422 (t - 1)/8                   (* A120486 *)
%Y A211422 Cf. A120486.
%K A211422 nonn
%O A211422 0,2
%A A211422 _Clark Kimberling_, Apr 10 2012