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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 (list; graph; refs; listen; history; text; internal format)
 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 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 A211437 ... 2w+3x+5y ....... (t-1)/2 A211438 ... 2w+2x+2y ....... (t-1)/2, A118277 A001844 ... w+x+2y ......... (t-1)/4, A000217 A211439 ... 2+3x+3y ........ (t-1)/2 A211440 ... 2x+3x+3y ....... (t-1)/2 A028896 ... w+x+y-1 ........ t/6, A000217 A211411 ... 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 ... 2+2x+3y-n A028896 ... (w+x+y)^3-1..... t/6, A000217 A211482 ... w*x+w*y+x*y-w*x*z 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+4y-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 A074148 ... 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+w+y|<=1 A211616 ... |w+w+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+w+y|<=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 A211637 ... w^2>x^2+y^2 A211638 ... w^2+x^2+y^2n 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 A063649 ... w^2+x^2+y^2=2n A211648 ... w^2+x^2+y^2=3n A211650 ... w^3x^3+y^3; see Comments at A211790 A211652 ... w^4x^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 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 Adjacent sequences:  A211419 A211420 A211421 * A211423 A211424 A211425 KEYWORD nonn AUTHOR Clark Kimberling, Apr 10 2012 STATUS approved

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Last modified November 17 22:58 EST 2018. Contains 317279 sequences. (Running on oeis4.)