A211435 -id:A211435 - 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 *)

A211547 The squares n^2, n >= 0, each one written three times.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16, 25, 25, 25, 36, 36, 36, 49, 49, 49, 64, 64, 64, 81, 81, 81, 100, 100, 100, 121, 121, 121, 144, 144, 144, 169, 169, 169, 196, 196, 196, 225, 225, 225, 256, 256, 256, 289, 289, 289, 324, 324, 324, 361, 361, 361

Offset: 0

Clark Kimberling, Apr 15 2012

Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=3x+3y.

For a guide to related sequences, see A211422.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A075561, A211422, A211435 (triply repeated triangular numbers).

t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 60}](*A211547, squares thrice*)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,2,-2,0,-1,1},{0,0,0,1,1,1,4},60] (* Ray Chandler, Aug 02 2015 *)

concat(vector(3), Vec(x^3*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^40))) \\\ Colin Barker, Dec 02 2017

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).
G.f.: x^3*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x + x^2)^2). - Colin Barker, Dec 02 2017
a(n) = A075561(n-2) for n > 2. - Georg Fischer, Oct 23 2018
E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 3*x*(x - 1)) - 5*cos(sqrt(3)*x/2) - sqrt(3)*(3 + 4*x)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Oct 17 2022

Definition simplified by N. J. A. Sloane, Nov 17 2020. Also the old version said "squares repeated three times", which was at best ambiguous, and strictly speaking was incorrect, since "squares repeated" is 0, 0, 1, 1, 4, 4, 9, 9, ... .

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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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