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
Keywords
A042965 Nonnegative integers not congruent to 2 mod 4.
0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92
Offset: 1
Comments
Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence (starting at 3) gives values of AUB, sorted and duplicates removed. Values of AUBUC give same sequence. - David W. Wilson
These are the nonnegative integers that can be written as a difference of two squares, i.e., n = x^2 - y^2 for integers x,y. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 25 2002. Equivalently, nonnegative numbers represented by the quadratic form x^2-y^2 of discriminant 4. The primes in this sequence are all the odd primes. - N. J. A. Sloane, May 30 2014
Numbers n such that Kronecker(4,n) == mu(gcd(4,n)). - Jon Perry, Sep 17 2002
Count, sieving out numbers of the form 2*(2*n+1) (A016825, "nombres pair-impairs"). A generalized Chebyshev transform of the Jacobsthal numbers: apply the transform g(x) -> (1/(1+x^2)) g(x/(1+x^2)) to the g.f. of A001045(n+2). Partial sums of 1,2,1,1,2,1,.... - Paul Barry, Apr 26 2005
The sequence 1,1,3,4,5,... is the image of A001045(n+1) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005
With offset 0 starting (1, 3, 4,...) = INVERT transform of A009531 starting (1, 2, -1, -4, 1, 6,...) with offset 0.
Apparently these are the regular numbers modulo 4 [Haukkanan & Toth]. - R. J. Mathar, Oct 07 2011
Numbers of the form x*y in nonnegative integers x,y with x+y even. - Michael Somos, May 18 2013
Numbers that are the sum of zero or more consecutive odd positive numbers. - Gionata Neri, Sep 01 2015
Numbers that are congruent to {0, 1, 3} mod 4. - Wesley Ivan Hurt, Jun 10 2016
Nonnegative integers of the form (2+(3*m-2)/4^j)/3, j,m >= 0. - L. Edson Jeffery, Jan 02 2017
This is { x^2 - y^2; x >= y >= 0 }; with the restriction x > y one gets the same set without zero; with the restriction x > 0 (i.e., differences of two nonzero squares) one gets the set without 1. An odd number 2n-1 = n^2 - (n-1)^2, a number 4n = (n+1)^2 - (n-1)^2. - M. F. Hasler, May 08 2018
Examples
G.f. = x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 11*x^9 + 12*x^10 + ...
References
- R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D9.
- J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 83.
Links
- Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
- Pentti Haukkanen and László Tóth, An analogue of Ramanujan's sum with respect to regular integers (mod r), Ramanujan J., Vol. 27, No. 1 (2012), pp. 71-88.
- Ron Knott, Pythagorean Triples and Online Calculators.
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Crossrefs
Programs
-
Haskell
a042965 = (`div` 3) . (subtract 3) . (* 4) a042965_list = 0 : 1 : 3 : map (+ 4) a042965_list -- Reinhard Zumkeller, Nov 09 2012
-
Magma
[n: n in [0..100] | not n mod 4 in [2]]; // Vincenzo Librandi, Sep 03 2015
-
Maple
a_list := proc(len) local rec; rec := proc(n) option remember; ifelse(n <= 4, [0, 1, 3, 4][n], rec(n-1) + rec(n-3) - rec(n-4)) end: seq(rec(n), n=1..len) end: a_list(76); # Peter Luschny, Aug 06 2022
-
Mathematica
nn=100; Complement[Range[0,nn], Range[2,nn,4]] (* Harvey P. Dale, May 21 2011 *) f[n_]:=Floor[(4*n-3)/3]; Array[f,70] (* Robert G. Wilson v, Jun 26 2012 *) LinearRecurrence[{1, 0, 1, -1}, {0, 1, 3, 4}, 70] (* L. Edson Jeffery, Jan 21 2015 *) Select[Range[0, 100], ! MemberQ[{2}, Mod[#, 4]] &] (* Vincenzo Librandi, Sep 03 2015 *)
-
PARI
a(n)=(4*n-3)\3 \\ Charles R Greathouse IV, Jul 25 2011
-
Python
def A042965(n): return (n<<2)//3-1 # Chai Wah Wu, Feb 10 2025
Formula
Recurrence: a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = n - 1 + (3n-3-sqrt(3)*(1-2*cos(2*Pi*(n-1)/3))*sin(2*Pi*(n-1)/3))/9. Partial sums of the period-3 sequence 0, 1, 1, 2, 1, 1, 2, 1, 1, 2, ... (A101825). - Ralf Stephan, May 19 2013
G.f.: A(x) = x^2*(1+x)^2/((1-x)^2*(1+x+x^2)); a(n)=Sum{k=0..floor(n/2)}, binomial(n-k-1, k)*A001045(n-2*k), n>0. - Paul Barry, Jan 16 2005, R. J. Mathar, Dec 09 2009
a(n) = floor((4*n-3)/3). - Gary Detlefs, May 14 2011
A214546(a(n)) != 0. - Reinhard Zumkeller, Jul 20 2012
From Michael Somos, May 18 2013: (Start)
Euler transform of length 3 sequence [3, -2, 1].
a(2-n) = -a(n). (End)
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = (12*n-12+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 4k-1, a(3k-1) = 4k-3, a(3k-2) = 4k-4. (End)
a(n) = round((4*n-4)/3). - Mats Granvik, Sep 24 2016
The g.f. A(x) satisfies (A(x)/x)^2 + A(x)/x = x*B(x)^2, where B(x) is the o.g.f. of A042968. - Peter Bala, Apr 12 2017
Sum_{n>=2} (-1)^n/a(n) = log(sqrt(2)+2)/sqrt(2) - (sqrt(2)-1)*log(2)/4. - Amiram Eldar, Dec 05 2021
From Peter Bala, Aug 03 2022: (Start)
a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 2, with a(2) = 1 and a(3) = 3.
E.g.f.: (9 + 12*exp(x)*(x - 1) + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 05 2023
Extensions
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Peter Pein and Ralf Stephan, Jun 17 2007
Typos fixed in Gary Detlefs's formula and in PARI program by Reinhard Zumkeller, Nov 09 2012
A000969 Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^2*(1 - x^3)).
1, 3, 7, 12, 18, 26, 35, 45, 57, 70, 84, 100, 117, 135, 155, 176, 198, 222, 247, 273, 301, 330, 360, 392, 425, 459, 495, 532, 570, 610, 651, 693, 737, 782, 828, 876, 925, 975, 1027, 1080, 1134, 1190, 1247, 1305, 1365, 1426, 1488, 1552, 1617, 1683, 1751, 1820, 1890
Offset: 0
Comments
From Paul Curtz, Oct 07 2018: (Start)
Terms that are on the x-axis of the following spiral (without 0):
28--29--29--30--31--31--32
|
27 13--14--15--15--16--17
| | |
27 13 4---5---5---6 17
| | | | |
26 12 3 0---1 7 18
| | | | | |
25 11 3---2---1 7 19
| | | |
25 11--10---9---9---8 19
| |
24--23--23--22--21--21--20 (End)
Diagonal 1, 4, 8, 13, 20, 28, ... (without 0) is A143978. - Bruno Berselli, Oct 08 2018
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..10000
- David Applegate, Omar E. Pol, and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
- R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
- P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
- Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Crossrefs
Programs
-
Haskell
a000969 = flip div 3 . a014105 . (+ 1) -- Reinhard Zumkeller, Jun 23 2015
-
Magma
[Floor(Binomial(2*n+3,2)/3): n in [0..60]]; // G. C. Greubel, Apr 18 2023
-
Maple
A000969:=-(1+z+2*z**2)/(z**2+z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation
-
Mathematica
f[x_, y_]:= Floor[Abs[y/x -x/y]]; Table[f[3, 2n^2+n+2], {n,53}] (* Robert G. Wilson v, Aug 11 2010 *) CoefficientList[Series[(1+x+2*x^2)/((1-x)^2*(1-x^3)), {x, 0, 50}], x] (* Stefano Spezia, Oct 08 2018 *) -
PARI
a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,-2,1,-1,2]^n*[1;3;7;12;18])[1,1] \\ Charles R Greathouse IV, May 10 2016
-
SageMath
[(binomial(2*n+3,2)//3) for n in range(61)] # G. C. Greubel, Apr 18 2023
Formula
a(n) = floor( (2*n+3)*(n+1)/3 ). Or, a(n) = (2*n+3)*(n+1)/3 but subtract 1/3 if n == 1 mod 3. - N. J. A. Sloane, May 05 2010
a(2^k-2) = A139250(2^k-1), k >= 1. - Omar E. Pol, Feb 13 2010
a(n) = Sum_{i=0..n} floor(4*i/3). - Enrique Pérez Herrero, Apr 21 2012
a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-3) -2*a(n-4) +1*a(n-5). - Joerg Arndt, Apr 22 2012
Sum_{n>=0} 1/a(n) = 6 - Pi/sqrt(3) - 10*log(2)/3. - Amiram Eldar, Oct 01 2022
E.g.f.: (exp(x)*(8 + 21*x + 6*x^2) + exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 05 2023
A229154 The clubs patterns appearing in n X n coins, with rotation allowed.
1, 2, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936, 972, 1008, 1045
Offset: 2
Comments
On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 2..1000
- Kival Ngaokrajang, Illustration of initial terms
- Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Crossrefs
Programs
-
Mathematica
CoefficientList[Series[-(x^6 - 2 x^5 + x^4 - x^3 + 2 x^2 + 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 08 2013 *) -
PARI
Vec(-x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013
Formula
a(n) = floor(n^2/3), a(3) = 2.
From Colin Barker, Oct 08 2013: (Start)
a(n) = n^2/3 + (2/9)*cos((2*Pi*n)/3) - 2/9.
G.f.: -x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1) / ((x-1)^3*(x^2+x+1)). (End)
Extensions
More terms from Colin Barker, Oct 08 2013
A281026 a(n) = floor(3*n*(n+1)/4).
0, 1, 4, 9, 15, 22, 31, 42, 54, 67, 82, 99, 117, 136, 157, 180, 204, 229, 256, 285, 315, 346, 379, 414, 450, 487, 526, 567, 609, 652, 697, 744, 792, 841, 892, 945, 999, 1054, 1111, 1170, 1230, 1291, 1354, 1419, 1485, 1552, 1621, 1692, 1764, 1837, 1912, 1989, 2067, 2146
Offset: 0
Links
- Bruno Berselli, Table of n, a(n) for n = 0..1000
- Bruno Berselli, Illustration of the initial terms.
- Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).
Crossrefs
Programs
-
Magma
[3*n*(n+1) div 4: n in [0..60]];
-
Maple
A281026:=n->floor(3*n*(n+1)/4): seq(A281026(n), n=0..100); # Wesley Ivan Hurt, Jan 13 2017
-
Mathematica
Table[Floor[3 n (n + 1)/4], {n, 0, 60}] LinearRecurrence[{3,-4,4,-3,1},{0,1,4,9,15},60] (* Harvey P. Dale, Jun 04 2023 *) -
Maxima
makelist(floor(3*n*(n+1)/4), n, 0, 60);
-
PARI
vector(60, n, n--; floor(3*n*(n+1)/4))
-
Python
[int(3*n*(n+1)/4) for n in range(60)]
-
Sage
[floor(3*n*(n+1)/4) for n in range(60)]
Formula
O.g.f.: x*(1 + x + x^2)/((1 + x^2)*(1 - x)^3).
E.g.f.: -(1 - 6*x - 3*x^2)*exp(x)/4 - (1 + i)*(i - exp(2*i*x))*exp(-i*x)/8, where i=sqrt(-1).
a(n) = a(-n-1) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) = a(n-4) + 6*n - 9.
a(n) = 3*n*(n+1)/4 + (i^(n*(n+1)) - 1)/4. Therefore:
a(4*k+r) = 12*k^2 + 3*(2*r+1)*k + r^2, where 0 <= r <= 3.
a(n) = n^2 - floor((n-1)*(n-2)/4).
a(n) = A011865(3*n+2).
A202253 Number of zero-sum -n..n arrays of 3 elements with adjacent element differences also in -n..n.
3, 9, 17, 27, 41, 57, 75, 97, 121, 147, 177, 209, 243, 281, 321, 363, 409, 457, 507, 561, 617, 675, 737, 801, 867, 937, 1009, 1083, 1161, 1241, 1323, 1409, 1497, 1587, 1681, 1777, 1875, 1977, 2081, 2187, 2297, 2409, 2523, 2641, 2761, 2883, 3009, 3137, 3267
Offset: 1
Keywords
Comments
Row 3 of A202252.
It appears that A202253 is also the number of ordered triples (w,x,y) with all terms in {-n,...,n} such that w+2x+3y=0; see the Mathematica and Example sections. - Clark Kimberling, Apr 10 2012
Examples
Some solutions for n=10: 7 9 6 4 -2 3 -3 -8 3 8 0 -6 1 -6 -3 -5 0 0 -3 0 6 2 2 0 0 -2 -3 1 1 2 4 5 -7 -9 -3 -4 -4 -5 1 8 -3 -6 3 5 -2 4 -1 0 The a(2)=9 solutions (w,x,y) of w+2x+3y=0, as described in the Comments section, are as follows: (-2,-2,2), (-2,1,0), (-1,-1,1), (-1,2,-1), (0,0,0), (1,-2,1), (1,1,-1), (2,-1,0), (2,2,-2). - _Clark Kimberling_, Apr 10 2012
Links
- R. H. Hardin, Table of n, a(n) for n = 1..210
Crossrefs
Cf. A202252.
Programs
-
Mathematica
t[n_]:=t[n]=Flatten[Table[w+2x+3y,{w,-n,n}, {x,-n,n},{y,-n,n}]] c[n_]:=Count[t[n],0] t=Table[c[n],{n,1,50}] (* A143978 ? *) (t-1)/2 (* A143978 *) (* Clark Kimberling, Apr 10 2012 *)
Formula
Empirical: a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
Conjecture: a(n) = 1+2*floor((2*n^2+2*n)/3). - Clark Kimberling, Apr 12 2012
Empirical g.f.: x*(3 + 3*x + 2*x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Mar 03 2018
A281151 a(n) = floor(4*n*(n+1)/5).
0, 1, 4, 9, 16, 24, 33, 44, 57, 72, 88, 105, 124, 145, 168, 192, 217, 244, 273, 304, 336, 369, 404, 441, 480, 520, 561, 604, 649, 696, 744, 793, 844, 897, 952, 1008, 1065, 1124, 1185, 1248, 1312, 1377, 1444, 1513, 1584, 1656, 1729, 1804, 1881, 1960, 2040, 2121, 2204, 2289
Offset: 0
Links
- Bruno Berselli, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).
Crossrefs
Programs
-
Magma
[4*n*(n+1) div 5: n in [0..60]];
-
Mathematica
Table[Floor[4 n (n + 1)/5], {n, 0, 60}] -
Maxima
makelist(floor(4*n*(n+1)/5), n, 0, 60);
-
PARI
vector(60, n, n--; floor(4*n*(n+1)/5))
-
Python
[int(4*n*(n+1)/5) for n in range(60)]
-
Sage
[floor(4*n*(n+1)/5) for n in range(60)]
Formula
O.g.f.: x*(1 + x^2)*(1 + x)^2/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(-n-1) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) = a(n-5) + 8*(n-2).
a(5*k+r) = 20*k^2 + 4*(2*r+1)*k + r^2, where 0 <= r <= 4. Example: for r=3, a(5*k+3) = (2*k+1)*(10*k+9), which gives: 9, 57, 145, 273, 441, 649 etc. Also, a(n) belongs to A047462, in fact: for r = 0 or 4, a(n) == 0 (mod 8); for r = 1 or 3, a(n) == 1 (mod 8); for r = 2, a(n) == 4 (mod 8).
a(n) = a(-n) + A047462(n).
a(n) = n^2 - floor((n-2)^2/5).
A320259 Terms that are on the y-axis of the square spiral built with 2*k, 2*k+1, 2*k+1 for k >= 0.
0, 2, 5, 9, 15, 22, 30, 40, 51, 63, 77, 92, 108, 126, 145, 165, 187, 210, 234, 260, 287, 315, 345, 376, 408, 442, 477, 513, 551, 590, 630, 672, 715, 759, 805, 852, 900, 950, 1001, 1053, 1107, 1162, 1218, 1276, 1335
Offset: 0
Keywords
Comments
a(n) mod 9 is of period 27.
The spiral:
28--29--29--30--31--31--32
|
27 13--14--15--15--16--17
| | |
27 13 4---5---5---6 17
| | | | |
26 12 3 0---1 7 18
| | | | | |
25 11 3---2---1 7 19
| | | |
25 11--10---9---9---8 19
| |
24--23--23--22--21--21--20
Examples
G.f. = 2*x + 5*x^2 + 9*x^3 + 15*x^4 + 22*x^5 + 30*x^6 + ... - _Michael Somos_, Nov 13 2018
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Crossrefs
Programs
-
GAP
a:=[0,2,5,9,15];; for n in [6..50] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-2*a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Oct 08 2018
-
Maple
seq(coeff(series(x^2*(2+x+x^2)/((1-x)^3*(1+x+x^2)),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 08 2018
-
Mathematica
LinearRecurrence[{2,-1,1,-2,1}, {0, 2, 5, 9, 15}, 50] (* or *) CoefficientList[Series[x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 09 2018 *) a[ n_] := Quotient[(n + 1) (2 n + 1), 3]; (* Michael Somos, Nov 13 2018 *) -
PARI
concat(0, Vec(x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Oct 08 2018
-
PARI
{a(n) = (n + 1) * (2*n + 1) \ 3}; /* Michael Somos, Nov 13 2018 */
Formula
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), a(0)=0, a(1)=2, a(2)=5, a(3)=9, a(4)=15.
a(n) = a(n-1) + A004772(n+1), a(0)=0, n>0.
a(n+15) = a(n-15) + 10*A004767(n).
a(-n-1) = ({0} U A000969(n)) = 0, 1, 3, 7, ... = b(n), the full x-axis terms.
a(-n-1) + a(n) = 0, 3, 8, 16, ... = A211480(n+1).
a(n) = b(n) + A004523(n+1).
G.f.: x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Oct 08 2018
a(n) = A000969(-2-n) for all n in Z. - Michael Somos, Nov 13 2018
Comments
Examples
Links
Crossrefs
Programs
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 *)