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%I #53 Sep 23 2022 08:46:22
%S 0,1,11,58,177,437,894,1659,2813,4502,6836,10008,14121,19449,26117,
%T 34372,44422,56597,71044,88160,108115,131328,158074,188773,223604,
%U 263172,307719,357715,413493,475690,544480,620632,704381,796413
%N Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.
%C Each sequence in the following guide counts 4-tuples
%C (w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}. The notation "m div" means that m divides every term of the sequence.
%C A211058 ... wx <= yz
%C A211787 ... wx <= 2yz
%C A211795 ... wx < 2yz
%C A211797 ... wx > 2yz
%C A211809 ... wx >= 2yz
%C A211812 ... wx <= 3yz
%C A211917 ... wx < 3yz
%C A211918 ... wx > 3yz
%C A211919 ... wx >= 3yz
%C A211920 ... 2wx < 3yz
%C A211921 ... 2wx <= 3yz
%C A211922 ... 2wx > 3yz
%C A211923 ... 2wx >= 3yz
%C A212019 ... wx = 2yz ..... 2 div
%C A212020 ... wx = 3yz ..... 2 div
%C A212021 ... 2wx = 3yz .... 2 div
%C A212047 ... wx = 4yz
%C A212048 ... 3wx = 4yz .... 2 div
%C A212049 ... wx = 5yz ..... 2 div
%C A212050 ... 2wx = 5yz .... 2 div
%C A212051 ... 3wx = 5yz .... 2 div
%C A212052 ... 4wx = 5yz .... 2 div
%C A209978 ... wx = yz + 1 .. 2 div
%C A212053 ... wx <= yz + 1
%C A212054 ... wx > yz + 1
%C A212055 ... wx <= yz + 2
%C A212056 ... wx > yz + 2
%C A197168 ... wx = yz + 2 .. 2 div
%C A061201 ... w = xyz
%C A212057 ... w < xyz
%C A212058 ... w >= xyz
%C A212059 ... w = xyz - 1
%C A212060 ... w = xyz - 2
%C A212061 ... wx = (yz)^2
%C A212062 ... w^2 = xyz
%C A212063 ... w^2 < xyz
%C A212064 ... w^2 >= xyz
%C A212065 ... w^2 <= xyz
%C A212066 ... w^2 > xyz
%C A212067 ... w^3 = xyz
%C A002623 ... w = 2x + y + z
%C A006918 ... w = 2x + 2y + z
%C A000601 ... w = x + 2y + 3z (except for initial 0's)
%C A212068 ... 2w = x + y + z
%C A212069 ... 3w = x + y + z (w = average{x,y,z})
%C A212088 ... 3w < x + y + z
%C A212089 ... 3w >= x + y + z
%C A212090 ... w < x + y + z
%C A000332 ... w >= x + y + z
%C A212145 ... w < 2x + y + z
%C A001752 ... w >= 2x + y + z
%C A001400 ... w = 2x +3y + 4z
%C A005900 ... w = -x + y + z
%C A192023 ... w = -x + y + z + 2
%C A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
%C A212087 ... w^2 + x^2 = y^2 + z^2
%C A212092 ... w^2 < x^2 + y^2 + z^2
%C A212093 ... w^2 <= x^2 + y^2 + z^2
%C A212094 ... w^2 > x^2 + y^2 + z^2
%C A212095 ... w^2 >= x^2 + y^2 + z^2
%C A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
%C A212097 ... w^3 < x^3 + y^3 + z^3
%C A212098 ... w^3 <= x^3 + y^3 + z^3
%C A212099 ... w^3 > x^3 + y^3 + z^3
%C A212100 ... w^3 >= x^3 + y^3 + z^3
%C A212101 ... wx^2 = yz^2
%C A212102 ... 1/w = 1/x + 1/y + 1/z
%C A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
%C A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
%C A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
%C A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
%C A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
%C A212133 ... median(w,x,y,z) = mean(w,x,y,z)
%C A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
%C A212135 ... median(w,x,y,z) > mean(w,x,y,z)
%C A212241 ... wx + yz > n
%C A212243 ... 2wx + yz = n
%C A212244 ... w = xyz - n
%C A212245 ... w = xyz - 2n
%C A212246 ... 2w = x + y + z - n
%C A212247 ... 3w = x + y + z + n
%C A212249 ... 3w < x + y + z + n
%C A212250 ... 3w >= x + y + z + n
%C A212251 ... 3w = x + y + z + n + 1
%C A212252 ... 3w = x + y + z + n + 2
%C A212254 ... w = x + 2y + 3z - n
%C A212255 ... w^2 = mean(x^2, y^2, z^2)
%C A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
%C In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent. In the list below, the same is true for expressions involving more than one relation.
%C A000332 ... w < x <= y < z .... C(n,4)
%C A000914 ... w < x <= y < z .... Stirling 1st kind
%C A000914 ... w < x <= y >= z ... Stirling 1st kind
%C A050534 ... w < x < y >= z .... tritriangular
%C A001296 ... w <= x <= y >= z .. 4-dim pyramidal
%C A006322 ... x < x > y >= z
%C A002418 ... w < x >= y < z
%C A050534 ... w < x >=y >= z
%C A212415 ... w < x >= y <= z
%C A001296 ... w < x >= y <= z
%C A212246 ... w <= x > y <= z
%C A006322 ... w <= x >= y <= z
%C A212501 ... w > x < y >= z
%C A212503 ... w < 2x and y < 2z ..... A (note below)
%C A212504 ... w < 2x and y > 2z ..... A
%C A212505 ... w < 2x and y >= 2z .... A
%C A212506 ... w <= 2x and y <= 2z ... A
%C A212507 ... w < 2x and y <= 2z .... B
%C A212508 ... w < 2x and y < 3z ..... C
%C A212509 ... w < 2x and y <= 3z .... C
%C A212510 ... w < 2x and y > 3z ..... C
%C A212511 ... w < 2x and y >= 3z .... C
%C A212512 ... w <= 2x and y < 3z .... C
%C A212513 ... w <= 2x and y <= 3z ... C
%C A212514 ... w <= 2x and y > 3z .... C
%C A212515 ... w <= 2x and y >= 3z ... C
%C A212516 ... w > 2x and y < 3z ..... C
%C A212517 ... w > 2x and y <= 3z .... C
%C A212518 ... w > 2x and y > 3z ..... C
%C A212519 ... w > 2x and y >= 3z .... C
%C A212520 ... w >= 2x and y < 3z .... C
%C A212521 ... w >= 2x and y <= 3z ... C
%C A212522 ... w >= 2x and y > 3z .... C
%C A212523 ... w + x < y + z
%C A212560 ... w + x <= y + z
%C A212561 ... w + x = 2y + 2z
%C A212562 ... w + x < 2y + 2z ....... B
%C A212563 ... w + x <= 2y + 2z ...... B
%C A212564 ... w + x > 2y + 2z ....... B
%C A212565 ... w + x >= 2y + 2z ...... B
%C A212566 ... w + x = 3y + 3z
%C A212567 ... 2w + 2x = 3y + 3z
%C A212570 ... |w - x| = |x - y| + |y - z|
%C A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
%C A212572 ... |w - x| <= |x - y| + |y - z| .. B
%C A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
%C A212574 ... |w - x| >= |x - y| + |y - z| .. B
%C A212575 ... 2|w - x| = |x - y| + |y - z|
%C A212576 ... |w - x| = 2|x - y| + 2|y - z|
%C A212577 ... |w - x| = 2|x - y| - |y - z|
%C A212578 ... 2|w - x| = |x - y| - |y - z|
%C A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
%C A212692 ... w = |x - y| + |y - z| ............... 2 div
%C A212568 ... w < |x - y| + |y - z| ............... 2 div
%C A212573 ... w <= |x - y| + |y - z| .............. 2 div
%C A212574 ... w > |x - y| + |y - z|
%C A212575 ... w >= |x - y| + |y - z|
%C A212676 ... w + x = |x - y| + |y - z| ......... H
%C A212677 ... w + y = |x - y| + |y - z|
%C A212678 ... w + x + y = |x - y| + |y - z|
%C A006918 ... w + x + y + z = |x - y| + |y - z| . H
%C A212679 ... |x - y| = |y - z| ................. H
%C A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
%C A212681 ... |x - y| < |y - z| ................... 2 div
%C A212682 ... |x - y| >= |y - z|
%C A212683 ... |x - y| = w + |y - z| ............... 2 div
%C A212684 ... |x - y| = n - w + |y - z|
%C A212685 ... |w - x| = w + |y - z|
%C A186707 ... |w - x| < w + |y - z| ... (Note D)
%C A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
%C A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
%C A212687 ... 2*|w - x| < n + |y - z| ......... B
%C A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
%C A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
%C A212690 ... 2*|w - x| <= n + |y - z| ........ B
%C A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
%C ...
%C In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
%C R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
%C A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
%C A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
%C A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
%C A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
%C A212744 ... w=range (=max-min) ............... E
%C A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
%C A212746 ... R is in {w,x,y,z} ................ E
%C A212569 ... R is not in {w,x,y,z} ............ E
%C A212749 ... w=R or x<R or y<R or z<R ......... A
%C A212750 ... w=R or x=R or y<R or z<R ......... A
%C A212751 ... w=R or x=R or y<R or z<R ......... A
%C A212752 ... w<R or x<R or y<R or z>R ......... A
%C A212753 ... w<R or x<R or y>R or z>R ......... D
%C A212754 ... w<R or x>R or y>R or z>R ......... D
%C A002415 ... w = x + R ........................ D
%C A212755 ... |w - x| = R ...................... D
%C A212756 ... 2w = x + R
%C A212757 ... 2w = R
%C A212758 ... w = floor(R/2)
%C A002413 ... w = floor((x+y+z/2))
%C A212759 ... w, x, y are even
%C A212760 ... w is even and x = y + z .......... E
%C A212761 ... w is odd and x and y are even .... F . 2 div
%C A212762 ... w and x are odd y is even ........ F . 2 div
%C A212763 ... w, x, y are odd .................. F
%C A212764 ... w, x, y are even and z is odd .... F
%C A030179 ... w and x are even and y and z odd
%C A212765 ... w is even and x,y,z are odd ...... F
%C A212766 ... w is even and x is odd ........... A . 2 div
%C A212767 ... w and x are even and w+x=y+z ..... E
%C A212889 ... R is even ........................ A
%C A212890 ... R is odd ......................... A . 2 div
%C A212742 ... w-x, x-y, y-z are all even ....... A
%C A212892 ... w-x, x-y, y-z are all odd ........ A
%C A212893 ... w-x, x-y, y-z have same parity ... A
%C A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
%C A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
%C A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
%C A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
%C A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
%C A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
%C A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
%C A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
%C A212901 ... |w-x| = |x-y| = |y-z|
%C A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
%C A212902 ... |w-x| < |x-y| < |y-z| ............ G
%C A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
%C A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
%C A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
%C ...
%C Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
%C B: 3,-1,-5,5,1,-3,1
%C C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
%C D: 4,-5,0,5,-4,1
%C E: 1,3,-3,-3,3,1,-1
%C F: 1,4,-4,-6,6,4,-4,-1,1
%C G: 2,1,-3,-1,1,3,-1,-2,1
%C H: 2,1,-4,1,2,-1
%D A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
%D P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.
%H Bo Gyu Jeong, <a href="/A211795/b211795.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = n^4 - A211809(n).
%e a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).
%t t = Compile[{{n, _Integer}}, Module[{s = 0},
%t (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
%t {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t Map[t[#] &, Range[0, 40]] (* A211795 *)
%t (* _Peter J. C. Moses_, Apr 13 2012 *)
%Y Cf. A000583 (n^4), A210000, A211809, A212959.
%K nonn
%O 0,3
%A _Clark Kimberling_, Apr 27 2012
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