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 A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z. 203
 0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) + A211809(n) = n^4. Each sequence in the following guide counts 4-tuples (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. A211058 ... wx <= yz A211787 ... wx <= 2yz A211795 ... wx < 2yz A211797 ... wx > 2yz A211809 ... wx >= 2yz A211812 ... wx <= 3yz A211917 ... wx < 3yz A211918 ... wx > 3yz A211919 ... wx >= 3yz A211920 ... 2wx < 3yz A211921 ... 2wx <= 3yz A211922 ... 2wx > 3yz A211923 ... 2wx >= 3yz A212019 ... wx = 2yz ..... 2 div A212020 ... wx = 3yz ..... 2 div A212021 ... 2wx = 3yz .... 2 div A212047 ... wx = 4yz A212048 ... 3wx = 4yz .... 2 div A212049 ... wx = 5yz ..... 2 div A212050 ... 2wx = 5yz .... 2 div A212051 ... 3wx = 5yz .... 2 div A212052 ... 4wx = 5yz .... 2 div A209978 ... wx = yz + 1 .. 2 div A212053 ... wx <= yz + 1 A212054 ... wx > yz + 1 A212055 ... wx <= yz + 2 A212056 ... wx > yz + 2 A197168 ... wx = yz + 2 .. 2 div A061201 ... w = xyz A212057 ... w < xyz A212058 ... w >= xyz A212059 ... w = xyz - 1 A212060 ... w = xyz - 2 A212061 ... wx = (yz)^2 A212062 ... w^2 = xyz A212063 ... w^2 < xyz A212064 ... w^2 >= xyz A212065 ... w^2 <= xyz A212066 ... w^2 > xyz A212067 ... w^3 = xyz A002623 ... w = 2x + y + z A006918 ... w = 2x + 2y + z A000601 ... w = x + 2y + 3z (except for initial 0s) A212068 ... 2w = x + y + z A212069 ... 3w = x + y + z (w = average{x,y,z}) A212088 ... 3w < x + y + z A212089 ... 3w >= x + y + z A212090 ... w < x + y + z A000332 ... w >= x + y + z A212145 ... w < 2x + y + z A001752 ... w >= 2x + y + z A001400 ... w = 2x +3y + 4z A005900 ... w = -x + y + z A192023 ... w = -x + y + z + 2 A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div A212087 ... w^2 + x^2 = y^2 + z^2 A212092 ... w^2 < x^2 + y^2 + z^2 A212093 ... w^2 <= x^2 + y^2 + z^2 A212094 ... w^2 > x^2 + y^2 + z^2 A212095 ... w^2 >= x^2 + y^2 + z^2 A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div A212097 ... w^3 < x^3 + y^3 + z^3 A212098 ... w^3 <= x^3 + y^3 + z^3 A212099 ... w^3 > x^3 + y^3 + z^3 A212100 ... w^3 >= x^3 + y^3 + z^3 A212101 ... wx^2 = yz^2 A212102 ... 1/w = 1/x + 1/y + 1/z A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z} A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m. A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m. A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m. A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m. A212133 ... median(w,x,y,z) = mean(w,x,y,z) A212134 ... median(w,x,y,z) <= mean(w,x,y,z) A212135 ... median(w,x,y,z) > mean(w,x,y,z) A212241 ... wx + yz > n A212243 ... 2wx + yz = n A212244 ... w = xyz - n A212245 ... w = xyz - 2n A212246 ... 2w = x + y + z - n A212247 ... 3w = x + y + z + n A212249 ... 3w < x + y + z + n A212250 ... 3w >= x + y + z + n A212251 ... 3w = x + y + z + n + 1 A212252 ... 3w = x + y + z + n + 2 A212254 ... w = x + 2y + 3z - n A212255 ... w^2 = mean(x^2, y^2, z^2) A212256 ... 4/w = 1/x + 1/y +1/z + 1/n 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. A000332 ... w < x <= y < z .... C(n,4) A000914 ... w < x <= y < z .... Stirling 1st kind A000914 ... w < x <= y >= z ... Stirling 1st kind A050534 ... w < x < y >= z .... tritriangular A001296 ... w <= x <= y >= z .. 4-dim pyramidal A006322 ... x < x > y >= z A002418 ... w < x >= y < z A050534 ... w < x >=y >= z A212415 ... w < x >= y <= z A001296 ... w < x >= y <= z A212246 ... w <= x > y <= z A006322 ... w <= x >= y <= z A212501 ... w > x < y >= z A212503 ... w < 2x and y < 2z ..... A (note below) A212504 ... w < 2x and y > 2z ..... A A212505 ... w < 2x and y >= 2z .... A A212506 ... w <= 2x and y <= 2z ... A A212507 ... w < 2x and y <= 2z .... B A212508 ... w < 2x and y < 3z ..... C A212509 ... w < 2x and y <= 3z .... C A212510 ... w < 2x and y > 3z ..... C A212511 ... w < 2x and y >= 3z .... C A212512 ... w <= 2x and y < 3z .... C A212513 ... w <= 2x and y <= 3z ... C A212514 ... w <= 2x and y > 3z .... C A212515 ... w <= 2x and y >= 3z ... C A212516 ... w > 2x and y < 3z ..... C A212517 ... w > 2x and y <= 3z .... C A212518 ... w > 2x and y > 3z ..... C A212519 ... w > 2x and y >= 3z .... C A212520 ... w >= 2x and y < 3z .... C A212521 ... w >= 2x and y <= 3z ... C A212522 ... w >= 2x and y > 3z .... C A212523 ... w + x < y + z A212560 ... w + x <= y + z A212561 ... w + x = 2y + 2z A212562 ... w + x < 2y + 2z ....... B A212563 ... w + x <= 2y + 2z ...... B A212564 ... w + x > 2y + 2z ....... B A212565 ... w + x >= 2y + 2z ...... B A212566 ... w + x = 3y + 3z A212567 ... 2w + 2x = 3y + 3z A212570 ... |w - x| = |x - y| + |y - z| A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div A212572 ... |w - x| <= |x - y| + |y - z| .. B A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div A212574 ... |w - x| >= |x - y| + |y - z| .. B A212575 ... 2|w - x| = |x - y| + |y - z| A212576 ... |w - x| = 2|x - y| + 2|y - z| A212577 ... |w - x| = 2|x - y| - |y - z| A212578 ... 2|w - x| = |x - y| - |y - z| A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|} A212692 ... w = |x - y| + |y - z| ............... 2 div A212568 ... w < |x - y| + |y - z| ............... 2 div A212573 ... w <= |x - y| + |y - z| .............. 2 div A212574 ... w > |x - y| + |y - z| A212575 ... w >= |x - y| + |y - z| A212676 ... w + x = |x - y| + |y - z| ......... H A212677 ... w + y = |x - y| + |y - z| A212678 ... w + x + y = |x - y| + |y - z| A006918 ... w + x + y + z = |x - y| + |y - z| . H A212679 ... |x - y| = |y - z| ................. H A212680 ... |x - y| = |y - z| + 1 ..............H 2 div A212681 ... |x - y| < |y - z| ................... 2 div A212682 ... |x - y| >= |y - z| A212683 ... |x - y| = w + |y - z| ............... 2 div A212684 ... |x - y| = n - w + |y - z| A212685 ... |w - x| = w + |y - z| A186707 ... |w - x| < w + |y - z| ... (Note D) A212714 ... |w - x| >= w + |y - z| .......... H . 2 div A212686 ... 2*|w - x| = n + |y - z| ............. 4 div A212687 ... 2*|w - x| < n + |y - z| ......... B A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div A212690 ... 2*|w - x| <= n + |y - z| ........ B A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div ... 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.   R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}. A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div A212744 ... w=range (=max-min) ............... E A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z} A212746 ... R is in {w,x,y,z} ................ E A212569 ... R is not in {w,x,y,z} ............ E A212749 ... w=R or xR ......... A A212753 ... wR or z>R ......... D A212754 ... wR or y>R or z>R ......... D A002415 ... w = x + R ........................ D A212755 ... |w - x| = R ...................... D A212756 ... 2w = x + R A212757 ... 2w = R A212758 ... w = floor(R/2) A002413 ... w = floor((x+y+z/2)) A212759 ... w, x, y are even A212760 ... w is even and x = y + z .......... E A212761 ... w is odd and x and y are even .... F . 2 div A212762 ... w and x are odd y is even ........ F . 2 div A212763 ... w, x, y are odd .................. F A212764 ... w, x, y are even and z is odd .... F A030179 ... w and x are even and y and z odd A212765 ... w is even and x,y,z are odd ...... F A212766 ... w is even and x is odd ........... A . 2 div A212767 ... w and x are even and w+x=y+z ..... E A212889 ... R is even ........................ A A212890 ... R is odd ......................... A . 2 div A212742 ... w-x, x-y, y-z are all even ....... A A212892 ... w-x, x-y, y-z are all odd ........ A A212893 ... w-x, x-y, y-z have same parity ... A A005915 ... min{|w-x|, |x-y|, |y-z|} = 0 A212894 ... min{|w-x|, |x-y|, |y-z|} = 1 A212895 ... min{|w-x|, |x-y|, |y-z|} = 2 A179824 ... min{|w-x|, |x-y|, |y-z|} > 0 A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1 A212897 ... min{|w-x|, |x-y|, |y-z|} > 1 A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2 A212899 ... min{|w-x|, |x-y|, |y-z|} > 2 A212901 ... |w-x| = |x-y| = |y-z| A212900 ... |w-x|, |x-y|, |y-z| are distinct . G A212902 ... |w-x| < |x-y| < |y-z| ............ G A212903 ... |w-x| <= |x-y| <= |y-z| .......... G A212904 ... |w-x| + |x-y| + |y-z| = n ........ H A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H ... Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1. B: 3,-1,-5,5,1,-3,1 C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1 D: 4,-5,0,5,-4,1 E: 1,3,-3,-3,3,1,-1 F: 1,4,-4,-6,6,4,-4,-1,1 G: 2,1,-3,-1,1,3,-1,-2,1 H: 2,1,-4,1,2,-1 REFERENCES A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152. P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797. LINKS Bo Gyu Jeong, Table of n, a(n) for n = 0..200 EXAMPLE 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) MATHEMATICA t = Compile[{{n, _Integer}}, Module[{s = 0},     (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},       {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]]; Map[t[#] &, Range[0, 40]] (* A211795 *) (* Peter J. C. Moses, Apr 13 2012 *) CROSSREFS Cf. A210000, A212959. Sequence in context: A048366 A107425 A211921 * A256226 A290360 A073720 Adjacent sequences:  A211792 A211793 A211794 * A211796 A211797 A211798 KEYWORD nonn AUTHOR Clark Kimberling, Apr 27 2012 STATUS approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)