A212959 - OEIS

%I #81 Dec 28 2021 00:20:03

%S 1,4,11,20,33,48,67,88,113,140,171,204,241,280,323,368,417,468,523,

%T 580,641,704,771,840,913,988,1067,1148,1233,1320,1411,1504,1601,1700,

%U 1803,1908,2017,2128,2243,2360,2481,2604,2731,2860,2993,3128,3267

%N Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

%C In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):

%C   A: 2,  0, -2,  1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);

%C   B: 3, -2, -2,  3, -1;

%C   C: 4, -6,  4, -1;

%C   D: 1,  2, -2, -1,  1;

%C   E: 2,  1, -4,  1,  2, -1;

%C   F: 2, -1,  1, -2,  1;

%C   G: 2, -1,  0,  1, -2,  1;

%C   H: 2, -1,  2, -4,  2, -1,  2, -1;

%C   I: 3, -3,  2, -3,  3, -1;

%C   J: 4, -7,  8, -7,  4, -1.

%C   ...

%C A212959 ... |w-x|=|x-y| ...... recurrence type A

%C A212960 ... |w-x| != |x-y| ................... B

%C A212683 ... |w-x| < |x-y| .................... B

%C A212684 ... |w-x| >= |x-y| ................... B

%C A212963 ... see entry for definition ......... B

%C A212964 ... |w-x| < |x-y| < |y-w| ............ B

%C A006331 ... |w-x| < y ........................ C

%C A005900 ... |w-x| <= y ....................... C

%C A212965 ... w = R ............................ D

%C A212966 ... 2*w = R

%C A212967 ... w < R ............................ E

%C A212968 ... w >= R ........................... E

%C A077043 ... w = x > R ........................ A

%C A212969 ... w != x and x > R ................. E

%C A212970 ... w != x and x < R ................. E

%C A055998 ... w = x + y - 1

%C A011934 ... w < floor((x+y)/2) ............... B

%C A182260 ... w > floor((x+y)/2) ............... B

%C A055232 ... w <= floor((x+y)/2) .............. B

%C A011934 ... w >= floor((x+y)/2) .............. B

%C A212971 ... w < floor((x+y)/3) ............... B

%C A212972 ... w >= floor((x+y)/3) .............. B

%C A212973 ... w <= floor((x+y)/3) .............. B

%C A212974 ... w > floor((x+y)/3) ............... B

%C A212975 ... R is even ........................ E

%C A212976 ... R is odd ......................... E

%C A212978 ... R = 2*n - w - x

%C A212979 ... R = average{w,x,y}

%C A212980 ... w < x + y and x < y .............. B

%C A212981 ... w <= x+y and x < y ............... B

%C A212982 ... w < x + y and x <= y ............. B

%C A212983 ... w <= x + y and x <= y ............ B

%C A002623 ... w >= x + y and x <= y ............ B

%C A087811 ... w = 2*x + y ...................... A

%C A008805 ... w = 2*x + 2*y .................... D

%C A000982 ... 2*w = x + y ...................... F

%C A001318 ... 2*w = 2*x + y .................... F

%C A001840 ... w = 3*x + y

%C A212984 ... 3*w = x + y

%C A212985 ... 3*w = 3*x + y

%C A001399 ... w = 2*x + 3*y

%C A212986 ... 2*w = 3*x + y

%C A008810 ... 3*x = 2*x + y .................... F

%C A212987 ... 3*w = 2*x + 2*y

%C A001972 ... w = 4*x + y ...................... G

%C A212988 ... 4*w = x + y ...................... G

%C A212989 ... 4*w = 4*x + y

%C A008812 ... 5*w = 2*x + 3*y

%C A016061 ... n < w + x + y <= 2*n ............. C

%C A000292 ... w + x + y <=n .................... C

%C A000292 ... 2*n < w + x + y <= 3*n ........... C

%C A212977 ... n/2 < w + x + y <= n

%C A143785 ... w < R < x ........................ E

%C A005996 ... w < R <= x ....................... E

%C A128624 ... w <= R <= x ...................... E

%C A213041 ... R = 2*|w - x| .................... A

%C A213045 ... R < 2*|w - x| .................... B

%C A087035 ... R >= 2*|w - x| ................... B

%C A213388 ... R <= 2*|w - x| ................... B

%C A171218 ... M < 2*m .......................... B

%C A213389 ... R < 2|w - x| ..................... E

%C A213390 ... M >= 2*m ......................... E

%C A213391 ... 2*M < 3*m ........................ H

%C A213392 ... 2*M >= 3*m ....................... H

%C A213393 ... 2*M > 3*m ........................ H

%C A213391 ... 2*M <= 3*m ....................... H

%C A047838 ... w = |x + y - w| .................. A

%C A213396 ... 2*w < |x + y - w| ................ I

%C A213397 ... 2*w >= |x + y - w| ............... I

%C A213400 ... w < R < 2*w

%C A069894 ... min(|w-x|,|x-y|) = 1

%C A000384 ... max(|w-x|,|x-y|) = |w-y|

%C A213395 ... max(|w-x|,|x-y|) = w

%C A213398 ... min(|w-x|,|x-y|) = x ............. A

%C A213399 ... max(|w-x|,|x-y|) = x ............. D

%C A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D

%C A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E

%C A006918 ... |w-x| + |x-y| > w+x+y ............ E

%C A213481 ... |w-x| + |x-y| <= w+x+y ........... E

%C A213482 ... |w-x| + |x-y| < w+x+y ............ E

%C A213483 ... |w-x| + |x-y| >= w+x+y ........... E

%C A213484 ... |w-x|+|x-y|+|y-w| = w+x+y

%C A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J

%C A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J

%C A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J

%C A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J

%C A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J

%C A213490 ... w,x,y,|w-x|,|x-y| distinct

%C A213491 ... w,x,y,|w-x|,|x-y| not distinct

%C A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct

%C A213495 ... w = min(|w-x|,|x-y|,|w-y|)

%C A213492 ... w != min(|w-x|,|x-y|,|w-y|)

%C A213496 ... x != max(|w-x|,|x-y|)

%C A213498 ... w != max(|w-x|,|x-y|,|w-y|)

%C A213497 ... w = min(|w-x|,|x-y|)

%C A213499 ... w != min(|w-x|,|x-y|)

%C A213501 ... w != max(|w-x|,|x-y|)

%C A213502 ... x != min(|w-x|,|x-y|)

%C ...

%C A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties.  Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.

%C Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - _Philippe Deléham_, Mar 16 2014

%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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

%F G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).

%F a(n) + A212960(n) = (n+1)^3.

%F a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - _Luce ETIENNE_, Apr 05 2014

%F a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - _Ayoub Saber Rguez_, Aug 31 2021

%e a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).

%e Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...

%e a(0) = 1;

%e a(1) = 1 + 3 = 4;

%e a(2) = 1 + 3 + 7 = 11;

%e a(3) = 1 + 3 + 7 + 9 = 20;

%e a(4) = 1 + 3 + 7 + 9 + 13 = 33;

%e a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

%t t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[Abs[w - x] == Abs[x - y], s = s + 1],

%t {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];

%t m = Map[t[#] &, Range[0, 50]]   (* A212959 *)

%o (PARI) a(n)=(6*n^2+8*n+3)\/4 \\ _Charles R Greathouse IV_, Jul 28 2015

%Y Cf. A047241, A211795.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jun 01 2012