A212959 -id:A212959 - cp's OEIS frontend

A213391 Number of (w,x,y) with all terms in {0,...,n} and 2max(w,x,y) < 3min(w,x,y).

0, 1, 2, 3, 10, 17, 24, 43, 62, 81, 118, 155, 192, 253, 314, 375, 466, 557, 648, 775, 902, 1029, 1198, 1367, 1536, 1753, 1970, 2187, 2458, 2729, 3000, 3331, 3662, 3993, 4390, 4787, 5184, 5653, 6122, 6591, 7138, 7685, 8232, 8863, 9494, 10125

Offset: 0

Clark Kimberling, Jun 11 2012

For a guide to related sequences, see A212959.

Also the number of (w,x,y) with all terms in {0,...,n-1} and 2*max(w,x,y) <= 3*min(w,x,y).

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A212959, A213392.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2*Max[w, x, y] < 3*Min[w, x, y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213391 *)

a(n) + A213392(n) = (n+1)^3.
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) - a(n-5) + 2*a(n-6) - a(n-7).
G.f.: (x + 4*x^4 + x^7)/(((1 - x)^4)*(1 + x + x^2)^2).
a(n) = (n^3 + 6*n*(((n+1) mod 3 + 1) mod 2) - 2 + 2*((n+1) mod 3))/9. - Ayoub Saber Rguez, Feb 01 2022
From Jon E. Schoenfield, Feb 02 2022: (Start)
a(n) =  n^3/9            if n == 0 (mod 3),
       (n^3 + 6*n + 2)/9 if n == 1 (mod 3),
       (n^3 + 6*n - 2)/9 if n == 2 (mod 3).
(End)

A213482 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| > w+x+y.

Original entry on oeis.org

0, 3, 14, 41, 87, 161, 265, 409, 594, 831, 1120, 1473, 1889, 2381, 2947, 3601, 4340, 5179, 6114, 7161, 8315, 9593, 10989, 12521, 14182, 15991, 17940, 20049, 22309, 24741, 27335, 30113, 33064, 36211, 39542, 43081, 46815, 50769, 54929

Offset: 0

Clark Kimberling, Jun 13 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A212959, A213479, A213481, A213483.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y > Abs[w - x] + Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213482 *)

a(n) + A213483(n) = (n+1)^3.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: (3*x + 8*x^2 + 10*x^3 + 3*x^4 - x^5)/((1 - x)^4*(1 + x)^2).
From Ayoub Saber Rguez, Dec 29 2021: (Start)
a(n) = A213481(n) - A213479(n).
a(n) = (23*n^3 + 39*n^2 + n + 9 - (3*n+9)*((n+1) mod 2))/24. (End)

A213484 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| + |y-w| >= w+x+y.

Original entry on oeis.org

1, 4, 7, 10, 16, 25, 34, 43, 55, 70, 85, 100, 118, 139, 160, 181, 205, 232, 259, 286, 316, 349, 382, 415, 451, 490, 529, 568, 610, 655, 700, 745, 793, 844, 895, 946, 1000, 1057, 1114, 1171, 1231, 1294, 1357, 1420, 1486, 1555, 1624, 1693, 1765

Offset: 0

Clark Kimberling, Jun 13 2012

a(n) + A213485(n) = (n+1)^3.

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y == Abs[w - x] + Abs[x - y] + Abs[y - w],
s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213484 *)

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5).
G.f.: (1 + x - x^2 + x^3 + x^4)/((1 - x)^3 (1 + x^2)).
From Ayoub Saber Rguez, Dec 31 2021: (Start)
a(n) + A213485(n) = (n+1)^3.
a(n) = 3*A054925(n+1) + 1.
a(n) = 3*(A192447(n+1)/2) + 1.
a(n) = (3*n^2 + 3*n + 4 + 3*((n+1) mod 4 - (n+1) mod 2))/4. (End)

A238410 a(n) = floor((3(n-1)^2 + 1)/2).

Original entry on oeis.org

0, 2, 6, 14, 24, 38, 54, 74, 96, 122, 150, 182, 216, 254, 294, 338, 384, 434, 486, 542, 600, 662, 726, 794, 864, 938, 1014, 1094, 1176, 1262, 1350, 1442, 1536, 1634, 1734, 1838, 1944, 2054, 2166, 2282, 2400, 2522, 2646, 2774, 2904, 3038, 3174, 3314, 3456, 3602, 3750, 3902, 4056, 4214, 4374, 4538, 4704

Offset: 1

Emeric Deutsch, Feb 27 2014

a(n) = the eccentric connectivity index of the path P[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined to be the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(4)=14 because the vertices of P[4] have degrees 1,2,2,1 and eccentricities 3,2,2,3; we have 1*3 + 2*2 + 2*2 + 1*3 = 14.
From Paul Curtz, Feb 23 2023: (Start)
East spoke of the hexagonal spiral using A004526 with a single 0:
.
            43  42  42  41  41  40
          43  28  28  27  27  26  40
        44  29  17  16  16  15  26  39
      44  29  17   8   8   7  15  25  39
    45  30  18   9   3   2   7  14  25  38
  45  30  18   9   3   0---2---6--14--24--38-->
    31  19  10   4   1   1   6  13  24  37
      31  19  10   4   5   5  13  23  37
        32  20  11  11  12  12  23  36
          32  20  21  21  22  22  36
            33  33  34  34  35  35
.
Other spokes are A032528, A005449, A227017, A045943, A005448, A212959, A000326, A282513, A143689, and A104249. (End)

Matthew House, Table of n, a(n) for n = 1..10000
M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311, 2011, 1229-1234.
B. Zhou and Zh. Du, On eccentric connectivity index, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A004526, A077043.

(Cf. A056105, A056107.)

a := proc (n) options operator, arrow: floor((3/2)*(n-1)^2+1/2) end proc: seq(a(n), n = 1 .. 70);

Table[Floor[(3(n-1)^2+1)/2],{n,80}]  (* or *) LinearRecurrence[{2,0,-2,1},{0,2,6,14},80] (* Harvey P. Dale, Apr 30 2022 *)

a(n)=(3*(n-1)^2 + 1)\2 \\ Charles R Greathouse IV, Feb 15 2017

a(n) = (3*n)^2/6 for n even and a(n) = ((3*n)^2 + 3)/6 for n odd. - Miquel Cerda, Jun 17 2016
From Ilya Gutkovskiy, Jun 17 2016: (Start)
G.f.: 2*x^2*(1 + x + x^2)/((1 - x)^3*(1 + x)).
a(n) = (6*n^2 - 12*n + 7 + (-1)^n)/4.
a(n) = 2* A077043(n-1). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Matthew House, Feb 15 2017
Sum_{n>=2} 1/a(n) = Pi^2/36 + tanh(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Mar 12 2023

A212967 Number of (w,x,y) with all terms in {0,...,n} and w < range{w,x,y}.

Original entry on oeis.org

0, 3, 10, 26, 50, 89, 140, 212, 300, 415, 550, 718, 910, 1141, 1400, 1704, 2040, 2427, 2850, 3330, 3850, 4433, 5060, 5756, 6500, 7319, 8190, 9142, 10150, 11245, 12400, 13648, 14960, 16371, 17850, 19434, 21090, 22857, 24700, 26660, 28700

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A212959, A212968, A036666.

List([1..45],n->Sum([1..n],k->(10*k*(k-1)+(2*k-1)*(-1)^k+1)/8)); # Muniru A Asiru, Nov 28 2018

[(n+1)*(10*n*(n+2) - 3*(-1)^n+3)/24: n in [0..50]]; // Vincenzo Librandi, Nov 29 2018

A212967:=n->(n+1)*(10*n*(n+2)-3*(-1)^n+3)/24: seq(A212967(n), n=0..100); # Wesley Ivan Hurt, Apr 28 2017

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < (Max[w, x, y] - Min[w, x, y]), s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A212967 *)
Accumulate[Accumulate[Table[n + LCM[n, 2], {n, 0, 60}]]] (* Jon Maiga, Nov 28 2018 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 3, 10, 26, 50, 89}, 50] (* Vincenzo Librandi, Nov 29 2018 *)

a(n) + A212968(n) = (n + 1)^3.
a(n) = (n + 1)*(10*n*(n + 2) - 3*(-1)^n + 3)/24.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(3 + 4*x + 3*x^2) and g(x) = ((1 - x)^4)(1 + x)^2.
a(n) = Sum_{k=1..n} A036666(k). - Jon Maiga, Nov 28 2018
E.g.f.: (exp(x)*(3 + 63*x + 60*x^2 + 10*x^3) - 3*exp(-x)*(1 - x))/24. - Franck Maminirina Ramaharo, Nov 29 2018

A212968 Number of (w,x,y) with all terms in {0,...,n} and w>=range{w,x,y}.

Original entry on oeis.org

1, 5, 17, 38, 75, 127, 203, 300, 429, 585, 781, 1010, 1287, 1603, 1975, 2392, 2873, 3405, 4009, 4670, 5411, 6215, 7107, 8068, 9125, 10257, 11493, 12810, 14239, 15755, 17391, 19120, 20977, 22933, 25025, 27222, 29563, 32015, 34619, 37340

Offset: 0

Clark Kimberling, Jun 02 2012

a(n)+A212967(n) = (n+1)^3.

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w >= (Max[w, x, y] - Min[w, x, y]), s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A212968 *)

a(n) = (n+1)*(14*n*(n+2)+3*(-1)^n+21)/24.
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).
G.f.: f(x)/g(x), where f(x) = 1 + 3*x + 6*x^2 + 3*x^3 + x^4 and g(x) = ((1-x)^4)(1+x)^2.

A212969 Number of (w,x,y) with all terms in {0,...,n} and w != x and x > range(w,x,y).

Original entry on oeis.org

0, 0, 2, 10, 26, 56, 100, 166, 252, 368, 510, 690, 902, 1160, 1456, 1806, 2200, 2656, 3162, 3738, 4370, 5080, 5852, 6710, 7636, 8656, 9750, 10946, 12222, 13608, 15080, 16670, 18352, 20160, 22066, 24106, 26250, 28536, 30932, 33478, 36140

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A212959, A045991, A212970.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w != x > (Max[w, x, y] - Min[w, x, y]),
  s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212969 *)
m/2 (* integers *)

a(n) = (n-1)*(2*n*(7*n-2) - 3*(-1)^n + 3)/24.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = 2*(x^2)*(1 + 3*x + 2*x^2 + x^3) and g(x) = ((1-x)^4)*(1+x)^2.
a(n) = A045991(n) - A212970(n-1). - Ayoub Saber Rguez, Mar 31 2023

A212970 Number of (w,x,y) with all terms in {0,...,n} and w != x and x < range(w,x,y).

Original entry on oeis.org

0, 2, 8, 22, 44, 80, 128, 196, 280, 390, 520, 682, 868, 1092, 1344, 1640, 1968, 2346, 2760, 3230, 3740, 4312, 4928, 5612, 6344, 7150, 8008, 8946, 9940, 11020, 12160, 13392, 14688, 16082, 17544, 19110, 20748, 22496, 24320, 26260, 28280

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Twice the partial sums of A210977. - J. M. Bergot, Aug 10 2013

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A088003, A210977, A212959.

Cf. A045991, A212969.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w != x < (Max[w, x, y] - Min[w, x, y]),
   s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212970 *)
m/2 (* essentially A088003 *)

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 2*x + 2*x^2) and g(x) = ((1-x)^4)(1+x)^2.
a(n) = 2 * A088003(n) for n>0.
From Ayoub Saber Rguez, Mar 31 2023: (Start)
a(n) + A212969(n+1) = A045991(n+1).
a(n) = (10*n^3 + 24*n^2 + 8*n + (6*n)*(n mod 2))/24. (End)

Typo in name corrected by Ayoub Saber Rguez, Mar 31 2023

A212971 Number of triples (w,x,y) with all terms in {0,...,n} and w < floor((x+y)/3).

Original entry on oeis.org

0, 0, 3, 11, 25, 48, 82, 128, 189, 267, 363, 480, 620, 784, 975, 1195, 1445, 1728, 2046, 2400, 2793, 3227, 3703, 4224, 4792, 5408, 6075, 6795, 7569, 8400, 9290, 10240, 11253, 12331, 13475, 14688, 15972, 17328, 18759, 20267, 21853, 23520

Offset: 0

Clark Kimberling, Jun 03 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A212959, A212972.

Cf. A045991, A212974.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < Floor[(x + y)/3], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212971*)
LinearRecurrence[{3,-3,2,-3,3,-1},{0,0,3,11,25,48},50] (* Harvey P. Dale, Aug 24 2021 *)

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
G.f.: (x^2)*(3 + 2*x + x^2)/((1 + x + x^2)*(1-x)^4).
a(n) = (n+1)^3 - A212972(n).
From Ayoub Saber Rguez, Dec 11 2023: (Start)
a(n) = A045991(n+1) - A212974(n).
a(n) = (n^3 + n^2 - n - 1 + (((n+1) mod 3) mod 2))/3. (End)

A212975 Number of (w,x,y) with all terms in {0,...,n} and even range.

Original entry on oeis.org

1, 2, 15, 28, 65, 102, 175, 248, 369, 490, 671, 852, 1105, 1358, 1695, 2032, 2465, 2898, 3439, 3980, 4641, 5302, 6095, 6888, 7825, 8762, 9855, 10948, 12209, 13470, 14911, 16352, 17985, 19618, 21455, 23292, 25345, 27398, 29679, 31960

Offset: 0

Clark Kimberling, Jun 03 2012

a(n)+A212976(n) = (n+1)^3.

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A212959, A212976.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Mod[Max[w, x, y] - Min[w, x, y], 2] == 0,
   s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212975 *)

a(n) = (n+1)*(2*n*(n+2)+3*(-1)^n+1)/4.
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).
G.f.: f(x)/g(x), where f(x) = 1 + 10*x^2 + x^4 and g(x) = ((1-x)^4)*(1+x)^2.

cp's OEIS Frontend

A213391 Number of (w,x,y) with all terms in {0,...,n} and 2*max(w,x,y) < 3*min(w,x,y).

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A213482 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| > w+x+y.

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A213484 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| + |y-w| >= w+x+y.

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A238410 a(n) = floor((3(n-1)^2 + 1)/2).

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A212967 Number of (w,x,y) with all terms in {0,...,n} and w < range{w,x,y}.

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A212968 Number of (w,x,y) with all terms in {0,...,n} and w>=range{w,x,y}.

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A212969 Number of (w,x,y) with all terms in {0,...,n} and w != x and x > range(w,x,y).

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A212970 Number of (w,x,y) with all terms in {0,...,n} and w != x and x < range(w,x,y).

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A213391 Number of (w,x,y) with all terms in {0,...,n} and 2max(w,x,y) < 3min(w,x,y).