A095794 -id:A095794 - cp's OEIS frontend

A216172 Number of all possible tetrahedra of any size, having reverse orientation to the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts.

0, 0, 1, 4, 10, 21, 39, 66, 105, 159, 231, 325, 445, 595, 780, 1005, 1275, 1596, 1974, 2415, 2926, 3514, 4186, 4950, 5814, 6786, 7875, 9090, 10440, 11935, 13585, 15400, 17391, 19569, 21945, 24531, 27339, 30381, 33670, 37219, 41041, 45150, 49560, 54285, 59340

Offset: 1

V.J. Pohjola, Sep 03 2012

The number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron is given by A000332(n+3).

Create a sequence wherein the sum of three consecutive numbers is a triangular number: 0,0,0,1,2,3,5,7...; then find the partial sums of this sequence: 0,0,0,1,3,6,11,18...; then take the partial sums of this sequence: 0,0,0,1,4,10,21,39,66... and after dropping the first two zeros, you get this sequence. - J. M. Bergot, Apr 14 2016

			For n=9 the numbers of the reversely oriented tetrahedra, starting from the smallest size, are A000292(7)=84, A000292(4)=20, and A000292(1)=1, the sum being a(9)=105.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Cf. A000292, A000332, A216173, A216175.

I:=[0, 0, 1, 4, 10, 21, 39]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Sep 12 2012

nnn = 100; Tev[n_] := (n - 2) (n - 1) n/6; Table[Sum[Tev[n - nn], {nn, 0, n - 1, 3}], {n, nnn}]
Table[(1/72) (-6 n - 5 n^2 + 2 n^3 + n^4 + 4 - 4 (-1)^Mod[n, 3]), {n, 50}]
CoefficientList[Series[x^2 / ((1 - x)^5*(1 + x + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 12 2012 *)
LinearRecurrence[{4,-6,5,-5,6,-4,1},{0,0,1,4,10,21,39},50] (* Harvey P. Dale, Feb 18 2018 *)

a(n)=(n^4+2*n^3-5*n^2-6*n+4-4*(-1)^(n%3))/72 \\ Charles R Greathouse IV, Sep 12 2012

a(n) = (1/72)*(-6*n -5*n^2 +2*n^3 +n^4 +4 -4*(-1)^(n mod 3)).
G.f.: x^3/((1-x)^5*(1+x+x^2)). - Bruno Berselli, Sep 11 2012
a(3*n-1) = A000217(A115067(n)); a(3*n) = A000217(A095794(n)); a(3*n+1) = A000217(A143208(n+2)) + A000217(n). - J. M. Bergot, Apr 14 2016
E.g.f.: (1/216)*(8 - 24*x + 24*x^2 + 24*x^3 + 3*x^4)*exp(x) - (1/27)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))*exp(-x/2). - Ilya Gutkovskiy, Apr 14 2016

A000338 Expansion of x^3(5-2x)*(1-x^3)/(1-x)^4.

Original entry on oeis.org

5, 18, 42, 75, 117, 168, 228, 297, 375, 462, 558, 663, 777, 900, 1032, 1173, 1323, 1482, 1650, 1827, 2013, 2208, 2412, 2625, 2847, 3078, 3318, 3567, 3825, 4092, 4368, 4653, 4947, 5250, 5562, 5883, 6213, 6552, 6900, 7257, 7623, 7998, 8382, 8775, 9177, 9588, 10008, 10437, 10875, 11322, 11778

Offset: 3

N. J. A. Sloane

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

T. D. Noe, Table of n, a(n) for n = 3..1000
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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

ff := n->9/2*n^2-15/2*n; seq(ff(n), n=3..60); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001, sequence without a(3).

nn = 100; CoefficientList[Series[(5 - 2 x) (1 - x^3)/(1 - x)^4, {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
LinearRecurrence[{3,-3,1},{5,18,42,75},60] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = 3*A095794(n-2), n>3. - R. J. Mathar, May 30 2022
G.f.: (1+x+x^2)*(5-2*x)*x^3/(1-x)^3. - Simon Plouffe in his 1992 dissertation
Sum_{n>=3} 1/a(n) = log(3)/5 + Pi*sqrt(3)/45 = 0.3406424... - R. J. Mathar, Apr 22 2024
a(n) = 5*A005448(n-2) -2*A005448(n-3). - R. J. Mathar, Apr 22 2024

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

Original entry on oeis.org

4, 14, 25, 30, 65, 64, 52, 120, 152, 121, 80, 190, 264, 275, 196, 114, 275, 400, 462, 434, 289, 154, 375, 560, 682, 714, 629, 400, 200, 490, 744, 935, 1036, 1020, 860, 529, 252, 620, 952, 1221, 1400, 1462, 1380, 1127, 676, 310, 765, 1184, 1540, 1806, 1955, 1960

Offset: 1

Gary W. Adamson, Jun 17 2004

Matrix square of the matrix B(n,m) = 2+3*(m-1), B containing the first terms of A016789

in its row n, n>0, 1<=m<=n.

			The matrix B starts as
  2 ;
  2,5 ;
  2,5,8 ;
  2,5,8,11 ;
  2,5,8,11,14 ;
and interpreting this as a lower triangular matrix, its square T = B^2 starts
  4;
  14,25;
  30,65,64;
  52,120,152,121;

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A024212, A096037, A011379, A002411, A096038, A095794.

A094930 := proc(n,m) (3*m-1)*(3*m+3*n-2)*(n+1-m)/2 ; end: seq(seq(A094930(n,m),m=1..n),n=1..20) ; # R. J. Mathar, Oct 09 2009

T(n,m) = sum_{k=m..n} B(n,k)*B(k,m) = (3*m-1)*(3*m+3*n-2)*(n+1-m)/2.
Row sums: sum_{m=1..n} T(n,m) = A024212(n).
G.f. as triangle: x*y*(4+2*x+13*x*y-16*x^2*y+x^2*y^2-4*x^3*y^2)/((1-x)*(1-x*y))^3. - Robert Israel, May 06 2019

Edited and extended by R. J. Mathar, Oct 09 2009

A096037 Triangle T(n,m) = (3n+3m-2)*(n+1-m)/2 read by rows.

Original entry on oeis.org

2, 7, 5, 15, 13, 8, 26, 24, 19, 11, 40, 38, 33, 25, 14, 57, 55, 50, 42, 31, 17, 77, 75, 70, 62, 51, 37, 20, 100, 98, 93, 85, 74, 60, 43, 23, 126, 124, 119, 111, 100, 86, 69, 49, 26, 155, 153, 148, 140, 129, 115, 98, 78, 55, 29, 187, 185, 180, 172, 161, 147, 130, 110, 87, 61, 32

Offset: 1

Gary W. Adamson, Jun 17 2004

			The triangle starts in row n=1 as
2;
7,5;
15,13,8;
26,24,19,11;

Cf. A094930, A024212, A011379, A002411, A096038, A095794.

def A096037(n,m):
    return (3*n+3*m-2)*(n+1-m)//2
print( [A096037(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

T(n,m) = (3*n+3*m-2)*(n+1-m)/2 .
T(n,m) = A094930(n,m)/(3*m-1).
T(n,1) = A005449(n).
T(n,n) = A016768(n-1).
Row sums: sum_{m=1..n} T(n,m) = n^2*(n+1) = A011379(n).

Edited and extended, A-numbers corrected by R. J. Mathar, Oct 11 2009

A143208 a(1)=2; for n>1, a(n) = (4-9n+3n^2)/2.

Original entry on oeis.org

2, -1, 2, 8, 17, 29, 44, 62, 83, 107, 134, 164, 197, 233, 272, 314, 359, 407, 458, 512, 569, 629, 692, 758, 827, 899, 974, 1052, 1133, 1217, 1304, 1394, 1487, 1583, 1682, 1784, 1889, 1997, 2108, 2222, 2339, 2459, 2582, 2708, 2837, 2969, 3104, 3242, 3383, 3527, 3674

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 20 2008

Old Name was: A sequence based on odd numbers of the type 3*n + 2: a(n) = a(n - 1) + n - 1; A000096; f(n) = 3*a(n)+2.

			G.f. = 2*x - x^2 + 2*x^3 + 8*x^4 + 17*x^5 + 29*x^6 + 44*x^7 + 62*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A095794.

[n eq 1 select 2 else (4-9*n+3*n^2)/2: n in [1..60]]; // G. C. Greubel, Jul 19 2024

a[0] = 0; a[1] = -1; a[n_] := a[n] = a[n - 1] + n - 1; a1 = Table[a[n], {n, 0, 30}]; f[n_] := 3*a[n] + 2; Table[f[n], {n, 0, 50}]
LinearRecurrence[{3,-3,1},{2,-1,2,8},60] (* Harvey P. Dale, Mar 22 2018 *)

Vec(x*(3*x^3-11*x^2+7*x-2)/(x-1)^3 + O(x^100)) \\ Colin Barker, Apr 14 2014

[(4-9*n+3*n^2)/2 + 3*int(n==1) for n in range(1,61)] # G. C. Greubel, Jul 19 2024

From Colin Barker, Apr 14 2014: (Start)
a(n) = (4 - 9*n + 3*n^2)/2 for n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
G.f.: x*(2 - 7*x + 11*x^2 - 3*x^3)/ (1-x)^3. (End).
a(n) = (n-2)*A095794(n) - (n-1)*A095794(n-1) for n>1. [Bruno Berselli, May 19 2015]
E.g.f.: (1/2)*(4 - 6*x + 3*x^2)*exp(x) - 2 + 3*x. - G. C. Greubel, Jul 19 2024

Better name and edits by Colin Barker and Joerg Arndt, Apr 14 2014

A341740 a(n) is the maximum value of the magic constant in a normal magic triangle of order n.

Original entry on oeis.org

12, 23, 37, 54, 74, 97, 123, 152, 184, 219, 257, 298, 342, 389, 439, 492, 548, 607, 669, 734, 802, 873, 947, 1024, 1104, 1187, 1273, 1362, 1454, 1549, 1647, 1748, 1852, 1959, 2069, 2182, 2298, 2417, 2539, 2664, 2792, 2923, 3057, 3194, 3334, 3477, 3623, 3772, 3924

Offset: 3

Stefano Spezia, Feb 18 2021

Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005449, A016777, A095794, A130808, A140090, A179805, A285009.

LinearRecurrence[{3,-3,1},{12,23,37},49]

O.g.f.: x^3*(12 - 13*x + 4*x^2)/(1 - x)^3.
E.g.f.: 3 + x - 2*x^2 - exp(x)*(6 - 4*x - 3*x^2)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
a(n) = (3*n^2 + n - 6)/2 for n > 2.
a(n) = A285009(n) + A016777(n-2) - 1 for n > 3.
a(n) = A095794(n) - 2 = A140090(n-1) - 1. - Hugo Pfoertner, Feb 18 2021

A027625 Numerator of n(n+5)/((n+2)(n+3)).

Original entry on oeis.org

0, 1, 7, 4, 6, 25, 11, 14, 52, 21, 25, 88, 34, 39, 133, 50, 56, 187, 69, 76, 250, 91, 99, 322, 116, 125, 403, 144, 154, 493, 175, 186, 592, 209, 221, 700, 246, 259, 817, 286, 300, 943, 329, 344, 1078, 375, 391, 1222

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A027626 (denominator), A095794, A115067, A179436.

[Numerator(n*(n+5)/((n+2)*(n+3))): n in [0..50]]; // Vincenzo Librandi, Mar 04 2014

CoefficientList[Series[x*(1+7*x+4*x^2+3*x^3+4*x^4-x^5-x^6-2*x^7)/(1-x^3)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 04 2014 *)
Numerator[25*Binomial[Range[0, 50]/5 +1, 2]/3] (* G. C. Greubel, Aug 05 2022 *)

a(n) = numerator(n*(n+5)/6); \\ Altug Alkan, Apr 18 2018

[numerator(n*(n+5)/6) for n in (0..50)] # G. C. Greubel, Aug 05 2022

G.f.: x*(1 + 7*x + 4*x^2 + 3*x^3 + 4*x^4 - x^5 - x^6 - 2*x^7)/(1 - x^3)^3.
a(n) = numerator of n*(n+5)/6. - Altug Alkan, Apr 18 2018
From Peter Bala, Aug 06 2022: (Start)
a(n) is quasi-polynomial in n:
  a(3*n) = (1/2)*n*(3*n+5) = A115067(n+1).
  a(3*n+1) = (1/2)*(n+2)*(3*n+1) = A095794(n+1).
  a(3*n+2) = (1/2)*(3*n+2)*(3*n+7) = A179436(n). (End)
Sum_{n>=1} 1/a(n) = 4*Pi/(15*sqrt(3)) + 87/50. - Amiram Eldar, Aug 11 2022

A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).

Original entry on oeis.org

-3, 1, 0, 1, 0, 3, 1, 0, 2, 0, 1, 0, 1, 0, -3, 1, 0, 0, 0, -5, 0, 1, 0, -1, 0, -6, 0, 3, 1, 0, -2, 0, -6, 0, 8, 0, 1, 0, -3, 0, -5, 0, 14, 0, -3, 1, 0, -4, 0, -3, 0, 20, 0, -11, 0, 1, 0, -5, 0, 0, 0, 25, 0, -25, 0, 3, 1, 0, -6

Offset: 0

Paul Curtz, Jul 14 2011

Consider an array with recurrence BB(m,n) = BB(m,n-1) + BB(m-1,n), m >= 0:
  3, -1, -1, -1,  -1,  -1,  -1,  -1,  -1,  -1,   -1,
  3,  2,  1,  0,  -1,  -2,  -3,  -4,  -5,  -6,   -7,
  3,  5,  6,  6,   5,   3,   0,  -4,  -9, -15,  -22,
  3,  8, 14, 20,  25,  28,  28,  24,  15,   0,  -22,
  3, 11, 25, 45,  70,  98, 126, 150, 165, 165,  143,
  3, 14, 39, 84, 154, 252, 378, 528, 693, 858, 1001,
with BB(m,n) = (3m-n)*binomial(n+m-1,n)/m if m > 0. So the BB are polynomials of degree m in n:
BB(1,n) = -(n-3)/1,
BB(2,n) = -(n-6)*(n+1)/2,    (see A055999)
BB(3,n) = -(n-9)*(n+1)*(n+2)/6,
BB(4,n) = -(n-12)*(n+1)*(n+2)*(n+3)/24,
BB(5,n) = -(n-15)*(n+1)*(n+2)*(n+3)*(n+4)/120.
Columns in the array are A010701, A016789, A095794, A005564, A059302.
T(n,k) is a zero-padded, column-shifted, sign-modified transpose of this array.

			Triangle begins
  -3;
   1,   0;
   1,   0,   3;
   1,   0,   2,   0;
   1,   0,   1,   0,  -3;
   1,   0,   0,   0,  -5,   0;
   1,   0,  -1,   0,  -6,   0,   3;
   1,   0,  -2,   0,  -6,   0,   8,   0;
   1,   0,  -3,   0,  -5,   0,  14,   0,  -3;
   1,   0,  -4,   0,  -3,   0,  20,   0, -11,   0;

Cf. A174559.

BB := proc(m,n) if m=0 then if n= 0 then 3 ; else -1; end if; else (3*m-n)*binomial(n+m-1,n)/m ; end if; end proc:
A193002 := proc(n,k) if type(k,'odd') then 0; else (-1)^(1+k/2)*BB(k/2,n-k) ; end if; end proc:
seq(seq(A193002(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 30 2011

Sum_{k=0..n} T(n,k) = A130806(n+5). (row sums)
Sum_{k=0..n} (-1)^(k/2)*T(n,k) = -A000032(n-2). (alternating row sums)
T(n,k) = (-1)^(1+k/2)*BB(k/2,n-k). - R. J. Mathar, Aug 30 2011
T(n,2k) = (-1)^(1+k)*(5-n/k)*binomial(n-k-1,k-1), k > 0. - R. J. Mathar, Aug 30 2011

A236343 Expansion of (1 - x + 2x^2 - x^3) / ((1 - x)^2 (1 - x^3)) in powers of x.

Original entry on oeis.org

1, 1, 3, 5, 6, 9, 12, 14, 18, 22, 25, 30, 35, 39, 45, 51, 56, 63, 70, 76, 84, 92, 99, 108, 117, 125, 135, 145, 154, 165, 176, 186, 198, 210, 221, 234, 247, 259, 273, 287, 300, 315, 330, 344, 360, 376, 391, 408, 425, 441, 459, 477, 494, 513, 532, 550, 570, 590

Offset: 0

Michael Somos, Jan 22 2014

The sequence is a quasi-polynomial sequence.

Given a sequence of Laurent polynomials defined by b(n) = (b(n-2)^2 - b(n-1)*b(n-3) * 2/x) / b(n-4), b(-2) = x, b(-4) = -b(-3) = -b(-1) = 1. Then the denominator of b(n) is x^a(n).

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 6*x^4 + 9*x^5 + 12*x^6 + 14*x^7 + 18*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A236337. Trisections are A000326, A095794, A045943.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+2*x^2-x^3)/((1-x)^2*(1-x^3)))); // G. C. Greubel, Aug 07 2018

seq(coeff(series((1-x+2*x^2-x^3)/((1-x)^2*(1-x^3)),x,n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Feb 12 2019

CoefficientList[Series[(1-x+2*x^2-x^3)/((1-x)^2*(1-x^3)), {x, 0, 60}], x] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = (n * (n+5) + [6, 0, 4][n%3 + 1]) / 6};

{a(n) = if( n<0, polcoeff( x^2 * (-1 + 2*x - x^2 + x^3) / ((1 - x)^2 * (1 - x^3)) + x * O(x^-n), -n), polcoeff( (1 - x + 2*x^2 - x^3) / ((1 - x)^2 * (1 - x^3)) + x * O(x^n), n))};

((1-x+2*x^2-x^3)/((1-x)^2*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z.
G.f.: (1 - x + 2*x^2 - x^3) / ((1 - x)^2 * (1 - x^3)).
Second difference is period 3 sequence [2, 0, -1, ...].
a(n) = 2*a(n-3) + a(n-6) + 3 = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(-6-n) = A236337(n).
From Peter Bala, Feb 11 2019: (Start)
a(3*n)   = (1/2)*(n + 1)*(3*n + 2);
a(3*n+1) = (1/2)*(n + 1)*(3*n + 4) - 1;
a(3*n+2) = (1/2)*(n + 1)*(3*n + 6). (End)

A272058 Start with all terms set to 0. Then add n to the next n+3 terms for n=0,1,2,... .

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 14, 20, 25, 33, 39, 49, 56, 68, 76, 90, 99, 115, 125, 143, 154, 174, 186, 208, 221, 245, 259, 285, 300, 328, 344, 374, 391, 423, 441, 475, 494, 530, 550, 588, 609, 649, 671, 713, 736, 780, 804, 850, 875, 923, 949, 999, 1026, 1078, 1106

Offset: 0

Wesley Ivan Hurt, Apr 19 2016

			n  | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10, ...
__________________________________________
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
+       0, 0, 0
+          1, 1, 1, 1,
+             2, 2, 2, 2, 2
+                3, 3, 3, 3, 3, 3
+                   4, 4, 4, 4, 4, 4, 4
+                      5, 5, 5, 5, 5, 5, 5, 5
+                         6, 6, 6, 6, 6, 6, 6, 6, 6
+                            7, 7, 7, 7, 7, 7, 7, 7, 7, 7
+                               8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
+                                  9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
+                                     ...
__________________________________________
a(n)|0, 0, 1, 3, 6,10,14,20,25,33,39, ...

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

Cf. A095794, A258087.

[0,0] cat [(6*n^2+6*n-23+(7-2*n)*(-1)^n)/16 : n in [2..100]];

A272058:=n->(6*n^2+6*n-23+(7-2*n)*(-1)^n)/16: 0,0,seq(A272058(n),n=2..100);

CoefficientList[Series[x^2*(1 + 2 x + x^2 - x^4)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1, 0, 0}, {0, 0, 1, 3, 6, 10, 14}, 60]

G.f.: x^2*(1 + 2*x + x^2 - x^4)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 6.
a(n) = (6*n^2 + 6*n - 23 + (7 - 2*n)*(-1)^n)/16 for n > 1.
a(n) = floor((n+3)/4) * floor((3*n-4)/2) + (floor((n-1)/2) mod 2) * floor((3*n-3)/4) for n > 1.
For n > 1, a(2n) = A095794(n). - Jon E. Schoenfield, Feb 19 2022

cp's OEIS Frontend

A216172 Number of all possible tetrahedra of any size, having reverse orientation to the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts.

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A000338 Expansion of x^3*(5-2*x)*(1-x^3)/(1-x)^4.

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A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

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A096037 Triangle T(n,m) = (3*n+3*m-2)*(n+1-m)/2 read by rows.

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A143208 a(1)=2; for n>1, a(n) = (4-9*n+3*n^2)/2.

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A341740 a(n) is the maximum value of the magic constant in a normal magic triangle of order n.

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A027625 Numerator of n*(n+5)/((n+2)*(n+3)).

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A000338 Expansion of x^3(5-2x)*(1-x^3)/(1-x)^4.

A096037 Triangle T(n,m) = (3n+3m-2)*(n+1-m)/2 read by rows.

A143208 a(1)=2; for n>1, a(n) = (4-9n+3n^2)/2.

A027625 Numerator of n(n+5)/((n+2)(n+3)).

A236343 Expansion of (1 - x + 2x^2 - x^3) / ((1 - x)^2 (1 - x^3)) in powers of x.