A000292 -id:A000292 - cp's OEIS frontend

A140236 a(n) = A000292(A000292(n)).

0, 1, 20, 220, 1540, 7770, 30856, 102340, 295240, 762355, 1798940, 3939936, 8104460, 15803060, 29426320, 52636760, 90889616, 152112005, 247574180, 392991060, 609896980, 927341646, 1383960600, 2030479100, 2932714200, 4175145975, 5865135276, 8137872120

Offset: 0

Artur Jasinski, May 14 2008

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row n=3 of A331436.

Cf. A000292.

a:= (n-> binomial(n+2,3))@@2:
seq(a(n), n=0..29);  # Alois P. Heinz, Mar 11 2024

Table[(n (1 + n) (2 + n) (3 + n) (2 + n^2) (12 + n (1 + n) (2 + n)))/1296,{n,0,20}]

b(n)=n*(n+1)*(n+2)/6;
a(n)=b(b(n));
vector(25,n,a(n-1)) \\ Joerg Arndt, Mar 11 2024

a(n) = (n*(1 + n)*(2 + n)*(3 + n)*(2 + n^2)*(12 + n*(1 + n)*(2 + n)))/1296.

G.f.: x*(1+10*x+65*x^2+120*x^3+80*x^4+4*x^5)/(1-x)^10. - Colin Barker, Apr 30 2012.

A118392 Denominator of sum of reciprocals of first n tetrahedral numbers A000292.

Original entry on oeis.org

1, 4, 20, 5, 7, 56, 24, 15, 55, 44, 52, 91, 35, 80, 272, 51, 57, 380, 140, 77, 253, 184, 200, 325, 117, 252, 812, 145, 155, 992, 352, 187, 595, 420, 444, 703, 247, 520, 1640, 287, 301, 1892, 660, 345, 1081, 752, 784, 1225, 425, 884, 2756, 477, 495, 3080

Offset: 1

Jonathan Vos Post, Apr 27 2006

Numerators are A118391.

			a(1) = 1 = denominator of 1/1.
a(2) = 4 = denominator of 5/4 = 1/1 + 1/4.
a(3) = 20 = denominator of 27/20 = 1/1 + 1/4 + 1/10.
a(4) = 5 = denominator of 7/5 = 1/1 + 1/4 + 1/10 + 1/20.
a(5) = 7 = denominator of 10/7 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35.
a(20) = 77 = denominator of 115/77 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35 + 1/56 + 1/84 + 1/120 + 1/165 + 1/220 + 1/286 + 1/364 + 1/455 + 1/560 + 1/680 + 1/816 + 1/969 + 1/1140 + 1/1330 + 1/1540.
Fractions are: 1/1, 5/4, 27/20, 7/5, 10/7, 81/56, 35/24, 22/15, 81/55, 65/44, 77/52, 135/91, 52/35, 119/80, 405/272, 76/51, 85/57, 567/380, 209/140, 115/77, 378/253, 275/184, 299/200, 486/325, 175/117, 377/252, 1215/812, 217/145, 232/155, 1485/992.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000292, A022998, A026741, A118391.

[Denominator(3*n*(n+3)/(2*(n+1)*(n+2))): n in [1..60]]; // G. C. Greubel, Feb 18 2021

A118392:= n -> denom(3*n*(n+3)/(2*(n+1)*(n+2)));
seq(A118392(n), n = 1..60); # G. C. Greubel, Feb 18 2021

Accumulate[1/Binomial[Range[70]+2,3]]//Denominator (* Harvey P. Dale, Jun 07 2018 *)

s=0;for(i=3,50,s+=1/binomial(i,3);print(denominator(s))) /* Phil Carmody, Mar 27 2012 */

[denominator(3*n*(n+3)/(2*(n+1)*(n+2))) for n in (1..60)] # G. C. Greubel, Feb 18 2021

A118391(n)/A118392(n) = Sum_{i=1..n} 1/A000292(n).
A118391(n)/A118392(n) = Sum_{i=1..n} 1/C(n+2,3).
A118391(n)/A118392(n) = Sum_{i=1..n} 6/(n*(n+1)*(n+2)).
a(n) = denominator( 3*n*(n+3)/(2*(n+1)*(n+2)) ). - G. C. Greubel, Feb 18 2021

More terms from Harvey P. Dale, Jun 07 2018

A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

Original entry on oeis.org

0, 0, 4, 24, 84, 224, 516, 1068, 2016, 3528, 5832, 9256, 14208, 21180, 30728, 43488, 60192, 81660, 108828, 142764, 184708, 236088, 298476, 373652, 463524, 570228, 696012, 843312, 1014720, 1213096, 1441512, 1703352, 2002196, 2341848, 2726400, 3160272, 3648180

Offset: 0

Peter Kagey, May 06 2020

nonn

a(n) >= 4 * A269747(n).
a(n) >= 4 * A000389(n+3) = A210569(n+2).
a(n) >= 4 * (n-1) + 4 * a(n-1) - 6 * a(n-2) + 4 * a(n-3) - a(n-4) for n >= 4.

Peter Kagey, Table of n, a(n) for n = 0..1000 (Computed via Anders Kaseorg's program in the Stack Exchange link.)
Code Golf Stack Exchange, Triangles in a tetrahedron

Cf. A000292, A000389, A210569.
Cf. A000332 (equilateral triangles in triangular grid), A269747 (regular tetrahedra in a tetrahedral grid), A102698 (equilateral triangles in cube), A103158 (regular tetrahedra in cube).
Cf. A077435, A085582, A187452, A189412-A189418, A271910.

A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 4, 1, 1, 0, 4, 2, 1, 10, 4, 5, 3, 1, 0, 10, 8, 7, 4, 1, 20, 10, 14, 13, 10, 5, 1, 0, 20, 20, 22, 20, 14, 6, 1, 35, 20, 30, 34, 35, 30, 19, 7, 1, 0, 35, 40, 50, 56, 55, 44, 25, 8, 1, 56, 35, 55, 70, 84, 91, 85, 63, 32, 9, 1, 0, 56, 70, 95, 120, 140, 146, 129, 88, 40, 10, 1

Offset: 0

Henry Bottomley, Nov 20 2000

			Rows are (1,0,4,0,10,0,20,...), (1,1,4,4,10,10,20,...), (1,2,5,8,14,20,30,...), (1,3,7,13,22,34,50,...), (1,4,10,20,35,56,84,...) etc.

Rows are A000292 with zeros, A058187 (A000292 with terms duplicated), A006918, A002623, A000292, A000330, A005900, A001845, A008412.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(4, 1)=4, T(0, 2n)=T(4, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^4.

A113067 Expansion of -x/((x^2+x+1)(x^2+3x+1)); invert transform gives signed version of tetrahedral numbers A000292.

Original entry on oeis.org

0, -1, 4, -11, 28, -72, 188, -493, 1292, -3383, 8856, -23184, 60696, -158905, 416020, -1089155, 2851444, -7465176, 19544084, -51167077, 133957148, -350704367, 918155952, -2403763488, 6293134512, -16475640049, 43133785636, -112925716859, 295643364940, -774004377960

Offset: 0

Creighton Dement, Oct 13 2005

Invert(a(n)) gives (0, -1, 4, -10, 20, -35, ...) = A000292 (with alternating signs).
Binomial(a(n)) gives (0, -1, 2, -2, 4, -7, 10, ...) = A094686 (with alternating signs, from 2nd term).
Floretion Algebra Multiplication Program, FAMP Code: 2basei[C*F]; C = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; F = + .5'i + .5'ii' + .5'ij' + .5'ik'

Creighton Dement, Floretion Integer Sequences (work in progress).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,-5,-4,-1).

Cf. A000292, A113067, A113068, A094686, A001906, A109961, A290890.

-x/((x^2+x+1)*(x^2+3*x+1)) + O[x]^30 // CoefficientList[#, x]& (* Jean-François Alcover, Jun 15 2017 *)

concat(0, Vec(-x / ((1 + x + x^2)*(1 + 3*x + x^2)) + O(x^30))) \\ Colin Barker, May 11 2019

[((lucas_number1(n,3,1)-lucas_number1(n,1,1)))/(-2) for n in range(1,32)] # Zerinvary Lajos, Jul 06 2008

a(n) + a(n+1) + a(n+2) = (-1)^n *A001906(n+2) = (-1)^n*F(2n+4).
a(n) + 3*a(n+1) + 3*a(n+2) + a(n+3) = ((-1)^(n+1))*A109961(n+2).
(|a(n)|) = A290890(n) for n >= 0, this being the p-INVERT of (1,2,3,4,...), where p(S) = 1 - S^2.  - Clark Kimberling, Aug 21 2017
a(n) = -4*a(n-1) - 5*a(n-2) - 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, May 11 2019
2*a(n) = (-1)^n*A001906(n+1) - A049347(n). - R. J. Mathar, Sep 20 2020

A185904 Multiplication table for the tetrahedral numbers (A000292), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 10, 16, 10, 20, 40, 40, 20, 35, 80, 100, 80, 35, 56, 140, 200, 200, 140, 56, 84, 224, 350, 400, 350, 224, 84, 120, 336, 560, 700, 700, 560, 336, 120, 165, 480, 840, 1120, 1225, 1120, 840, 480, 165, 220, 660, 1200, 1680, 1960, 1960, 1680, 1200, 660, 220, 286, 880, 1650, 2400, 2940, 3136, 2940, 2400, 1650, 880, 286, 364, 1144, 2200, 3300, 4200, 4704, 4704, 4200, 3300, 2200, 1144, 364, 455, 1456, 2860, 4400, 5775, 6720, 7056, 6720, 5775, 4400, 2860, 1456, 455, 560, 1820, 3640, 5720, 7700, 9240, 10080, 10080, 9240, 7700, 5720, 3640, 1820, 560

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185906 < A000007 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,  4,  10,  20,  35
   4, 16,  40,  80, 140
  10, 40, 100, 200, 350
  20, 80, 200, 400, 700

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000007, A003991, A098358, A144112, A185905, A185906, A185907.

Row 1 = Column 1 = A000292.

(* This program generates A098358 and its accumulation array, A185904. *)
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A098358 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185904 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185904 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
T[n_, k_] := Binomial[k + 2, 3]*Binomial[n + 2, 3]; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

T(n,k) = binomial(k+2,3)*binomial(n+2,3), k >= 1, n >= 1.

A080249 Stirling-like number triangle defined by the sequence A000292=C(n+3,3).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 21, 15, 1, 1, 85, 171, 35, 1, 1, 341, 1795, 871, 70, 1, 1, 1365, 18291, 19215, 3321, 126, 1, 1, 5461, 184275, 402591, 135450, 10377, 210, 1, 1, 21845, 1848211, 8236095, 5143341, 716562, 28017, 330, 1, 1, 87381, 18503955, 166570111, 188253030, 45270813, 3069990, 67617, 495, 1

Offset: 0

Paul Barry, Feb 17 2003

Columns include A002450, A016225. The defining sequence A000292=C(n+3,3) is the sequence of partial sums of the defining sequence for number triangle A080248.

			Triangle begins:
1;
1,    1;
1,    5,      1;
1,   21,     15,      1;
1,   85,    171,     35,      1;
1,  341,   1795,    871,     70,     1;
1, 1365,  18291,  19215,   3321,   126,   1;
1, 5461, 184275, 402591, 135450, 10377, 210, 1;
For example, 171 = 21+10*15, 35 = 15+20*1.

Cf. A080248, A008277.

T(n,k) = T(n-1,k-1) + A000292(k)*T(n-1,k). Columns are generated by 1/product{k=0..n, 1-C(k+3,3)*x}.

A127774 Triangle read by rows: row n consists of n-1 zeros followed by A000292(n).

Original entry on oeis.org

1, 0, 4, 0, 0, 10, 0, 0, 0, 20, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 0, 0, 0, 0, 165, 0, 0, 0, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 364

Offset: 1

Gary W. Adamson, Jan 28 2007

Essentially triangle T(n,k), read by rows, given by (0,0,0,0,0,0,0,...) DELTA (4,-3/2,5/6,-1/3,3/5,-1/10,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 14 2011

			First few rows of the triangle are:
  1;
  0,  4;
  0,  0, 10;
  0,  0,  0, 20;
  0,  0,  0,  0, 35;
  ...

Cf. A000292, A127648, A127773.

from math import isqrt
from sympy.ntheory.primetest import is_square
def A127774(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(a+1)*(a+2)//6 if is_square((n<<3)+1) else 0 # Chai Wah Wu, Jun 09 2025

G.f.: 1/((x*y-1)^4). - R. J. Mathar, Aug 12 2015

More terms from Michel Marcus, Jun 10 2025

A147621 The 3rd Witt transform of A000292.

Original entry on oeis.org

0, 0, 0, 0, 4, 26, 120, 455, 1456, 4122, 10608, 25194, 55980, 117572, 235144, 450681, 832048, 1485800, 2575368, 4345965, 7158060, 11532402, 18209100, 28224105, 43008120, 64512240, 95365920, 139075245, 200268432, 284997384, 401107356

Offset: 0

R. J. Mathar, Nov 08 2008

The 2nd Witt transform is essentially in A032094.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Index entries for linear recurrences with constant coefficients, signature (8,-28,60,-102,168,-258,336,-393,452,-484,452,-393, 336,-258,168,-102,60,-28,8,-1).

Cf. A000292, A032094, A049347, A099254, A128504.

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) )); // G. C. Greubel, Oct 24 2022

CoefficientList[Series[x^4(2*x^2 - x + 2)(2*x^4 - 2*x^3 + 9*x^2 - 2*x+2)/((1-x)^12 * (1 + x + x^2)^4), {x, 0, 40}],  x] (* Vincenzo Librandi  Dec 13 2012 *)

def A147621_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) ).list()
A147621_list(40) # G. C. Greubel, Oct 24 2022

G.f.: x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4).

a(n) = (1/729)*(b(n) + c(n)), where b(n) = n*(n+3)*(n+6)*(3*n^8 +72*n^7 +618*n^6 + 2052*n^5 +207*n^4 -11772*n^3 -14268*n^2 +9648*n -232960)/492800 and c(n) = 9*A049347(n) +5*A049347(n-1) +9*(-1)^n*(A099254(n) -A099254(n-1)) -18(-1)^n*A128504(n) +27*(-1)^n*Sum_{k=0..n} A099254(n-k)*A099254(k-1). - G. C. Greubel, Oct 24 2022

A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)(n-k+1)(n-k+2)/2, read by rows.

Original entry on oeis.org

1, 4, 1, 10, 3, 2, 20, 6, 6, 3, 35, 10, 12, 9, 4, 56, 15, 20, 18, 12, 5, 84, 21, 30, 30, 24, 15, 6, 120, 28, 42, 45, 40, 30, 18, 7, 165, 36, 56, 63, 60, 50, 36, 21, 8, 220, 45, 72, 84, 84, 75, 60, 42, 24, 9, 286, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10, 364, 66, 110, 135, 144, 140, 126, 105, 80, 54, 30, 11

Offset: 1

Gary W. Adamson & Roger L. Bagula, Mar 28 2009

The triangle can also be defined by multiplying the triangles A(n,k)=1 and A158823(n,k), that is, this here are the partial column sums of A158823.

			First few rows of the triangle are:
    1;
    4,  1;
   10,  3,   2;
   20,  6,   6,   3;
   35, 10,  12,   9,   4;
   56, 15,  20,  18,  12,   5;
   84, 21,  30,  30,  24,  15,   6;
  120, 28,  42,  45,  40,  30,  18,   7;
  165, 36,  56,  63,  60,  50,  36,  21,   8;
  220, 45,  72,  84,  84,  75,  60,  42,  24,  9;
  286, 55,  90, 108, 112, 105,  90,  70,  48, 27, 10;
  364, 66, 110, 135, 144, 140, 126, 105,  80, 54, 30, 11;
  455, 78, 132, 165, 180, 180, 168, 147, 120, 90, 60, 33, 12;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A062707, A104633, A158823.

Row sums: A000332.

A158824:= func< n,k | k eq 1 select Binomial(n+2,3) else (k-1)*Binomial(n-k+2,2) >; [A158824(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021

T[n_, k_]:= If[k==1, Binomial[n+2, 3], (k-1)*Binomial[n-k+2, 2]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

def A158824(n,k): return binomial(n+2,3) if k==1 else (k-1)*binomial(n-k+2,2)
flatten([[A158824(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

T(n,k) = binomial(n+2,3) if k = 1 otherwise (k-1)*binomial(n-k+2, 2).

Sum_{k=1..n} T(n, k) = binomial(n+3, 4) = A000332(n+3). - G. C. Greubel, Apr 01 2021

cp's OEIS Frontend

A140236 a(n) = A000292(A000292(n)).

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A118392 Denominator of sum of reciprocals of first n tetrahedral numbers A000292.

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A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

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A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

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A113067 Expansion of -x/((x^2+x+1)*(x^2+3*x+1)); invert transform gives signed version of tetrahedral numbers A000292.

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A185904 Multiplication table for the tetrahedral numbers (A000292), by antidiagonals.

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A080249 Stirling-like number triangle defined by the sequence A000292=C(n+3,3).

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A127774 Triangle read by rows: row n consists of n-1 zeros followed by A000292(n).

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A113067 Expansion of -x/((x^2+x+1)(x^2+3x+1)); invert transform gives signed version of tetrahedral numbers A000292.

A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)(n-k+1)(n-k+2)/2, read by rows.