%I A000292 M3382 N1363
%S A000292 0,1,4,10,20,35,56,84,120,165,220,286,364,455,560,680,816,969,1140,
%T A000292 1330,1540,1771,2024,2300,2600,2925,3276,3654,4060,4495,4960,5456,5984,
%U A000292 6545,7140,7770,8436,9139,9880,10660,11480,12341,13244,14190,15180
%N A000292 Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
%C A000292 The number of balls in a triangular pyramid in which each edge contains
n+1 balls. The sum of the first n triangular numbers (A000217).
%C A000292 Also (1/6)*(n^3+3*n^2+2*n) is the number of ways to color vertices of
a triangle using <= n colors, allowing rotations and reflections.
Group is the dihedral group D_6 with cycle index (x1^3+2*x3+3*x1*x2)/
6.
%C A000292 Also the convolution of the natural numbers with themselves - Felix Goldberg
(felixg(AT)tx.technion.ac.il), Feb 01 2001
%C A000292 Connected with the Eulerian numbers (1,4,1) via 1*a(x-2)+4*a(x-1)+1*a(x)
= x^3. - Gottfried Helms (helms(AT)uni-kassel.de), Apr 15 2002
%C A000292 a(n) = sum |i-j| for all 1 <= i <= j <= n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
Aug 05 2002
%C A000292 a(n) = sum of the all possible products p*q where (p,q) are ordered pairs
and p+q = n+1. a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
May 29 2003
%C A000292 Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry
(perry(AT)globalnet.co.uk), Jun 14 2003
%C A000292 Number of permutations of n+3 which have exactly 1 descent and avoid
the pattern 1324. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca),
Nov 05 2004
%C A000292 Schlaefli symbol for this polyhedron: {3,3}
%C A000292 Transform of n^2 under the Riordan array (1/(1-x^2),x). - Paul Barry
(pbarry(AT)wit.ie), Apr 16 2005
%C A000292 a(n) = -A108299(n+5,6) = A108299(n+6,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 01 2005
%C A000292 a(n) = -A110555(n+4,3). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jul 27 2005
%C A000292 a(n) is a perfect square only for n = {1, 2, 48}. a(48) = 19600 = 140^2.
- Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 24 2006
%C A000292 a(n+1) is the number of terms in the expansion of (a_1+a_2+a_3+a_4)^n
- Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007. (Corrected
by Graeme McRae (g_m(AT)mcraefamily.com), Aug 28 2007)
%C A000292 This is also the average "permutation entropy", sum((pi(n)-n)^2)/n!,
over the set of all possible n! permutations pi. - Jeff Boscole (jazzerciser(AT)hotmail.com),
Mar 20 2007
%C A000292 a(n)=diff(S(n,x),x)|_{x=2}. First derivative of Chebyshev S-polynomials
evaluated at x=2. See A049310. W. Lang, Apr 04 2007.
%C A000292 If X is an n-set and Y a fixed (n-1)-subset of X then a(n-2) is equal
to the number of 3-subsets of X intersecting Y. - Milan R. Janjic
(agnus(AT)blic.net), Aug 15 2007
%C A000292 Number of n-permutations (n=4) of 2 objects u, v, with repetition allowed,
containing exactly three (3) u's. Example: a(2)=4 because we have:
uuuv, uuvu, uvuu and vuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 12 2008
%C A000292 Complement of A145397; A023533(a(n))=1; A014306(a(n))=0. [From Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 14 2008]
%C A000292 Equals row sums of triangle A152205 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 29 2008]
%C A000292 a(n) is the number of gifts received from the lyricist's true love up
to and including day n in the song "The Twelve Days of Christmas".
a(12)=364, almost the number of days in the year. [From Bernard Hill
(bernard(AT)braeburn.co.uk), Dec 05 2008]
%C A000292 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07
2009: (Start)
%C A000292 Sequence of the absolute values of the z^1 coefficients of the polynomials
in the GF2 denominators of A156925. See A157703 for background information.
%C A000292 (End)
%C A000292 Starting with 1 = row sums of triangle A158823 [From Gary W. Adamson
(qntmpkt(AT)yahoo.com), Mar 28 2009]
%C A000292 Wiener index of the path graph P_n [From Eric W. Weisstein (eric(AT)weisstein.com),
Apr 30 2009]
%C A000292 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)
%C A000292 This is a 'Matryoshka doll' sequence with alpha=0, the multiplicative
counterpart is A000178
%C A000292 seq(add(add(i,i=alpha..k),k=alpha..n),n=alpha..50); (End)
%C A000292 a(n) is the number of non-decreasing, three-element permutations of n
distinct numbers. [From Samuel Savitz, Sep 12 2009]
%C A000292 a(n+4) = Number of different partitions of number n on sum of 4 elements
a(6)=a(2+4)becuse we have 10 different partionions 2 on sum of 4
elements 2=2+0+0+0=1+1+0+0=0+2+0+0=1+0+1+0=0+1+1+0=0+0+2+0=1+0+0+1=0+1+0+1=0+0+1+1=0+0+0+2
[From Artur Jasinski (grafix(AT)csl.pl), Nov 30 2009]
%D A000292 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000292 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000292 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 828.
%D A000292 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964,
p. 194.
%D A000292 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY,
1996, p. 83.
%D A000292 H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J.
Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical
and political essays in honor of Dirk J. Struik, Reidel, Dordrecht,
1974.
%D A000292 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public.
256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see
vol. 2, p. 4.
%D A000292 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq.
(1).
%D A000292 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society
Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000292 A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et
al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.
%D A000292 D. Wells, The Penguin Dictionary of Curious and interesting Numbers,
pp. 126-7 Penguin Books 1987.
%H A000292 N. J. A. Sloane, <a href="b000292.txt">Table of n, a(n) for n = 0..10000</
a>
%H A000292 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000292 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A000292 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000292 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A000292 O. Aichholzer and H. Krasser, <a href="http://www.ist.tugraz.at/publications/
oaich/psfiles/ak-psotd-01.ps.gz">The point set order type data base:
a collection of applications and results</a>, pp. 17-20 in Abstracts
13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo,
Aug. 13-15, 2001.
%H A000292 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000292 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/
abs/math.NT/0509316">On the Integrality of n-th Roots of Generating
Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H A000292 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviTetra">
First 2500 Tetrahedral numbers</a>
%H A000292 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
On Regular Polytope Numbers</a>
%H A000292 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</
a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000292 N. J. A. Sloane, <a href="a000292.gif">Illustration of initial terms</
a>
%H A000292 N. J. A. Sloane, <a href="a000292a.jpg">Pyramid of 20 balls corresponding
to a(3)=20.</a>
%H A000292 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
Geometri/Tetraedr.htm">Nombres Tetraedriques</a>
%H A000292 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TetrahedralNumber.html">Link to a section of The World of Mathematics
(1).</a>
%H A000292 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Composition.html">Link to a section of The World of Mathematics (2).</
a>
%H A000292 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000292 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
WienerIndex.html">Wiener Index</a> [From Eric W. Weisstein (eric(AT)weisstein.com),
Apr 30 2009]
%F A000292 Partial sums of the triangular numbers (A000217).
%F A000292 G.f.: x/(1-x)^4. a(-4-n)=-a(n).
%F A000292 a(n)=(n+3)/n*a(n-1) - Ralf Stephan, Apr 26 2003
%F A000292 Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26
2003
%F A000292 a(n)=C[1, 2, ]+C[2, 2]+...+C[n-1, 2]+C[n, 2]; n=5: a(5)=0+1+3+6+10=20.
- Labos E. (labos(AT)ana.sote.hu), May 09 2003
%F A000292 a(n)=sum{k=0..n, k(n-k)} (offset 1). - Paul Barry (pbarry(AT)wit.ie),
Jul 21 2003
%F A000292 Determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+2, i)
- Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003
%F A000292 The sum of a series constructed by the products of the index and the
length of the series (n) minus the index (i): a(n) = sum[i(n-i)].
Also the sum of n terms of A000217. - Martin Steven McCormick (mathseq(AT)wazer.net),
Apr 06 2005
%F A000292 a(n)=sum{k=0..floor((n-1)/2), (n-2k)^2} [offset 0]; a(n+1)=sum{k=0..n,
k^2*(1-(-1)^(n+k-1))/2} [offset 0]; - Paul Barry (pbarry(AT)wit.ie),
Apr 16 2005
%F A000292 C(3+n,3)-C(2+n,2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May
08 2006
%F A000292 Values of the Verlinde formula for SL_2, with g=2: a(n)=sum(j=1, n-1,
n/(2*sin^2(j*Pi/n))) - Simone Severini (ss54(AT)york.ac.uk), Sep
25 2006
%F A000292 a(n) = Sum[ Sum[ k, {k,1,m} ], {m,1,n} ]. - Alexander Adamchuk (alex(AT)kolmogorov.com),
Oct 28 2006
%F A000292 a(n)=Sum{k=1..n} binomial(n*k+1,n*k-1), with a(0)=0. - Paolo P. Lava
(ppl(AT)spl.at), Apr 13 2007
%F A000292 a(n)=numbperm(n,3)/6, n>=2 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 26 2007
%F A000292 a(n-1) = 1/(1!*2!)*sum {1 <= x_1, x_2 <= n} |det V(x_1,x_2)| = 1/2*sum
{1 <= i,j <= n} |i-j|, where V(x_1,x_2} is the Vandermonde matrix
of order 2. Column 2 of A133112. - Peter Bala (pbala(AT)toucansurf.com),
Sep 13 2007
%F A000292 Starting with 1 = binomial transform of [1, 3, 3, 1,...]; e.g. a(4) =
20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - Gary W. Adamson
(qntmpkt(AT)yahoo.com), Nov 04 2007
%F A000292 a(n) = A006503(n) - A002378(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Sep 24 2008]
%F A000292 a(0)=0, a(1)=1, a(2)=4, a(3)=10, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4)
for n>=4. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com),
Nov 18 2008]
%F A000292 sum_{n=1..infinity} 1/a(n) = 3/2, case x=1 in Gradstein-Ryshik 1.513.7.
[From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009]
%F A000292 E.g.f.:((x^3)/6+x^2+x)*exp(x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org),
Feb 21 2009]
%e A000292 a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of
balls, 6 in a triangle at the bottom, 3 in the middle layer and 1
on top.
%e A000292 Consider the square array
%e A000292 1 2 3 4 5 6...
%e A000292 2 4 6 8 10 12...
%e A000292 3 6 9 12 16 20...
%e A000292 4 8 12 16 20 24...
%e A000292 5 10 15 20 25 30...
%e A000292 ...
%e A000292 then a(n) = sum of n-th antidiagonal. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
Apr 06 2003
%p A000292 A000292 := n->binomial(n+3,3);
%p A000292 Or, f:=n->(1/6)*(n^3+3*n^2+2*n);
%p A000292 a:=n->sum ((j+n)*(n+2)/9,j=0..n): seq(a(n),n=0..44); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Dec 17 2006
%p A000292 Table[((n^3 - n)/6), {n,1, 45}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Mar 21 2007
%p A000292 ZL := [S, {S=Prod(B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL,
size=n), n=3..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Mar 13 2007
%p A000292 a:=n->sum(numbperm (n,2)/6, j=0..n): seq(a(n), n=1..45); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007
%p A000292 seq(numbperm (n,3)/6, n=2..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 26 2007
%p A000292 seq(sum(binomial(n,k+1),k=2..2),n=2..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Dec 14 2007
%p A000292 a:=n->sum(j^2-j, j=0..n): seq(a(n)/2, n=1..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 08 2008
%p A000292 seq(binomial(n+3,3)*1^n,n=-1..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 12 2008
%p A000292 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07
2009: (Start)
%p A000292 nmax:=44; for n from 0 to nmax do fz(n):=product((1-m*z)^(n+1-m),m=1..n);
c(n):= abs(coeff(fz(n),z,1)); end do: a:=n-> c(n): seq(a(n), n=0..nmax);
%p A000292 (End)
%p A000292 restart: G(x):=x^3*exp(x): f[0]:=G(x): for n from 1 to 56 do f[n]:=diff(f[n-1],
x) od: x:=0: seq(f[n]/3!,n=2..46);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 05 2009]
%t A000292 Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[50]]]]]
%t A000292 Table[Sum[(n - i)*i, {i, 0, n}], {n, 1, 55}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jul 11 2009]
%t A000292 Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[0, 44]]]]] [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
%t A000292 lst = {}; Do[AppendTo[lst, GegenbauerC[n, 2, 1]], {n, -1, 43}]; lst [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 14 2009]
%t A000292 a=0;Table[(a=n^2-n+a)/2,{n,90}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com),
Nov 18 2009]
%t A000292 b = {}; Do[a = {}; Do[Do[Do[Do[If[m + n + p + r == k, AppendTo[a, {m,
n, p, r}]], {m, 0, k}], {n, 0, k}], {p, 0, k}], {r, 0, k}]; AppendTo[b,
Length[a]], {k, 0, 10}]; b [From Artur Jasinski (grafix(AT)csl.pl),
Nov 30 2009]
%o A000292 (PARI) a(n)=(n)*(n+1)*(n+2)/6
%Y A000292 Sums of 2 consecutive terms give A000330.
%Y A000292 a(3n-3)=A006566(n). A000447(n)=a(2n-2). A002492(n)=a(2n+1).
%Y A000292 First differences give triangular numbers.
%Y A000292 Column 0 of triangle A094415.
%Y A000292 Cf. A000217, A001044, A003991, A061552.
%Y A000292 Cf. A040977, A133111, A133112.
%Y A000292 A152205 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]
%Y A000292 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07
2009: (Start)
%Y A000292 Cf. A156925, A157703.
%Y A000292 (End)
%Y A000292 A158823 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]
%Y A000292 Sequence in context: A138778 A038409 A090579 this_sequence A101552 A038419
A057319
%Y A000292 Adjacent sequences: A000289 A000290 A000291 this_sequence A000293 A000294
A000295
%K A000292 nonn,core,easy,nice,new
%O A000292 0,3
%A A000292 N. J. A. Sloane (njas(AT)research.att.com).
%E A000292 More terms from Michael Somos
%E A000292 Corrected PARI program. - Harry J. Smith (hjsmithh(AT)sbcglobal.net),
Dec 22 2008
%E A000292 Multiplied g.f. with x to match the offset R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Apr 23 2009
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