%I A000330 M3844 N1574
%S A000330 0,1,5,14,30,55,91,140,204,285,385,506,650,819,1015,1240,1496,1785,2109,
%T A000330 2470,2870,3311,3795,4324,4900,5525,6201,6930,7714,8555,9455,10416,11440,
%U A000330 12529,13685,14910,16206,17575,19019,20540,22140,23821,25585,27434,29370
%N A000330 Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
%C A000330 The sequence contains exactly one square greater than 1, namely 4900
(according to Gardner). - Jud McCranie (j.mccranie(AT)comcast.net),
Mar 19 2001, Mar 22 2007
%C A000330 Number of rhombi in an n X n rhombus. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be),
May 14 2000
%C A000330 Number of acute triangles made from the vertices of a regular n-polygon
when n is odd (cf. A007290). - Sen-Peng You (giawgwan(AT)single.url.com.tw),
Apr 05 2001
%C A000330 Gives number of squares formed from an n X n square. In a 1 X 1 square,
one is formed. In a 2 X 2 square, five squares are formed. In a 3
X 3 square, 14 squares are formed and so on. - Kristie Smith (kristie10spud(AT)hotmail.com),
Apr 16 2002
%C A000330 a(n-1)=B_3(n)/3 where B_3(x)=x(x-1)(x-1/2) is the third Bernoulli polynomial.
- Michael Somos Mar 13 2004
%C A000330 Number of permutations avoiding 13-2 that contain the pattern 32-1 exactly
once.
%C A000330 Since 3*r = (r+1)+r+(r-1) = T(r+1)-T(r-2), where T(r) = r-th triangular
number r*(r+1)/2, we have 3*r^2 = r*{T(r+1)-T(r-2)} = f(r+1)-f(r-1)......(i),
where f(r) = (r-1)*T(r) = (r+1)*T(r-1). Summing over n, R.H.S. of
relation (i) telescopes to f(n+1)+f(n) = T(n)*{(n+2)+(n-1)}, whence
result sum_(1, n)r^2 = n*(n+1)*(2*n+1)/6 immediately follows. - Lekraj
Beedassy (blekraj(AT)yahoo.com), Aug 06 2004
%C A000330 Also as a(n)=(1/6)*(2*n^3+3*n^2+n), n>0: structured trigonal diamond
numbers (vertex structure 5) (Cf. A006003 = alternate vertex; A000447
= structured diamonds; A100145 for more on structured numbers). -
James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.
%C A000330 Number of triples of integers from {1,2,...,n} whose last component is
greater than or equal to the others.
%C A000330 Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Jun 12 2005
%C A000330 Sum of the first n squares, or square pyramidal numbers. - Cino Hilliard
(hillcino368(AT)hotmail.com), Jun 18 2007
%C A000330 Maximal number of cubes of side 1 in a right pyramid with a square base
of side n and height n - Pasquale CUTOLO (p.cutolo(AT)inwind.it),
Jul 09 2007
%C A000330 If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then
a(n-3) is the number of 4-subsets of X intersecting both Y and Z.
- Milan R. Janjic (agnus(AT)blic.net), Sep 19 2007
%C A000330 We also have the identity (1+(1+4)+(1+4+9)+..+(1+4+9+16+ .. + n^2)=n(n+1)(n+2)[n+(n+1)+(n+2)]/
36; .. and in general the k-fold nested sum of squares can be expressed
as n(n+1)...(n+k)[n+(n+1)+...+(n+k)]/((k+2)!(k+1)/2 ) - Alexander
R. Povolotsky (pevnev(AT)juno.com), Nov 21 2007
%C A000330 The terms of this sequence are coefficients of the Engel expansion of
the following converging sum: 1/(1^2) + (1/1^2)*(1/(1^2+2^2)) + (1/
1^2)*(1/(1^2+2^2))*(1/(1^2+2^2+3^2)) + .. - Alexander R. Povolotsky
(pevnev(AT)juno.com), Dec 10 2007
%C A000330 Starting (1, 5, 14, 30,...) = binomial transform of [1, 4, 5, 2, 0, 0,
0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2008
%C A000330 Starting (1,5,14,30,...) = second partial sums of binomial transform
of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+2,i+2)*b(i)}, where b(i)=1,
2,0,0,0,... [From Borislav St. Borisov (b.st.borisov(AT)abv.bg),
Mar 05 2009]
%C A000330 Convolution of A001477 with A005408: a(n)=SUM((2*k+1)*(n-k):0<=k<=n).
[From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 07
2009]
%C A000330 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07
2009: (Start)
%C A000330 Sequence of the absolute values of the z^1 coefficients of the polynomials
in the GF1 denominators of A156921. See A157702 for background information.
%C A000330 (End)
%D A000330 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000330 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000330 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques
Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%D A000330 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 813.
%D A000330 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964,
p. 194.
%D A000330 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of
combinatorial proof, M.A.A. 2003, id. 215,223.
%D A000330 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 122, see #19 (3(1)),
I(n); p. 155.
%D A000330 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 A000330 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons,
Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
%D A000330 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. 2.
%D A000330 M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, pg
293.
%D A000330 Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order
Symmetric Polynomials, Applicable Algebra in Engineering, Communication
and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%H A000330 T. D. Noe, <a href="b000330.txt">Table of n, a(n) for n = 0..1000</a>
%H A000330 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000330 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A000330 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000330 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 A000330 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000330 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000330 H. Bottomley, <a href="a330.gif">Illustration of initial terms</a>
%H A000330 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviPiram">
First 2500 Pyramidal numbers</a>
%H A000330 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0202219">Restricted
permutations by patterns of type 2-1</a>.
%H A000330 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/
grid-squares">Square Counting</a>
%H A000330 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SquarePyramidalNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000330 G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">
Operate on"n^2"</a>
%H A000330 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000330 a(n)=binomial(2(n+1), 3)/4 - Paul Barry (pbarry(AT)wit.ie), Jul 19 2003
%F A000330 a(n)=[(n^4-(n-1)^4)-(n^2-(n-1)^2)]/12 - Xavier Acloque Oct 16 2003
%F A000330 a(n) = Sqrt[Sum[Sum[(i*j)^2, {i, 1, n}], {j, 1, n}]. a(n) = Sum[Sum[Sum[(i*j*k)^2,
{i, 1, n}], {j, 1, n}], {k, 1, n}]^(1/3). - Alexander Adamchuk (alex(AT)kolmogorov.com),
Oct 26 2004
%F A000330 a(n)=sum(i=1, n, i*(2*n-2*i+1)) - sum of squares gives 1+(1+3)+(1+3+5)+...
- Jon Perry (perry(AT)globalnet.co.uk), Dec 08 2004
%F A000330 a(n+1) = A000217(n+1) + 2*A000292(n-1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Mar 10 2005
%F A000330 Sum_{ n>0} 1/a(n) = 6(3 - 4log(2)) and Sum_{ n>0} = (-1)^(n+1)*1/a(n)
= 6(Pi - 3) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31
2005
%F A000330 Sum of two consecutive tetrahedral (or pyramidal) numbers A000292: C(n+3,
3) = (n+1)(n+2)(n+3)/6: a(n) = A000292(n-1) + A000292(n-2). - Alexander
Adamchuk (alex(AT)kolmogorov.com), May 17 2006
%F A000330 Euler transform of length 2 sequence [ 5, -1]. - Michael Somos Sep 04
2006
%F A000330 G.f.: x(1+x)/(1-x)^4. E.g.f.: (x+3/2*x^2+1/3*x^3)*exp(x).
%F A000330 a(n)=n(n+1)(2n+1)/6=binomial(n+2, 3)+binomial(n+1, 3)=-a(-1-n).
%F A000330 a(n) = a(n-1) + n^2 - Rolf Pleisch (r.pleisch(AT)gmx.ch), Jul 22 2007
%F A000330 a(n) = A132121(n,0). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Aug 12 2007
%F A000330 Hankel transform of C(2n-3,n-1) is -a(n). - Paul Barry (pbarry(AT)wit.ie),
Feb 12 2008
%F A000330 Convolution of A000290 with A000012 - Sergio Falcon (sfalcon(AT)dma.ulpgc.es),
Feb 05 2008
%F A000330 Starting n (-1,0,1,2,...) a(n)=C(n+2,2)+2*C(n+2,3) [From Borislav St.
Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
%p A000330 A000330 := n->n*(n+1)*(2*n+1)/6;
%p A000330 a:=n->(1/6)*n*(n+1)*(2*n+1): seq(a(n),n=0..53); (Deutsch)
%p A000330 a:=n->sum ((j-n-1)^2,j=1..n): seq(a(n),n=0..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Dec 17 2006
%p A000330 a:=array(0...44): a[0]:=0: a[1]:=1:print(0,a[0]); print(1,a[1]); for
i from 2 to 40 do a[i]:= a[i-1]+(i^2):print(i,a[i]); od: - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Mar 27 2007
%p A000330 A000330:=(1+z)/(z-1)^4; [S. Plouffe in his 1992 dissertation, sequence
starting at a(1).]
%p A000330 with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r),
right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=1,
stack): seq(count(subs(r=2, ZL), size=m*2), m=1..45) ; - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jan 02 2008
%p A000330 a:=n->sum(k^2, k=1..n):seq(a(n), n=0...44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 15 2008
%p A000330 with(finance):seq(add(futurevalue(k,3,2)*effectiverate(k,1),k=0..n)/16,
n=0..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 15
2008
%p A000330 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07
2009: (Start)
%p A000330 nmax:=44; for n from 0 to nmax do fz(n):= product( (1-(2*m-1)*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 A000330 (End)
%t A000330 Table[Binomial[w+2, 3]+Binomial[w+1, 3], {w, 1, 30}]
%t A000330 s = 0; lst = {s}; Do[s += n^2; AppendTo[lst, s], {n, 1, 60, 1}]; lst
[From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
%t A000330 Table[Sum[i^2, {i, 0, n - 1}], {n, 1, 50}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jul 10 2009]
%o A000330 (PARI) a(n)=n*(n+1)*(2*n+1)/6
%o A000330 Floretion Algebra Multiplication Program, FAMP Code: 1vesforseq[ + .5'i
+ .5i' + .5'jk' + .5'kj' + e ], ForType: 1B, LoopType: tes (2nd iteration).
(Dement)
%o A000330 (PARI) sumsq(n) = for(x=0,n,y=x*(x+1)*(2*x+1)/6;(print1(y","))) - Cino
Hilliard (hillcino368(AT)hotmail.com), Jun 18 2007
%o A000330 (PARI) a(n)=sum(m=1,n,sum(i=1,m,(2*i-1))) - Alexander R. Povolotsky (pevnev(AT)juno.com),
Nov 04 2007
%Y A000330 A006331(n)=2*A000330(n). Cf. A000217, A050446, A050447, A000537, A006003.
%Y A000330 Cf. A000292, A033994, A132124, A132112, A050409.
%Y A000330 Sums of 2 consecutive terms give A005900.
%Y A000330 Column 0 of triangle A094414. Column 1 of triangle A008955. Right side
of triangle A082652. Row 2 of array A103438.
%Y A000330 Partial sums of A000290. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg),
Mar 05 2009]
%Y A000330 Cf. A156921, A157702.
%Y A000330 Sequence in context: A096893 A074784 A109678 this_sequence A166068 A070129
A081861
%Y A000330 Adjacent sequences: A000327 A000328 A000329 this_sequence A000331 A000332
A000333
%K A000330 nonn,easy,core,nice
%O A000330 0,3
%A A000330 N. J. A. Sloane (njas(AT)research.att.com).
%E A000330 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Mar 11 2009
|
