0,3

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

Number of rhombi in an n X n rhombus. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

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

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

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

Number of permutations avoiding 13-2 that contain the pattern 32-1 exactly once.

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

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.

Number of triples of integers from {1,2,...,n} whose last component is greater than or equal to the others.

Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 12 2005

Sum of the first n squares, or square pyramidal numbers. - Cino Hilliard (hillcino368(AT)hotmail.com), Jun 18 2007

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

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

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

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

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

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]

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]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)

Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF1 denominators of A156921. See A157702 for background information.

(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 215,223.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 122, see #19 (3(1)), I(n); p. 155.

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.

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).

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.

M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, pg 293.

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.

T. D. Noe, Table of n, a(n) for n = 0..1000

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Two Enumerative Functions

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

H. Bottomley, Illustration of initial terms

R. Jovanovic, First 2500 Pyramidal numbers

T. Mansour, Restricted permutations by patterns of type 2-1.

T. Sillke, Square Counting

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

G. Xiao, Sigma Server, Operate on"n^2"

Index entries for "core" sequences

a(n)=binomial(2(n+1), 3)/4 - Paul Barry (pbarry(AT)wit.ie), Jul 19 2003

a(n)=[(n^4-(n-1)^4)-(n^2-(n-1)^2)]/12 - Xavier Acloque Oct 16 2003

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

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

a(n+1) = A000217(n+1) + 2*A000292(n-1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Mar 10 2005

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

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

Euler transform of length 2 sequence [ 5, -1]. - Michael Somos Sep 04 2006

G.f.: x(1+x)/(1-x)^4. E.g.f.: (x+3/2*x^2+1/3*x^3)*exp(x).

a(n)=n(n+1)(2n+1)/6=binomial(n+2, 3)+binomial(n+1, 3)=-a(-1-n).

a(n) = a(n-1) + n^2 - Rolf Pleisch (r.pleisch(AT)gmx.ch), Jul 22 2007

a(n) = A132121(n,0). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 12 2007

Hankel transform of C(2n-3,n-1) is -a(n). - Paul Barry (pbarry(AT)wit.ie), Feb 12 2008

Convolution of A000290 with A000012 - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 05 2008

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]

A000330 := n->n*(n+1)*(2*n+1)/6;

a:=n->(1/6)*n*(n+1)*(2*n+1): seq(a(n), n=0..53); (Deutsch)

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

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

A000330:=(1+z)/(z-1)^4; [S. Plouffe in his 1992 dissertation, sequence starting at a(1).]

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

a:=n->sum(k^2, k=1..n):seq(a(n), n=0...44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 15 2008

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

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)

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);

(End)

Table[Binomial[w+2, 3]+Binomial[w+1, 3], {w, 1, 30}]

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]

Table[Sum[i^2, {i, 0, n - 1}], {n, 1, 50}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

(PARI) a(n)=n*(n+1)*(2*n+1)/6

Floretion Algebra Multiplication Program, FAMP Code: 1vesforseq[ + .5'i + .5i' + .5'jk' + .5'kj' + e ], ForType: 1B, LoopType: tes (2nd iteration). (Dement)

(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

(PARI) a(n)=sum(m=1, n, sum(i=1, m, (2*i-1))) - Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 04 2007

A006331(n)=2*A000330(n). Cf. A000217, A050446, A050447, A000537, A006003.

Cf. A000292, A033994, A132124, A132112, A050409.

Sums of 2 consecutive terms give A005900.

Column 0 of triangle A094414. Column 1 of triangle A008955. Right side of triangle A082652. Row 2 of array A103438.

Partial sums of A000290. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]

Cf. A156921, A157702.

Sequence in context: A096893 A074784 A109678 this_sequence A166068 A070129 A081861

Adjacent sequences: A000327 A000328 A000329 this_sequence A000331 A000332 A000333

nonn,easy,core,nice

N. J. A. Sloane (njas(AT)research.att.com).

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009