0,3

Also a(n)=Sum(tan^2((k - 1/2)*pi/(2n)), k, 1, n); - Ignacio Larrosa (ignacio.larrosa(AT)eresmas.net), Apr 17 2001

Number of edges in the join of two complete graphs, each of order n, K_n * K_n - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002

The power series expansion of the entropy function H(x) = (1+x)ln(1+x)+(1-x)ln(1-x) has 1/a_i as coefficient of x^(2i) (the odd terms being zero). - Tommaso Toffoli (tt(AT)bu.edu), May 06 2002

Partial sums of A016813 (4n+1). Also with offset = 0, a(n) = (2n+1)(n+1) = A005408 * A000027 = 2n^2 + 3n + 1, i.e. a(0) = 1. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Sep 29 2002

Sequence also refers to greatest semiperimeter of primitive Pythagorean triangles having inradius n-1. Such a triangle has consecutive longer sides, with short leg 2n-1, hypotenus a(n)-(n-1)=A001844(n) and area (n-1)*a(n)=6*A000330(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 23 2003

Number of divisors of 12^(n-1), i.e. A000005(A001021(n-1)). - Henry Bottomley (se16(AT)btinternet.com), Oct 22 2001

Number of standard tableaux of shape (2n-1,1,1) (n>=1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 30 2004

It is well known that for n>0, A014105(n) [0,3,10,21,...] is the first of 2n+1 consecutive integers such that the sum of the squares of the first n+1 such integers is equal to the sum of the squares of the last n; e.g. 10^2+11^2+12^2=13^2+14^2.

Less well known is that for n>1, a(n) [0,1,6,15,28... ] is the first of 2n consecutive integers such that sum of the squares of the first n such integers is equal to the sum of the squares of the last n-1 plus n^2; e.g. 15^2+16^2+17^2 = 19^2+20^2+3^2 - Charlie Marion, Dec 16 2006

a(n) is also a perfect number A000396 when n is an even superperfect number A061652. [From Omar E. Pol (info(AT)polprimos.com), Sep 05 2008]

Sequence arises from reading the line from 0, in the direction 0, 6,... and the line from 1, in the direction 1, 15,..., in the square spiral whose vertices are the triangular numbers A000217. [From Omar E. Pol (info(AT)polprimos.com), Jan 09 2009]

Also, let Hex(n)= hexagonal number, T(n)=triangular number, then Hex(n)= T(n)+3*T(n-1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 28 2009]

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).

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

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.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

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

Paul Cooijmans, Odds.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 340

Hyun Kwang Kim, On Regular Polytope Numbers

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.

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

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

E.g.f.: exp(x)(x+2x^2) - Paul Barry (pbarry(AT)wit.ie), Jun 09 2003

G.f.: x(1+3x)/(1-x)^3. a(n)=A000217(2n-1)=A014105(-n).

a(n)=4*A000217(n-1) + n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004

a(n) = right term of M^n * [1,0,0], where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,4,1]. Example: a(5) = 45 since M^5 *[1,0,0] = [1,5,45]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2006

Row sums of triangle A131914. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007

Row sums of n-th row, triangle A134234 starting (1, 6, 15, 28,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 14 2007

Starting with offset 1, = binomial transform of [1, 5, 4, 0, 0, 0,...]. Also, A004736 * [1, 4, 4, 4,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2007

a(n)^2+(a(n)+1)^2+...+(a(n)+n-1)^2=(a(n)+n+1)^2+...(a(n)+2n-1)^2+n^2; e.g., 6^2+7^2=9^2+2^2; 28^+29^2+30^2+31^2=33^2+34^2+35^2+4^2 - Charlie Marion (charliemath(AT)optonline.net), Nov 10 2007

a(n) = C(n+1,2) + 3 C(n,2)

a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=6 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]

a(n)=4*n+a(n-1)-7 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

For n=2, a(2)=4*2+0-7=1; n=3, a(3)=4*3+1-7=6; n=4, a(4)=4*4+6-7=15 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

[seq (stirling2(2*n, 1)*binomial(2*n, 2), n=0..48)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo), Dec 06 2006

a:=n->sum(n/2, j=2..n): seq(a(2*n), n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007

A000384:=-(1+3*z)/(z-1)^3; [S. Plouffe in his 1992 dissertation, dropping the initial zero.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+4 od: seq(a[n], n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 2008

with(finance):seq(add(cashflows([n, k, k], 0 ), k=0..n-1), n=0..51); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008

a:=n->sum(1+sum(2, k=1..n), k=0..n):seq(a(n), n=0...43); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]

Array[ #*(2*#-1)&, 20, 0] - Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 29 2008

Table[2*n^2 + 3*n + 1, {n, -1, 46}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

(PARI) a(n)=n*(2*n-1)

(Other) sage: [2*n*bernoulli_polynomial(n, 1) for n in xrange(0, 49)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2009]

Cf. A000217, A014105, A077616.

a(n)= A093561(n+1, 2), (4, 1)-Pascal column.

a(n)=A100345(n, n-1) for n>0.

Cf. A131914.

Cf. A134235.

Cf. A004736.

Cf. A000217, A000326, A000566.

Cf. A000396, A061652. [From Omar E. Pol (info(AT)polprimos.com), Sep 05 2008]

Cf. A014634, A014635. [From Omar E. Pol (info(AT)polprimos.com), Jan 09 2009]

Sequence in context: A094142 A081873 A096892 this_sequence A164000 A134978 A115742

Adjacent sequences: A000381 A000382 A000383 this_sequence A000385 A000386 A000387

nonn,easy,nice,new

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

More terms from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2006