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

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.

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

Milan Janjic, Two Enumerative Functions

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.

Paul Cooijmans, Odds.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 340

Hyun Kwang Kim, On Regular Polytope Numbers

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)

[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; [Conjectured by S. Plouffe in his 1992 dissertation.]

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

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

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.

Adjacent sequences: A000381 A000382 A000383 this_sequence A000385 A000386 A000387

Sequence in context: A094142 A081873 A096892 this_sequence A134978 A115742 A026102

nonn,easy,nice

njas

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