0,2

Note that when starting from {a(n)}^2, equality holds between series of first n+1 and next n consecutive squares : a(n)^2+(a(n)+1)^2+...+(a(n)+n)^2 = (a(n)+n+1)^2+(a(n)+n+2)^2+...(a(n)+2n)^2, e.g. 10^2+11^2+12^2 = 13^2+14^2 - Henry Bottomley (se16(AT)btinternet.com), Jan 22 2001

a(n) = sum of second set of n consecutive even numbers - sum of the first set of n consecutive odd numbers: a(1) = 4-1, a(3) = (8+10+12) - (1+3+5) = 21. - Amarnath Murthy, Nov 07 2002

a(n) = A084849(n) - 1; A100035(a(n)+1) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 31 2004

Also a(n)=3*Sum(tan^2(k*pi/(2(n+1))), k, 1, n); - Ignacio Larrosa (ignacio.larrosa(AT)eresmas.net), Apr 17 2001

If Y is a fixed 3-subset of a (2n+1)-set X then a(n) is the number of (2n-1)-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Oct 28 2007

Milan Janjic, Two Enumerative Functions

a(n)^2 = n*(a(n)+1 + a(n)+2 + ... + a(n)+2n), e.g. 10^2 = 2*(11 + 12 + 13 +14) - Charlie Marion (charliem(AT)bestweb.net), Jun 15 2003

G.f.: x(3+x)/(1-x)^3. E.g.f.: exp(x)(3x+2x^2). a(n)=A000217(2n)=A000384(-n).

Partial sums of odd numbers 3 mod 4, i.e. 3, 3+7, 3+7+11, ... Cf. A001107. - Jon Perry (perry(AT)globalnet.co.uk), Dec 18 2004

a(n) = A126890(n,k) + A126890(n,n-k), 0<=k<=n. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 30 2006

seq(binomial(2*n+1, 2), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 21 2007

a:=n->sum(j, j=0..n): seq(a(2*n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007

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

Cf. A000217, A000384.

Second column of array A094416.

Cf. A100040, A100041.

Adjacent sequences: A014102 A014103 A014104 this_sequence A014106 A014107 A014108

Sequence in context: A073604 A004194 A097590 this_sequence A027917 A038347 A117495

nonn,easy

njas