0,3

Floor[ arithmetic mean of next n numbers]. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 11 2003

Pairwise sums of repeated squares (A008794).

Also, number of topologies on n+1 unlabeled elements with exactly 4 elements in the topology. a(3) gives 4 elements a,b,c,d; the valid topologies are (0,a,ab,abcd), (0,a,abc,abcd), (0,ab,abc,abcd), (0,a,bcd,abcd) and (0,ab,cd,abcd), with a count of 5. - Jon Perry (perry(AT)globalnet.co.uk), Mar 05 2004

Euler transform of a(n+1) is length 4 sequence [2,2,0,-1].

Partition n in two parts, say r and s so that r^2 + s^2 is minimal, then a(n) = r^2 +s^2. Geometrical significance: folding a rod with length n units at right angles in such a way that the end points are at the least distance, which is given by a(n)^(1/2) as the hypotenus of a right angled triangle with the sum of the base and height = n units. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 18 2004

Convolution of A002061(n)-0^n and (-1)^n. Convolution of n (A001477) with {1,0,2,0,2,0,2...}. Partial sums of repeated odd numbers {0,1,1,3,3,5,5,...}. - Paul Barry (pbarry(AT)wit.ie), Jul 22 2004

The ratio of the sum of terms over the total number of terms in an n X n spiral. The sum of terms of an n X n spiral is A037270, or Sum{k=0..n^2,k} = (n^4 + n^2)/2 and the total number of terms is n^2. - William A. Tedeschi (fynmun(AT)hotmail.com), Feb 27 2008

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, 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.

S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ...

MathWorld, Topology

a(2n) = 2n^2, a(2n+1) = 2n^2 + 2n + 1.

a(n) = (2n^2 + 1 - (-1)^n) / 4. a(0)=0, a(1)=1, a(n+1)=n+1+max(2*floor(a(n)/2); 3*floor(a(n)/3)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 06 2002

G.f.: (x+x^2+x^3+x^4)/((1-x)*(1-x^2)^2) - Len Smiley (smiley(AT)math.uaa.alaska.edu).

a(n)=a(n-2)+2n-2. - Paul Barry (pbarry(AT)wit.ie), Jul 17 2004

G.f.: x(1+x^2)/((1-x^2)(1-x)^2)=x(1+x^2)/((1+x)(1-x)^3); a(n)=sum{k=0..n, (k^2-k+1-0^k)(-1)^(n-k) }; a(n)=sum{k=0..n, ((1+(-1)^(n-k))-0^(n-k))k }. - Paul Barry (pbarry(AT)wit.ie), Jul 22 2004

a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 1. a(n) = A116940(n) - A005843(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 27 2006

Starting with offset 1, = row sums of triangle A134444. Also, with offset 1, = binomial transform of [1, 1, 2, -2, 4, -8, 16, -32,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2007

a(n) = floor((n^2+1)/2) - William A. Tedeschi (fynmun(AT)hotmail.com), Feb 27 2008

Contribution from Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Sep 12 2008): (Start)

a(n) = A004526(n+2) + A000217 (n)

(End)

A000982:=-(1+z**2)/(z+1)/(z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]

with (combinat):seq(count(Partition((n^2+1)), size=2), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2008

Cf. A000096.

Cf. A134444.

Cf. A037270.

Adjacent sequences: A000979 A000980 A000981 this_sequence A000983 A000984 A000985

Sequence in context: A076145 A049617 A054074 this_sequence A122221 A083704 A111097

nonn,easy,new

njas