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A000982 a(n) = ceiling(n^2/2).
(Formerly M1348 N0517)
71
0, 1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288, 313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968, 1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Floor(arithmetic mean of next n numbers). - Amarnath Murthy, 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, Mar 05 2004

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 hypotenuse of a right angled triangle with the sum of the base and height = n units. - Amarnath Murthy, 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, 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, Feb 27 2008

Starting with offset 1 = row sums of triangle A158946. - Gary W. Adamson, Mar 31 2009

Partial sums of A109613. - Reinhard Zumkeller, Dec 05 2009

Also the number of compositions of even natural numbers into 2 parts <n. For example a(3)=5 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2) of even natural numbers into 2 parts <3. a(4)=8 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2), (1,3), (3,1), (3,3) of even natural numbers into 2 parts <4. - Adi Dani, Jun 05 2011

A001105 and A001844 interleaved. - Omar E. Pol, Sep 18 2011

Number of (w,x,y) having all terms in {0,...,n} and w=average(x,y). - Clark Kimberling, May 15 2012

For n>0, minimum number of lines necessary to get through all unit cubes of an n X n X n cube (see Kantor link). - Michel Marcus, Apr 13 2013

Sum_{n > 0} 1/a(n) = Sum_{n > 0} 1/(2*n^2) + Sum_{n >= 0} 1/(2*n + 2*n^2 + 1) = (zeta(2) + (Pi* tanh(Pi/2)))/2 = 2.26312655.... - Enrique Pérez Herrero, Jun 17 2013

REFERENCES

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

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..3000

M. Benoumhani, M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, t_{N0}(n,4) in theorem 5.

J. M. Kantor, Mathématiques venues d'ailleurs: divertissements mathématiques en U.R.S.S., Le cube transpercé, pp. 56-62, Belin, Paris, 1982.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.

MathWorld, Topology

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

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

G.f.: -x*(1+x^2) / ( (1+x)*(x-1)^3 ). - Simon Plouffe in his 1992 dissertation

a(n) = (2*n^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, Nov 06 2002

G.f.: (x+x^2+x^3+x^4)/((1-x)*(1-x^2)^2), not reduced. - Len Smiley.

a(n) = a(n-2) + 2n - 2. - Paul Barry, Jul 17 2004

From Paul Barry, Jul 22 2004: (Start)

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. (End)

From Reinhard Zumkeller, Feb 27 2006: (Start)

a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 1.

a(n) = A116940(n) - A005843(n). (End)

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, Oct 25 2007

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

a(n) = A004526(n+2) + A000217(n). - Yosu Yurramendi, Sep 12 2008

From Jaume Oliver Lafont, Dec 05 2008: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) + 2;

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). (End)

a(n) = A004526(n)^2 + A110654(n)^2. - Philippe Deléham, Mar 12 2009

a(n) = n^2 - floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013

Euler transform is length 4 sequence [ 2, 2, 0, -1].

a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015

a(n) is also the number of independent entries in a centrosymmetric n X n matrix: M(i, j) = M(n-i+1, n-j+1). - Wolfdieter Lang, Oct 12 2015

For n > 1, a(n+1)/a(n) = 3 - A081352(n-2)/a(n). - Miko Labalan, Mar 26 2016

EXAMPLE

G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 13*x^5 + 18*x^6 + 25*x^7 + 32*x^8 + ...

Centrosymmetric 3 X 3 matrix: [[a,b,c],[d,e,d],[c,b,a]], a(3) = 3*(3-1)/2 + (3-1)/2 + 1 = (3^2+1)/2  = 5 from a,b,c,d,e. 4 X 4 case: [[a,b,c,d],[e,f,g,h],[h,g,f,e],[d,c,b,a]],

  a(4) = 4*4/2 = 8. - Wolfdieter Lang, Oct 12 2015

MAPLE

seq( ceil(n^2/2), n=0..30) ; # R. J. Mathar, Jun 05 2011

MATHEMATICA

Table[Ceiling[n^2/2], {n, 0, 120}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)

PROG

(MAGMA) [(2*n^2 + 1 - (-1)^n) / 4: n in [0..60]]; // Vincenzo Librandi, Jun 16 2011

(Haskell)

a000982 = (`div` 2) . (+ 1) . (^ 2)  -- Reinhard Zumkeller, Jun 27 2013

(PARI) a(n)=(n^2+1)\2 \\ Charles R Greathouse IV, Sep 13 2013

(PARI) x='x+O('x^100); concat([0], Vec(x*(1+x^2)/((1+x)*(1-x)^3))) \\ Altug Alkan, Oct 12 2015

CROSSREFS

Cf. A000096, A134444, A037270, A158946, A001844, A001105, A007590.

Column 2 of A195040.

Sequence in context: A256829 A049617 A054074 * A200274 A122221 A083704

Adjacent sequences:  A000979 A000980 A000981 * A000983 A000984 A000985

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 30 06:51 EDT 2017. Contains 287302 sequences.