A093353 -id:A093353 - cp's OEIS frontend

A159915 a(n) = floor((n+1)/4)*floor(n/2).

0, 0, 0, 1, 2, 2, 3, 6, 8, 8, 10, 15, 18, 18, 21, 28, 32, 32, 36, 45, 50, 50, 55, 66, 72, 72, 78, 91, 98, 98, 105, 120, 128, 128, 136, 153, 162, 162, 171, 190, 200, 200, 210, 231, 242, 242, 253, 276, 288, 288, 300, 325, 338, 338, 351, 378, 392, 392, 406, 435, 450, 450

Offset: 0

M. F. Hasler, May 01 2009, May 03 2009

Half the number of (n-2)-element subsets of {1,...,n} with odd sum of the elements.
This is half the antepenultimate column of A159916, cf. formula.
The number of subsets of {1,...,n} with n-2 elements, adding up to an odd integer, is always even (cf. examples), so we divide it by 2.
We prefer to include a(0)=a(1)=a(2)=0, even if it might seem more natural to start only at n=2 or n=3.
From the rational g.f. it can be seen that the sequence is a linear recurrence with constant coefficients (3,-5,7,-7,5,-3,1) of order 7.
A quasipolynomial of order 4 and degree 2. - Charles R Greathouse IV, Sep 18 2024

			a(0)=a(1)=0 since there are no subsets with -2 or -1 elements.
a(2)=0 since the sum of the elements of a 0-element subset is zero.
a(3)=1 since for n=3 we have two singleton subsets of {1,2,3}, {1} and {3}, with odd sum of elements.
a(4)=2 since for n=4 we have four 2-element subsets of {1,2,3,4} with odd sum: {1,2}, {2,3}, {1,4}, {3,4}.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A093005, A093353, A159916.

A159915:= func< n | Floor((n+1)/4)*Floor(n/2) >;
[A159915(n): n in [0..70]]; // G. C. Greubel, Sep 18 2024

Table[Floor[(n+1)/4]*Floor[n/2], {n,0,70}] (* G. C. Greubel, Sep 18 2024 *)

A159915(n)= polcoeff( (1-x+x^2)/((1-x)^3*(1+x^2)^2) + O(x^(n-2)), n-3);
a(n,t=[0,0,0,1,2,2,3],c=[1,-3,5,-7,7,-5,3]~)=while(n-->5,t=concat(vecextract(t,"^1"),t*c));t[n+2] /* Note: a(n+1,[0,0,0,0,1,2,2]) gives the same result as a(n) */

A159915(n)=(n+1)\4*(n\2) \\ M. F. Hasler, May 03 2009

def A159915(n): return ((n+1)//4)*(n//2)
[A159915(n) for n in range(71)] # G. C. Greubel, Sep 18 2024

G.f.: x^3*(1 - x + x^2)/(1 - 3*x + 5*x^2 - 7*x^3 + 7*x^4 - 5*x^5 + 3*x^6 - x^7) = x^3*(1-x+x^2)/((1-x)^3*(1+x^2)^2).
a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n > 7.
For n > 2, a(n) = A159916(n*(n-1)/2 + n - 2)/2 = T(n,n-2)/2 as defined there.
From M. F. Hasler, May 03 2009: (Start)
a(n) = floor((n+1)/4)*floor(n/2).
a(2n+1) = A093005(n).
a(2n) = A093353(n-1) = floor(n/2)*n. (End)
a(n) ~ n^2/8. - Charles R Greathouse IV, Sep 18 2024

A238411 a(n) = 2nfloor(n/2).

Original entry on oeis.org

0, 4, 6, 16, 20, 36, 42, 64, 72, 100, 110, 144, 156, 196, 210, 256, 272, 324, 342, 400, 420, 484, 506, 576, 600, 676, 702, 784, 812, 900, 930, 1024, 1056, 1156, 1190, 1296, 1332, 1444, 1482, 1600, 1640, 1764, 1806, 1936, 1980, 2116, 2162, 2304, 2352, 2500

Offset: 1

Emeric Deutsch, Feb 27 2014

For n>=3, a(n) = the eccentric connectivity index of the cycle C[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined as the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(6)=36 because each vertex of C[6] has degree 2 and eccentricity 3; 6*2*3 = 36.

M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311, 2011, 1229-1234.
B. Zhou and Zh. Du, On eccentric connectivity index, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A093353.

[2*n*Floor(n/2): n in [1..50]]; // Bruno Berselli, Feb 25 2016

a := proc (n) options operator, arrow: 2*n*floor((1/2)*n) end proc: seq(a(n), n = 1 .. 70);

Table[2 n Floor[n/2], {n, 1, 50}] (* Bruno Berselli, Feb 25 2016 *)

makelist(2*n*floor(n/2), n, 1, 50); /* Bruno Berselli, Feb 25 2016 */

[2*n*floor(n/2) for n in (1..50)] # Bruno Berselli, Feb 25 2016

From Bruno Berselli, Feb 25 2016: (Start)
G.f.: 2*x*(2 + x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = n*(2*n + (-1)^n - 1)/2.
a(n+1) = 2*A093353(n). (End)

A178946 a(n) = n(n+1)(2n+1)/6 - nfloor(n/2).

Original entry on oeis.org

1, 3, 11, 22, 45, 73, 119, 172, 249, 335, 451, 578, 741, 917, 1135, 1368, 1649, 1947, 2299, 2670, 3101, 3553, 4071, 4612, 5225, 5863, 6579, 7322, 8149, 9005, 9951, 10928, 12001, 13107, 14315, 15558, 16909, 18297, 19799, 21340, 23001

Offset: 1

Gary W. Adamson, Dec 30 2010

nonn

Previous name was: A modified variant of A005900.
Let S(x) = (1, 3, 5, 7,...); then A178946 = (1/2) * ((S(x)^2 + S(x^2)).
If n is even, a(n) is the sum of the first n squares minus n^2/2.  If n is odd, a(n) is the sum of the first n squares minus n(n-1)/2. - Wesley Ivan Hurt, Sep 17 2013

			(1/2) *((1, 6, 19, 44, 85, 146, 231,...) + (1, 0, 3, 0, 5, 0, 7, 0, 9,...)) =
(1, 3, 11, 22, 45, 73, 119,...).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000330, A005900, A093353.

[n*(n+1)*(2*n+1)/6 - n*Floor(n/2): n in [1..50]]; // Vincenzo Librandi, Sep 17 2013

A005900 := proc(n) n*(2*n^2+1)/3 ; end proc:
A178946 := proc(n) if type(n,'even') then A005900(n)/2 ; else (A005900(n)+n)/2 ; end if;end proc:
seq(A178946(n),n=1..60) ; # R. J. Mathar, Jan 03 2011
seq(k*(k+1)*(2*k+1)/6 - k*floor(k/2), k=1..100); # Wesley Ivan Hurt, Sep 17 2013

Table[n(n+1)(2n+1)/6-n*Floor[n/2], {n,100}] (* Wesley Ivan Hurt, Sep 17 2013 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,3,11,22,45,73},50] (* Harvey P. Dale, Mar 20 2018 *)

a(2n) = A005900(2n)/2. a(2n+1) = (A005900(2n+1)+2n+1)/2.
a(n) = +2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6). G.f.: x*(1+x+4*x^2+x^4+x^3) / ( (1+x)^2*(x-1)^4 ). - R. J. Mathar, Jan 03 2011
a(n) = A000330(n+1) - A093353(n), n>0. - Wesley Ivan Hurt, Sep 17 2013

Better name using formula from Wesley Ivan Hurt, Joerg Arndt, Sep 17 2013

A182021 Achromatic number of n-cycle.

Original entry on oeis.org

3, 2, 3, 3, 3, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 12, 13

Offset: 3

N. J. A. Sloane, Apr 06 2012

Hare, W. R.; Hedetniemi, S. T.; Laskar, R.; Pfaff, J. Complete coloring parameters of graphs. Proceedings of the sixteenth Southeastern international conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1985). Congr. Numer. 48 (1985), 171--178. MR0830709 (87h:05088)

A093353 := proc(n)
    if n < 1 then
        0;
    else
        (n + modp(n,2))*(n+1)/2 ;
    end if;
end proc:
A182021 := proc(n)
    for m from 0 do
        sm := A093353(m-1) ;
        if sm >  n then
            m := m-1 ;
            sm := A093353(m-1) ;
            if type(m,'odd') and n = sm+1 then
                return m-1 ;
            else
                return m;
            end if;
        end if;
    end do:
end proc:
seq(A182021(n),n=3..80) ; # R. J. Mathar, Jul 12 2013

A093353[n_] := If[n < 1, 0, (n+Mod[n, 2])*(n+1)/2];
a[n_] := For[m = 0, True, m++, sm = A093353[m-1]; If[sm > n, m = m-1; sm = A093353[m-1]; If[OddQ[m] && n == sm+1, Return[m-1], Return[m]]]];
Table[a[n], {n, 3, 80}] (* Jean-François Alcover, Apr 15 2023, after R. J. Mathar *)

Let s_m = m^2/2 if m even, m(m-1)/2 if m odd. For m >= 0, the s_m sequence is 0, 0, 2, 3, 8, 10, 18, 21, 32, 36, 50, ... (A093353 with a different offset).

Suppose s_m <= n < s_{m+1}. If m is odd and n = s_m + 1 then a(n) = m-1, otherwise a(n) = m.

A226731 a(n) = (2n - 1)!/(2n).

Original entry on oeis.org

20, 630, 36288, 3326400, 444787200, 81729648000, 19760412672000, 6082255020441600, 2322315553259520000, 1077167364120207360000, 596585001666576384000000, 388888194657798291456000000

Offset: 3

Wesley Ivan Hurt, Jun 15 2013

For n < 3, the formula does not produce an integer.

The ratio of the product of the partition parts of 2n into exactly two parts to the sum of the partition parts of 2n into exactly two parts. For example, a(3) = 20, and 2*3 = 6 has 3 partitions into exactly two parts: (5,1), (4,2), (3,3). Forming the ratio of product to sum (of parts), we have (5*1*4*2*3*3)/(5+1+4+2+3+3) = 360/18 = 20. - Wesley Ivan Hurt, Jun 24 2013

			a(3) = (2*3 - 1)!/(2*3) = 5!/6 = 120/6 = 20.

Seiichi Manyama, Table of n, a(n) for n = 3..225
Index entries for sequences related to partitions.

Cf. A001105, A002674, A005843, A009445, A093353, A143623, A211374.

seq((2*k-1)!/(2*k),k=1..20); # Wesley Ivan Hurt, Jun 24 2013

Table[(2n-1)!/(2n),{n,3,20}] (* Harvey P. Dale, Jun 19 2013 *)

a(n) = (2*n-1)!/(2*n); \\ Michel Marcus, Nov 01 2016

a(n) = A009445(n-1)/A005843(n) = A002674(n)/A001105(n). - Wesley Ivan Hurt, Jun 24 2013
a(n) ~ sqrt(Pi)*2^(2*n-1)*n^(2*n-3/2)/exp(2*n). - Ilya Gutkovskiy, Nov 01 2016
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=3} 1/a(n) = e - 8/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = cos(1) + sin(1) - 4/3. (End)

A242371 Modified eccentric connectivity index of the cycle graph with n vertices, C[n].

Original entry on oeis.org

12, 32, 40, 72, 84, 128, 144, 200, 220, 288, 312, 392, 420, 512, 544, 648, 684, 800, 840, 968, 1012, 1152, 1200, 1352, 1404, 1568, 1624, 1800, 1860, 2048, 2112, 2312, 2380, 2592, 2664, 2888, 2964, 3200, 3280, 3528, 3612, 3872, 3960, 4232, 4324, 4608, 4704

Offset: 3

Nilanjan De, Jun 08 2014

nonn

The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph. This is a generalization of eccentric connectivity index.

a(n) = 4*A093353(n-1) = n*A168273(n) for n>2. - Alois P. Heinz, Jun 26 2014

			a(3) = 3*4 = 12 because there are 3 vertices and each vertex has eccentricity 1 and the total degree of neighboring vertices is 4.

Nilanjan De, Table of n, a(n) for n = 3..100
N. De, S. M. A. Nayeem and A. Pal, Bounds for modified eccentric connectivity index, Advanced Modeling and Optimization, 16(1) (2014) 133-142.
N. De, S. M. A. Nayeem and A. Pal, Bounds for modified eccentric connectivity index, arXiv:1402.1870 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Eccentricity

Cf. A093353, A168273, A206490, A238410, A238411.

a:= n-> n*(2*n-1+(-1)^n):
seq(a(n), n=3..60);  # Alois P. Heinz, Jun 26 2014

a[n_] := 2n(n-Boole[OddQ[n]]);
Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Nov 28 2018 *)

a(n) = if (n % 2, 2*n*(n-1), 2*n^2); \\ Michel Marcus, Jun 20 2014

a(n) = 2*n*(n-1) if n is odd; and a(n) = 2*n^2 if n is even (n>2).

G.f.: -4*x^3*(3+5*x-4*x^2-2*x^3+2*x^4)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 26 2014

cp's OEIS Frontend

A159915 a(n) = floor((n+1)/4)*floor(n/2).

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A238411 a(n) = 2*n*floor(n/2).

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A178946 a(n) = n*(n+1)*(2*n+1)/6 - n*floor(n/2).

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A182021 Achromatic number of n-cycle.

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A226731 a(n) = (2n - 1)!/(2n).

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A242371 Modified eccentric connectivity index of the cycle graph with n vertices, C[n].

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A238411 a(n) = 2nfloor(n/2).

A178946 a(n) = n(n+1)(2n+1)/6 - nfloor(n/2).