A028723 -id:A028723 - cp's OEIS frontend

A352665 Maximum number of induced copies of the 4-node path in an n-node graph.

0, 0, 0, 1, 5, 9, 16, 32, 48, 80, 112, 160

Offset: 1

Pontus von Brömssen, Mar 26 2022

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to the inducibility of the 4-node path, which is known to be between 1173/5824 = 0.201407... and 0.204513; see Even-Zohar and Linial (2015), who attribute the upper bound to Emil R. Vaughan.

			All optimal graphs (i.e., n-node graphs having a(n) induced copies of P_4) for 4 <= n <= 9 are listed below. Since P_4 is self-complementary, the optimal graphs come in complementary pairs. Here, ECB(n_1, ..., n_k) denotes the empty cyclic braid graph with cluster sizes n_1, ..., n_k, as defined by Morrison and Scott (2017), i.e., the graph obtained by arranging k clusters of n_1, ..., n_k nodes, respectively, in a cycle, and joining all pairs of nodes in neighboring clusters with edges.
  n = 4: P_4 (self-complementary).
  n = 5: C_5 (self-complementary).
  n = 6: ECB(1, 1, 1, 1, 2) and its complement.
  n = 7: 8 optimal graphs, among them ECB(1, 1, 1, 2, 2) and ECB(1, 1, 2, 1, 2), and their complements. In graph6 format, the optimal graphs are "F?o~_", "FCY^_", "FCpv?", "FCxv?", "FCxvO", "FQjRo", "FQyuo", and "FQyvO".
  n = 8: The antiprism graph and its complement (the Wagner graph).
  n = 9: 22 optimal graphs, among them all graphs that are supergraphs of ECB(1, 2, 2, 2, 2) and subgraphs of its complement (10 graphs altogether), and the 1-skeletons of the Johnson solids J10 (the gyroelongated square pyramid) and J51 (the triaugmented triangular prism) and their complements. In graph6 format, the optimal graphs are "H?bF`xw", "H?o}^_}", "H?o}^bp", "H?q`qjo", "H?q`v`[", "H?rF`zo", "H?rF`zq", "HCRbdO{", "HCXfczo", "HCXfczq", "HCXk~a]", "HCXk~bo", "HCXk~bp", "HCY^fXy", "HCrb`qi", "HCrb`rc", "HEhuTxm", "HEhutxm", "HQjUjqm", "HQyurjU", "HQyurji", and "HQyurzU".

Chaim Even-Zohar and Nati Linial, A Note on the inducibility of 4-vertex graphs, Graphs and Combinatorics 31 (2015), 1367-1380; arXiv version, arXiv:1312.1205 [math.CO], 2013-2014.
Falk Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 43e7869.
Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352666 (claw graph), A352667 (paw graph), A352668 (diamond graph), A352669 (cycles).

a(10)-a(12) added using tinygraph by Falk Hüffner, Apr 07 2022

A302647 a(n) = (2n^2(n^2 - 3) - (2n^2 + 1)(-1)^n + 1)/64.

Original entry on oeis.org

0, 0, 2, 6, 18, 36, 72, 120, 200, 300, 450, 630, 882, 1176, 1568, 2016, 2592, 3240, 4050, 4950, 6050, 7260, 8712, 10296, 12168, 14196, 16562, 19110, 22050, 25200, 28800, 32640, 36992, 41616, 46818, 52326, 58482, 64980, 72200, 79800, 88200, 97020, 106722

Offset: 1

Wesley Ivan Hurt, Apr 10 2018

Consider the partitions of n into two parts (s,t) where s <= t. Then a(n) represents the total volume of all rectangular prisms with dimensions s, t, and |t-s|.
Take a chessboard of (n+1) X (n+1) unit squares in which the a1 square is black. a(n) is the number of composite rectangles of p X q unit squares whose vertices are covered by white unit squares (1 < p <= n+1, 1 < q <= n+1). For example, in a 4 X 4 chessboard there are two such rectangles (for both rectangles p = q = 3) and the coordinates of their lower left vertices are a2 and b1, i.e., a(3) = 2. For the number of composite rectangles whose vertices are covered by black unit squares see A317714. - Ivan N. Ianakiev, Aug 22 2018
Also the graph crossing number of the (n+2)-barbell graph (assuming Guy's conjecture). - Eric W. Weisstein, May 17 2023

Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).
Index entries for sequences related to partitions.

Cf. A028723, A317714.

Positive terms are the third column of the triangle in A145118.

[(1/2)*Floor(n/2)*(1+Floor(n/2))*(Floor(n/2)-n)*(1-n+Floor(n/2)): n in [1..45]]; // Vincenzo Librandi, Apr 11 2018

Table[(1/2)*Floor[n/2]*(1 + Floor[n/2])*(Floor[n/2] - n)*(1 - n + Floor[n/2]), {n, 100}]
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 2, 6, 18, 36, 72, 120}, 20] (* Eric W. Weisstein, May 17 2023 *)
Table[(1 - (-1)^n - 2 (3 + (-1)^n) n^2 + 2 n^4)/64, {n, 20}] (* Eric W. Weisstein, May 17 2023 *)
CoefficientList[Series[-2 x^2 (1 + x + x^2)/((-1 + x)^5 (1 + x)^3), {x, 0, 20}], x] (* Eric W. Weisstein, May 17 2023 *)

a(n) = Sum_{k=1..floor(n/2)} k * (n-k) * (n-2*k).
a(n) = (1/2)*floor(n/2)*(1+floor(n/2))*(floor(n/2)-n)*(1-n+floor(n/2)).
From Colin Barker, Apr 11 2018: (Start)
G.f.: 2*x^3*(1 + x + x^2) / ((1 - x)^5*(1 + x)^3).
a(n) = n^2*(n-2)*(n+2) / 32 for n even.
a(n) = (n^2 - 1)^2 / 32 for n odd.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n>8. (End)
a(n) = 2 * A028723(n+2). - Alois P. Heinz, Apr 12 2018
a(n) = 2 * binomial(floor((n+1)/2),2) * binomial(floor((n+2)/2),2). - Bruno Berselli, Apr 12 2018
Sum_{n>=3} 1/a(n) = Pi^2/3 - 5/2. - Amiram Eldar, Jun 20 2025

A007333 An upper bound on the biplanar crossing number of the complete graph on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 7, 12, 18, 37, 53, 75, 100, 152, 198, 256, 320, 430, 530, 650, 780, 980, 1165, 1380, 1610, 1939, 2247, 2597, 2968, 3472, 3948, 4480, 5040, 5772, 6468, 7236, 8040, 9060, 10035, 11100, 12210, 13585, 14905, 16335, 17820, 19624, 21362

Offset: 1

N. J. A. Sloane

This bound in based on a particular decomposition of K_n (see Owens for details). The actual biplanar crossing number for K_9 is 1 (not 4 as given by this bound). - Sean A. Irvine, Dec 30 2019

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

Colin Barker, Table of n, a(n) for n = 1..1000
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280.
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1).

Cf. A000241, A028723.

LinearRecurrence[{2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1},{0,0,0,0,0,0,0,0,4,7,12,18,37,53},70] (* Harvey P. Dale, Feb 13 2022 *)

concat([0,0,0,0,0,0,0,0], Vec(x^9*(4 - x + 2*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3) + O(x^40))) \\ Colin Barker, Feb 02 2020

a(4*k) = k * (k-1) * (k-2) * (7*k-3) / 6, a(4*k+1) = k * (k-1) * (7*k^2-10*k+4) / 6, a(4*k+2) = k * (k-1) * (7*k^2-3*k-1) / 6, a(4*k+3) = k^2 * (k-1) * (7*k+4) / 6 [from Owens]. - Sean A. Irvine, Dec 30 2019; [typo corrected by Colin Barker, Feb 01 2020]
From Colin Barker, Jan 28 2020: (Start)
G.f.: x^9*(4 - x + 2*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3).
a(n) = 2*a(n-1) - a(n-2) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) - 3*a(n-8) + 6*a(n-9) - 3*a(n-10) + a(n-12) - 2*a(n-13) + a(n-14) for n>14.
(End)

More terms and title clarified by Sean A. Irvine, Dec 30 2019

A011845 a(n) = floor( binomial(n,8)/9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 18, 55, 143, 333, 715, 1430, 2701, 4862, 8398, 13996, 22610, 35530, 54479, 81719, 120175, 173586, 246675, 345345, 476905, 650325, 876525, 1168700, 1542684, 2017356, 2615091, 3362260, 4289780, 5433721, 6835972, 8544965, 10616471

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-8,28,-56,70,-56,28,-8,1).

Cf. A000031, A001037, A028723, A051168.

A column of triangle A011847.

[Binomial(n, 8) div 9: n in [0..45]]; // Vincenzo Librandi, Oct 15 2015

Table[Floor[Binomial[n, 8] / 9], {n, 1, 50}] (* Vincenzo Librandi, Oct 15 2015 *)

a(n) = binomial(n, 8)\9; \\ Michel Marcus, Oct 14 2015

a(n) = floor(binomial(n+1,9)/(n+1)). [Gary Detlefs, Nov 23 2011]

Definition corrected by Pedro Antonio, Oct 14 2015

More terms from Vincenzo Librandi, Oct 15 2015

A331575 a(n) is the number of subsets of {1..n} that contain 4 even and 4 odd numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 25, 75, 225, 525, 1225, 2450, 4900, 8820, 15876, 26460, 44100, 69300, 108900, 163350, 245025, 353925, 511225, 715715, 1002001, 1366365, 1863225, 2484300, 3312400, 4331600, 5664400, 7282800, 9363600, 11860560, 15023376, 18779220, 23474025, 28997325

Offset: 0

Enrique Navarrete, Jan 20 2020

			a(9)=5 and the 5 subsets are {1,2,3,4,5,6,7,8}, {1,2,3,4,5,6,8,9}, {1,2,3,4,6,7,8,9}, {1,2,4,5,6,7,8,9}, {2,3,4,5,6,7,8,9}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1).

Cf. A028723, A331574.

Cf. A288876 (even bisection, shifted).

[IsEven(n) select Binomial((n div 2),4)^2 else Binomial((n-1) div 2,4)*Binomial((n+1) div 2,4): n in [0..41]]; // Marius A. Burtea, Jan 21 2020

a:= n-> ((b, q)-> b(q, 4)*b(n-q, 4))(binomial, iquo(n, 2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 30 2020

a[n_] := If[OddQ[n], Binomial[(n - 1)/2, 4]*Binomial[(n + 1)/2, 4], Binomial[n/2, 4]^2]; Array[a, 42, 0] (* Amiram Eldar, Jan 21 2020 *)
LinearRecurrence[{2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1},{0,0,0,0,0,0,0,0,1,5,25,75,225,525,1225,2450},50] (* Harvey P. Dale, Jul 20 2025 *)

a(n) = if (n%2, binomial((n-1)/2,4)*binomial((n+1)/2,4), binomial(n/2,4)^2); \\ Michel Marcus, Jan 21 2020

concat([0,0,0,0,0,0,0,0], Vec(x^8*(1 + 3*x + 9*x^2 + 9*x^3 + 9*x^4 + 3*x^5 + x^6) / ((1 - x)^9*(1 + x)^7) + O(x^40))) \\ Colin Barker, Jan 21 2020

a(n) = binomial(n/2,4)^2, n even;
a(n) = binomial((n-1)/2,4)*binomial((n+1)/2,4), n odd.
From Colin Barker, Jan 21 2020: (Start)
G.f.: x^8*(1 + 3*x + 9*x^2 + 9*x^3 + 9*x^4 + 3*x^5 + x^6) / ((1 - x)^9*(1 + x)^7).
a(n) = 2*a(n-1) + 6*a(n-2) - 14*a(n-3) - 14*a(n-4) + 42*a(n-5) + 14*a(n-6) - 70*a(n-7) + 70*a(n-9) - 14*a(n-10) - 42*a(n-11) + 14*a(n-12) + 14*a(n-13) - 6*a(n-14) - 2*a(n-15) + a(n-16) for n>15.
(End)
E.g.f.: (cosh(x)-sinh(x))*(1575+1350*x+630*x^2+204*x^3+54*x^4+12*x^5+4*x^6+(-1575+1800*x-1080*x^2+456*x^3-156*x^4+48*x^5-16*x^6+8*x^7+2*x^8)*(cosh(2*x)+sinh(2*x)))/294912. - Stefano Spezia, Jan 27 2020

A172101 Triangle, read by rows, given by [0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...] DELTA [1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 4, 2, 1, 0, 1, 3, 6, 6, 3, 1, 0, 1, 3, 9, 9, 9, 3, 1, 0, 1, 4, 12, 18, 18, 12, 4, 1, 0, 1, 4, 16, 24, 36, 24, 16, 4, 1, 0, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 0, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 0, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1

Offset: 0

Philippe Deléham, Jan 25 2010

Number of symmetric Dyck paths of semilength n with k peaks.

			Triangle begins :
  1;
  0,  1;
  0,  1,  1;
  0,  1,  1,  1;
  0,  1,  2,  2,  1;
  0,  1,  2,  4,  2,   1;
  0,  1,  3,  6,  6,   3,   1;
  0,  1,  3,  9,  9,   9,   3,   1;
  0,  1,  4, 12, 18,  18,  12,   4,   1;
  0,  1,  4, 16, 24,  36,  24,  16,   4,  1;
  0,  1,  5, 20, 40,  60,  60,  40,  20,  5,  1;
  0,  1,  5, 25, 50, 100, 100, 100,  50, 25,  5,  1;
  0,  1,  6, 30, 75, 150, 200, 200, 150, 75, 30,  6,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001405 (row sums), A005566, A084938, A088518 (diagonal sums), A088855.

Column k: A000007 (k=0), A000012 (k=1), A008619 (k=2), A002620 (k=3), A028724 (k=4), A028723 (k=5), A028725 (k=6), A331574 (k=7).

[n eq 0 select 1 else (&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 08 2022

T[n_, k_]:= Product[Binomial[Floor[(n-j)/2], Floor[(k-j)/2]], {j,0,1}];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2022 *)

def A172101(n,k):
    if (n==0): return 1
    else: return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A172101(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 08 2022

Sum_{k=0..n} T(n,k) = A001405(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - [n=1] + A088518(n)*[n >= 1].
From G. C. Greubel, Apr 08 2022: (Start)
T(n, k) = binomial(floor((n-1)/2), floor((k-1)/2))*binomial(floor(n/2), floor(k/2)).
T(2*n, n) = [n=0] + A005566(n-1)*[n >= 1].
T(n-1, n-k) = T(n-1, k), n >= 1, 1 <= k <= n. (End)

A303231 Total volume of all rectangular prisms with dimensions q, p+q and |q-p| such that p and q are prime, n = p+q and p < q.

Original entry on oeis.org

0, 0, 0, 0, 15, 0, 105, 80, 315, 280, 0, 168, 1287, 1232, 2145, 3136, 0, 2664, 4845, 6320, 6783, 11176, 0, 11088, 12075, 17888, 0, 14448, 0, 17640, 24273, 27776, 29667, 62560, 0, 61632, 0, 28272, 50505, 76720, 0, 99120, 68757, 141944, 79335, 163024, 0

Offset: 1

Wesley Ivan Hurt, Apr 20 2018

Index entries for sequences related to partitions

Cf. A010051, A028723, A112742.

Table[n*Sum[(n - i) (n - 2 i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}], {n, 80}]

a(n) = n*sum(i=1, (n-1)\2, (n-i)*(n-2*i)*isprime(i)*isprime(n-i)); \\ Michel Marcus, Apr 21 2018

a(n) = n * Sum_{i=1..floor((n-1)/2)} (n-i) * (n-2*i) * c(i) * c(n-i), where c is the prime characteristic (A010051).

A331576 a(n) is the number of subsets of {1..n} that contain 5 even and 5 odd numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 36, 126, 441, 1176, 3136, 7056, 15876, 31752, 63504, 116424, 213444, 365904, 627264, 1019304, 1656369, 2576574, 4008004, 6012006, 9018009, 13117104, 19079424, 27029184, 38291344, 53018784, 73410624, 99628704, 135210384, 180280512, 240374016, 315490896

Offset: 0

Enrique Navarrete, Jan 20 2020

In general, the number of subsets of {1..n} that contain k even and k odd numbers is given by binomial(n/2, k)^2 for n even and binomial((n-1)/2, k)*binomial((n+1)/2, k) for n odd.

			a(11)=6 and the 6 subsets are {1,2,3,4,5,6,7,8,9,10}, {1,2,3,4,5,6,7,8,10,11}, {1,2,3,4,5,6,8,9,10,11}, {1,2,3,4,6,7,8,9,10,11}, {1,2,4,5,6,7,8,9,10,11}, {2,3,4,5,6,7,8,9,10,11}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,8,-18,-27,72,48,-168,-42,252,0,-252,42,168,-48,-72,27,18,-8,-2,1).

Cf. A028723 (k=2), A331574 (k=3), A331575 (k=4). See comment.

[IsOdd(n) select Binomial((n-1) div 2,5)*Binomial((n+1) div 2,5) else Binomial(n div 2,5)^2: n in [0..41]]; // Marius A. Burtea, Jan 21 2020

a:= n-> ((b, q)-> b(q, 5)*b(n-q, 5))(binomial, iquo(n, 2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 30 2020

a[n_] := If[OddQ[n], Binomial[(n - 1)/2, 5]*Binomial[(n + 1)/2, 5], Binomial[n/2, 5]^2]; Array[a, 42, 0] (* Amiram Eldar, Jan 21 2020 *)

concat([0,0,0,0,0,0,0,0,0,0], Vec(x^10*(1 + 4*x + 16*x^2 + 24*x^3 + 36*x^4 + 24*x^5 + 16*x^6 + 4*x^7 + x^8) / ((1 - x)^11*(1 + x)^9) + O(x^40))) \\ Colin Barker, Jan 21 2020

a(n) = binomial(n/2, 5)^2, for n even;
a(n) = binomial((n-1)/2, 5)*binomial((n+1)/2,5), for n odd.
From Colin Barker, Jan 21 2020: (Start)
G.f.: x^10*(1 + 4*x + 16*x^2 + 24*x^3 + 36*x^4 + 24*x^5 + 16*x^6 + 4*x^7 + x^8) / ((1 - x)^11*(1 + x)^9).
a(n) = 2*a(n-1) + 8*a(n-2) - 18*a(n-3) - 27*a(n-4) + 72*a(n-5) + 48*a(n-6) - 168*a(n-7) - 42*a(n-8) + 252*a(n-9) - 252*a(n-11) + 42*a(n-12) + 168*a(n-13) - 48*a(n-14) - 72*a(n-15) + 27*a(n-16) + 18*a(n-17) - 8*a(n-18) - 2*a(n-19) + a(n-20) for n>19.
(End)
E.g.f.: (cosh(x) - sinh(x))*(99225 + 88200*x + 40950*x^2 + 13050*x^3 + 3225*x^4 + 660*x^5 + 120*x^6 + 20*x^7 + 5*x^8 + (-99225 + 110250*x - 63000*x^2 + 24750*x^3 - 7575*x^4 + 1950*x^5 - 450*x^6 + 100*x^7 - 25*x^8 + 10*x^9 + 2*x^10)*(cosh(2*x) + sinh(2*x)))/29491200. - Stefano Spezia, Jan 27 2020

A307891 Rectilinear crossing number A014540(n) - crossing number A000241(n) of complete graph on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 3, 4, 9, 6, 15, 14, 21, 22, 37, 30, 53, 52, 69, 74, 102, 96

Offset: 1

Ed Pegg Jr, May 03 2019

			For 8 nodes the crossing number is 18 and the rectilinear crossing number is 19.  The difference for 8 nodes is 1.  Thus a(8)=1.

Cf. A000241, A014540, A028723.

cp's OEIS Frontend

A352665 Maximum number of induced copies of the 4-node path in an n-node graph.

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A302647 a(n) = (2*n^2*(n^2 - 3) - (2*n^2 + 1)*(-1)^n + 1)/64.

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A007333 An upper bound on the biplanar crossing number of the complete graph on n nodes.

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A011845 a(n) = floor( binomial(n,8)/9).

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A331575 a(n) is the number of subsets of {1..n} that contain 4 even and 4 odd numbers.

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A172101 Triangle, read by rows, given by [0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...] DELTA [1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, ...] where DELTA is the operator defined in A084938.

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A303231 Total volume of all rectangular prisms with dimensions q, p+q and |q-p| such that p and q are prime, n = p+q and p < q.

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A302647 a(n) = (2n^2(n^2 - 3) - (2n^2 + 1)(-1)^n + 1)/64.