A130130 -id:A130130 - cp's OEIS frontend

A382682 Number of integer partitions of n with origin-to-boundary graph-distance equal to 3.

0, 0, 0, 0, 0, 0, 1, 4, 8, 15, 23, 32, 43, 54, 67, 82, 97, 114, 133, 152, 173, 196, 219, 244, 271, 298, 327, 358, 389, 422, 457, 492, 529, 568, 607, 648, 691, 734, 779, 826, 873, 922, 973, 1024, 1077, 1132, 1187, 1244, 1303, 1362, 1423, 1486, 1549, 1614, 1681, 1748, 1817, 1888, 1959

Offset: 0

N Guru Sharan, Jun 03 2025

Also the number of partitions of n with a fixed Durfee triangle of size 3.

Column k=3 of the triangle in A325188.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A130130, A325168, A325188.

CoefficientList[Series[(q^6 + 2 q^7 + q^8 + 2 q^9 - q^10 - q^12 - q^13 + q^14)/((1 - q)^3 (1 + q + q^2)), {q, 0, 200}],q]

G.f.: q^6*(1 + 2*q + q^2 + 2*q^3 - q^4 - q^6 - q^7 + q^8)/((1 - q)^3*(1 + q + q^2)).

9*a(n) = 2*A099837(n+3)+6*n^2+59-45*n for n>9. - R. J. Mathar, Jun 24 2025

A262067 a(n) = n^n - (n-2)^n.

Original entry on oeis.org

2, 4, 26, 240, 2882, 42560, 745418, 15097600, 347066882, 8926258176, 253930611002, 7916100448256, 268352394448322, 9828088361009152, 386707997366768618, 16268790735900180480, 728714136550643404802, 34624041592426892361728

Offset: 1

Altug Alkan, Sep 10 2015

Inspired by multi-dimensional cubes: For n>1, the number of lattice points on the surface of a k-dimensional cube with side-length n is f(n,k) = n^k - (n-2)^k. a(n) = f(n,n).

			For n = 2, a(n) = n^n - (n-2)^n = 2^2 - (2-2)^2 = 4.

Cf. A000312, A008788.

For sequences with "Number of points on surface of k-dimensional cube," cf. A130130 (k=1), A008574 (k=2, shifted), A005897 (k=3), A008511 (k=4), A008512 (k=5), A008513 (k=6).

[n^n - (n-2)^n : n in [1..20]]; // Wesley Ivan Hurt, Sep 10 2015

A262067:=n->n^n - (n-2)^n: seq(A262067(n), n=1..20); # Wesley Ivan Hurt, Sep 10 2015

Array[#^# - (# - 2)^# &, {18}] (* Michael De Vlieger, Sep 10 2015 *)

a(n) = n^n - (n-2)^n;
vector(40, n, a(n))

a(n) = A000312(n) - A008788(n-2).

A384562 Number of integer partitions of n with origin-to-boundary graph-distance equal to 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 12, 24, 42, 66, 98, 135, 181, 233, 298, 367, 452, 543, 651, 765, 899, 1039, 1202, 1371, 1564, 1765, 1993, 2227, 2491, 2763, 3066, 3377, 3722, 4075, 4465, 4863, 5299, 5745, 6232, 6727, 7266, 7815, 8409, 9013, 9665, 10327, 11040, 11763, 12538, 13325, 14167, 15019, 15929, 16851, 17832, 18825, 19880, 20947, 22079, 23223, 24433, 25657, 26950

Offset: 0

N Guru Sharan, Jun 03 2025

This also counts the number of partitions of n with a fixed Durfee triangle of size 4. This is the column k=4 of the triangle in A325188.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A130130, A325168, A325188, A382682.

CoefficientList[Series[(q^10 (1 + 4q + 6 q^2 + 7 q^3 + 6 q^4 + 2 q^5 - 5 q^7 - 5 q^8 - 5 q^9 + q^11 + 3 q^12 + 2 q^13 - q^16))/((1 - q)(1 - q^2)(1 - q^3)(1 - q^4)), {q, 0, 50}], q]

G.f.: q^10*(1 + 4*q + 6*q^2 + 7*q^3 + 6*q^4 + 2*q^5 - 5*q^7 - 5*q^8 - 5*q^9 + q^11 + 3*q^12 + 2*q^13 - q^16)/((1 - q)*(1 - q^2)*(1 - q^3)*(1 - q^4)).

A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 2, 3, 2, 1, 0, 1, 2, 4, 4, 2, 2, 0, 1, 2, 5, 6, 4, 2, 1, 0, 1, 2, 6, 8, 7, 4, 2, 0, 0, 1, 2, 7, 10, 11, 8, 4, 2, 1, 0, 1, 2, 8, 12, 16, 14, 8, 4, 2, 2, 0, 1, 2, 9, 14, 22, 22, 15, 8, 4, 2, 1, 0, 1, 2, 10, 16, 29, 32, 26, 16, 8, 4, 2, 0

Offset: 0

Stefano Spezia, May 19 2024

A(n,k) is also the size of the Hamming ball in {0,1}^n of radius (k-1)/2 if k is odd and of the union of two Hamming balls in {0,1}^n of radius k/2-1 whose centers are of Hamming distance 1 if k is even.

			The array begins:
  1, 1, 2, 1,  0,  1,  2,  1, ...
  0, 1, 2, 2,  2,  2,  2,  2, ...
  0, 1, 2, 3,  4,  4,  4,  4, ...
  0, 1, 2, 4,  6,  7,  8,  8, ...
  0, 1, 2, 5,  8, 11, 14, 15, ...
  0, 1, 2, 6, 10, 16, 22, 26, ...
  0, 1, 2, 7, 12, 22, 32, 42, ...
  0, 1, 2, 8, 14, 29, 44, 64, ...
  ...

Noga Alon, Zhihan Jin, and Benny Sudakov, The Helly number of Hamming balls and related problems, arXiv:2405.10275 [math.CO], 2024. See p. 3.
S. L. Bezrukov, Specification of all maximal subsets of the unit cube with respect to given diameter. Problemy Peredachi Informatsii, pages 106-109, 1987. On ResearchGate.
G. Katona, Intersection theorems for systems of finite sets, Acta Mathematica Academiae Scientiarum Hungaricae 15, 329-337 (1964).
Daniel J. Kleitman, On a combinatorial conjecture of Erdös, Journal of Combinatorial Theory, Series A 1, 209-214, (1966).

Cf. A000007 (k=0), A000012 (k=1), A000124 (k=5), A000125 (k=7), A005843 (k=4), A006261 (k=11), A007395 (k=2), A008859 (k=13), A011782 (main diagonal), A014206, A046127 (k=8), A059173, A059174, A130130 (n=1), A158411 (n=2), A373006 (antidiagonal sums).

A[n_,k_]:=If[OddQ[k],Sum[Binomial[n,i],{i,0,(k-1)/2}], Binomial[n-1,k/2-1]+Sum[Binomial[n,i],{i,0,k/2-1}]]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(n,k) = Sum_{i=0..(k-1)/2} binomial(n,i) if k is odd;
A(n,k) = binomial(n-1,k/2-1) + Sum_{i=0..k/2-1} binomial(n,i) if k is even.
A(n,3) = n+1.
A(n,6) = A014206(n-1).
A(n,9) = A000127(n+1).
A(n,10) = A059173(n) for n > 0.
A(n,12) = A059174(n) for n > 0.
A(0,k) = A007877(k) for k > 0.

A322505 Factorial expansion of 1/sqrt(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

0, 1, 1, 0, 4, 5, 0, 6, 4, 9, 0, 11, 7, 3, 11, 10, 2, 2, 5, 16, 11, 3, 7, 18, 16, 19, 11, 12, 21, 19, 22, 5, 31, 21, 25, 30, 20, 6, 5, 21, 17, 41, 36, 14, 28, 13, 45, 16, 0, 33, 1, 2, 41, 1, 28, 43, 9, 15, 16, 28, 22, 19, 22, 13, 34, 61, 38, 40, 56, 44, 69, 25, 42, 44, 34, 73, 71, 42, 17

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			1/sqrt(2) = 0 + 1/2! + 1/3! + 0/4! + 4/5! + 5/6! + 0/7! + 6/8! + ...

Index entries for factorial base representation

Cf. A010503 (decimal expansion), A130130 (continued fraction).

Cf. A009949 (sqrt(2)).

SetDefaultRealField(RealField(250));  [Floor(1/Sqrt(2))] cat [Floor(Factorial(n)/Sqrt(2)) - n*Floor(Factorial((n-1))/Sqrt(2)) : n in [2..80]];

With[{b = 1/Sqrt[2]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 12 2018 *)

default(realprecision, 250); b = 1/sqrt(2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=1/sqrt(2);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

cp's OEIS Frontend

A382682 Number of integer partitions of n with origin-to-boundary graph-distance equal to 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A262067 a(n) = n^n - (n-2)^n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A384562 Number of integer partitions of n with origin-to-boundary graph-distance equal to 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A322505 Factorial expansion of 1/sqrt(2) = Sum_{n>=1} a(n)/n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage