A057979 -id:A057979 - cp's OEIS frontend

A196389 Triangle T(n,k), read by rows, given by (0,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 1

Offset: 0

Philippe Deléham, Oct 28 2011

Row sums are A028310; diagonal sums are A057979; column sums are A000027.

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 2, 1;
  0, 0, 0, 3, 1;
  0, 0, 0, 0, 4, 1;
  0, 0, 0, 0, 0, 5, 1;
  0, 0, 0, 0, 0, 0, 6, 1;
  0, 0, 0, 0, 0, 0, 0, 7, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 8, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 1; ...

Cf. A084938.

T(n,n)=1, T(n+1,n)=n.
G.f.: (1-x*y+x^2*y)/(1-x*y)^2. - Philippe Deléham, Oct 31 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A028310(n), A057711(n+1), A064017(n+1) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 31 2011

A109094 Length of the longest path (repeated edges not allowed) between two arbitrary, distinct nodes in K_n, the complete graph on n vertices.

Original entry on oeis.org

0, 1, 2, 5, 9, 13, 20, 25, 35, 41, 54, 61, 77, 85, 104, 113, 135, 145, 170, 181, 209, 221, 252, 265, 299, 313, 350, 365, 405, 421, 464, 481, 527, 545, 594, 613, 665, 685, 740, 761, 819, 841, 902, 925, 989, 1013, 1080, 1105, 1175, 1201, 1274, 1301, 1377, 1405

Offset: 1

Ryan Propper, Jun 18 2005

			a(4) = 5 because the length of the longest path between any two distinct vertices in K_4 is 5.

R. J. Mathar, Comments on this sequence
Eric Weisstein's World of Mathematics, Complete Graph.
Eric Weisstein's World of Mathematics, Euler Path.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A057979, A128223.

a(1)=0; a(2n+1) = n*(n-1)/2-1 = A014107(n+1), n>0; a(2n)=n*(n-2)/2+1= A001844(n-1). - Martin Fuller, R. J. Mathar and Mitch Harris, Dec 06 2007

O.g.f.: x^2*(x^4-2*x^3-x^2-x-1)/((-1+x)^3 *(x+1)^2) . - R. J. Mathar, Jan 17 2008

A179820 a(n) = n-th triangular number mod (n+2).

Original entry on oeis.org

0, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 1, 43, 1, 44, 1, 45

Offset: 0

Zak Seidov, Jul 28 2010

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

Cf. A000217, A057979, A007879, A158416.

Essentially the same as A133622.

Table[Mod[n(n+1)/2,n+2],{n,0,200}]
LinearRecurrence[{0,2,0,-1},{0,1,3,1,4},110] (* or *) Join[{0,1},Riffle[Range[3,50],1]] (* Harvey P. Dale, Apr 02 2024 *)

a(0)=0, afterwards if n is odd then a(n)=1 else a(n)=(n+4)/2
a(0)=0, afterwards a(n)=1 for odd n and n/2+2 for even n.
a(n)= +2*a(n-2) -a(n-4), n>4. a(n) = (6+n*((-1)^n+1)+2*(-1)^n)/4, n>0. G.f.: -x*(-1-3*x+x^2+2*x^3) / ( (x-1)^2*(1+x)^2 ). [From R. J. Mathar, Aug 03 2010]

A374440 Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 1, 1, 1, 1, 4, 1, 3, 2, 0, 1, 5, 1, 6, 3, 1, 1, 1, 6, 1, 10, 4, 4, 3, 0, 1, 7, 1, 15, 5, 10, 6, 1, 1, 1, 8, 1, 21, 6, 20, 10, 5, 4, 0, 1, 9, 1, 28, 7, 35, 15, 15, 10, 1, 1, 1, 10, 1, 36, 8, 56, 21, 35, 20, 6, 5, 0

Offset: 0

Peter Luschny, Jul 21 2024

Member of the family of Lucas-Fibonacci polynomials.

			Triangle starts:
  [ 0]  1;
  [ 1]  1,  0;
  [ 2]  1,  1,  1;
  [ 3]  1,  2,  1,  0;
  [ 4]  1,  3,  1,  1,  1;
  [ 5]  1,  4,  1,  3,  2,  0;
  [ 6]  1,  5,  1,  6,  3,  1,  1;
  [ 7]  1,  6,  1, 10,  4,  4,  3,  0;
  [ 8]  1,  7,  1, 15,  5, 10,  6,  1,  1;
  [ 9]  1,  8,  1, 21,  6, 20, 10,  5,  4,  0;
  [10]  1,  9,  1, 28,  7, 35, 15, 15, 10,  1, 1;

Cf. A374441.

Cf. A000032 (Lucas), A001611 (even sums, Fibonacci + 1), A000071 (odd sums, Fibonacci - 1), A001911 (alternating sums, Fibonacci(n+3) - 2), A025560 (row lcm), A073028 (row max), A117671 & A025174 (central terms), A057979 (subdiagonal), A000217 (column 3).

T := proc(n, k) option remember; if k = 0 or k = 2 then 1 elif k > n then 0
elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9);
T := (n, k) -> ifelse(k = 0, 1, binomial(n - floor(k/2), ceil(k/2)) -
binomial(n - ceil((k + irem(k + 1, 2))/2), floor(k/2))):

T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling((k + even(k))/2), floor(k/2)) if k > 0, T(n, 0) = 1, where even(k) = 1 if k is even, otherwise 0.

Columns with odd index agree with the odd indexed columns of A374441.

A115955 Product of A115952 and summing matrix (1/(1-x),x).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Paul Barry, Feb 02 2006

Row sums are A057979(n+3).

			Triangle begins
1,
0, 1,
0, 1, 1,
0, 0, 0, 1,
0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1

A368684 Number of partitions of n into 2 parts such that the smaller part divides both n and floor(n/2).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 5, 1, 2, 1, 6, 1, 2, 1, 6, 1, 4, 1, 4, 1, 2, 1, 8, 1, 3, 1, 4, 1, 4, 1, 6, 1, 2, 1, 8, 1, 2, 1, 6, 1, 4, 1, 4, 1, 4, 1, 9, 1, 2, 1, 4, 1, 4, 1, 8, 1, 2, 1, 8, 1, 2, 1, 6, 1, 6

Offset: 1

Wesley Ivan Hurt, Jan 03 2024

Essentially, A000005 interspersed with 1's [prepend 0].

Number of divisors of A057979(n+1) for n >= 2.

Index entries for sequences related to partitions.

Bisections: A060576, A000005.

Cf. A001620, A057979, A091954.

with(numtheory): 0, seq(2*tau(n) - tau(2*n) + (n mod 2), n=2..100); # Ridouane Oudra, Jan 18 2025

Join[{0}, Table[DivisorSigma[0, (n+2+(n-2)*(-1)^n)/4], {n, 2, 100}]]

a(n) = if(n == 1, 0, numdiv((n+2+(n-2)*(-1)^n)/4)); \\ Amiram Eldar, Jan 28 2025

a(n) = A000005(A057979(n+1)) for n >= 2.
a(2n-1) = A060576(n), a(2n) = A000005(n).
a(n) = d(floor((n+1)/2))^((n+1) mod 2), for n >= 2.
a(n) = d( (n+2+(n-2)*(-1)^n)/4 ) for n >= 2.
a(n) = Sum_{k=1..floor(n/2)} c(n/k) * c(floor(n/2)/k), where c(m) = 1 - ceiling(m) + floor(m).
a(n) = A000005(n) - A091954(n), for n > 1. - Ridouane Oudra, Jan 18 2025
Sum_{k=1..n} a(k) ~ (log(n/2) + 2*gamma)*n/2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 28 2025

cp's OEIS Frontend

A196389 Triangle T(n,k), read by rows, given by (0,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A109094 Length of the longest path (repeated edges not allowed) between two arbitrary, distinct nodes in K_n, the complete graph on n vertices.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A179820 a(n) = n-th triangular number mod (n+2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A374440 Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A115955 Product of A115952 and summing matrix (1/(1-x),x).

Views

Author

Keywords

Comments

Examples

A368684 Number of partitions of n into 2 parts such that the smaller part divides both n and floor(n/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula