A193974 -id:A193974 - cp's OEIS frontend

A205002 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

2, 2, 3, 4, 3, 5, 4, 8, 4, 6, 6, 5, 7, 5, 6, 16, 9, 6, 10, 6, 8, 7, 12, 9, 7, 8, 7, 11, 15, 8, 16, 32, 8, 10, 8, 12, 19, 11, 9, 10, 21, 9, 22, 9, 10, 13, 24, 17, 10, 12, 11, 10, 27, 10, 13, 11, 12, 16, 30, 11, 31, 17, 11, 64, 11, 17, 34, 12, 14, 13, 36, 12, 37, 20, 12, 13, 12, 18, 40, 18, 13, 22, 42, 14, 13, 23, 17, 13, 45, 13, 16, 15

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

See A204892 for a discussion and guide to related sequences.

			(See example at A205001.)

Antti Karttunen, Table of n, a(n) for n = 1..11001

Cf. A000217, A204892, A193974, A205006, A205007.

s[n_] := s[n] = Binomial[n + 1, 2]; z1 = 500; z2 = 60;
Table[s[n], {n, 1, 30}]  (* A000217 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]  (* A193974 ? *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]      (* A205001 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]      (* A205002 *)
Table[j[n], {n, 1, z2}]      (* A205003 *)
Table[s[k[n]], {n, 1, z2}]   (* A205004 *)
Table[s[j[n]], {n, 1, z2}]   (* A205005 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205006 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205007 *)

A205002(n) = for(k=2,oo,my(sk=binomial(k+1,2)); for(j=1,k-1,if(!((sk-binomial(j+1,2))%n),return(k)))); \\ Antti Karttunen, Sep 27 2018

More terms from Antti Karttunen, Sep 27 2018

A193973 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=xp(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=xp(n-1,x)+1 with p(0,x)=1.

Original entry on oeis.org

2, 3, 5, 4, 7, 9, 5, 9, 12, 14, 6, 11, 15, 18, 20, 7, 13, 18, 22, 25, 27, 8, 15, 21, 26, 30, 33, 35, 9, 17, 24, 30, 35, 39, 42, 44, 10, 19, 27, 34, 40, 45, 49, 52, 54, 11, 21, 30, 38, 45, 51, 56, 60, 63, 65, 12, 23, 33, 42, 50, 57, 63, 68, 72, 75, 77, 13, 25, 36, 46

Offset: 0

Clark Kimberling, Aug 10 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

This array show the differences of the sequence of triangular numbers (A000217); viz., row n consists of t(n) - t(n-k) for k=1..n-1. - Clark Kimberling, Apr 15 2017

			First six rows:
  2;
  3,  5;
  4,  7,  9;
  5,  9, 12, 14;
  6, 11, 15, 18, 20;
  7, 13, 18, 22, 25, 27;
  ...

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A000217, A193664, A193842, A193974.

a000217 := proc(n) n*(n+1)/2 end:
seq(print(seq(a000217(n+2) - a000217(n+1-k),k=0..n)),n=0..5); # Georg Fischer, May 03 2022

z = 13;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193973 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193974 *)

T(n, k) = A000217(n + 2) - A000217(n + 1 - k), 0 <= k <= n. - Georg Fischer, May 03 2022

A205001 Least k such that n divides the k-th difference between distinct triangular numbers.

Original entry on oeis.org

1, 1, 3, 6, 2, 8, 5, 28, 4, 11, 14, 8, 20, 7, 13, 120, 35, 12, 44, 11, 26, 18, 65, 34, 17, 25, 16, 49, 104, 24, 119, 496, 23, 42, 22, 58, 170, 52, 31, 41, 209, 30, 230, 29, 40, 75, 275, 134, 39, 62, 50, 38, 350, 37, 74, 49, 61, 117, 434, 48

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

			The triangular numbers: s(k)=k(k+1)/2.
Their differences, ordered as in A193974:
u(1)=s(2)-s(1)=2
u(2)=s(3)-s(1)=5
u(3)=s(3)-s(2)=3
u(4)=s(4)-s(1)=9
u(5)=s(4)-s(2)=7
u(6)=s(4)-s(3)=4.
a(1)=1 because 1 divides u(1)
a(2)=1 because 2 divides u(1)
a(3)=3 because 3 divides u(3)
a(4)=6 because 4 divides u(6)
a(5)=2 because 5 divides u(2)
a(6)=8 because 6 divides u(8)

Cf. A205002, A204892.

Mathematica
```
(See the program at A205002.)
```

cp's OEIS Frontend

A205002 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

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A193973 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=xp(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=xp(n-1,x)+1 with p(0,x)=1.

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A205001 Least k such that n divides the k-th difference between distinct triangular numbers.

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cp's OEIS Frontend

A205002 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

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Author

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Comments

Examples

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Programs

Mathematica

PARI

Extensions

A193973 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=x*p(n-1,x)+1 with p(0,x)=1.

Views

Author

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Maple

Mathematica

Formula

A205001 Least k such that n divides the k-th difference between distinct triangular numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A193973 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=xp(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=xp(n-1,x)+1 with p(0,x)=1.