A106465 -id:A106465 - cp's OEIS frontend

A106468 Absolute value of inverse of number triangle A106465.

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

Offset: 0

Paul Barry, May 03 2005

A 'mixed' sequence array : rows alternate between the rows of the sequence array for the sequence (1,0,1,0,0,0...) and the sequence array for the sequence (1,1,1,1,0,0,0,...). Column 2k has g.f. x^2k(1+x+x^2+x^3); column 2k+1 has g.f. x^(2k+1)(1+x^2). Row sums are A106469. A106468=mod(A106465^(-1),2).

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

A106467 Inverse of number triangle A106465.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 1, -1, -1, 1, 0, 0, -1, 0, 1, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1

Offset: 0

Paul Barry, May 03 2005

A 'mixed' sequence array : rows alternate between the rows of the sequence array for the sequence (1,0,-1,0,0,0...) and the sequence array for the sequence (1,-1,-1,1,0,0,0,...). Column 2k has g.f. x^2k(1-x-x^2+x^3); column 2k+1 has g.f. x^(2k+1)(1-x^2). Row sums are 0^n=binomial(0,n)=(1,0,0,0,....). Diagonal sums are (1,-1,0,1,0,-1,...) with g.f. (1-x+x^2)/(1+x^2).

			Triangle begins
1;
-1,1;
-1,0,1;
1,-1,-1,1;
0,0,-1,0,1;
0,0,1,-1,1,1;
0,0,0,0,-1,0,1;

A106466 Interleave 1,2,3,.. with 1,1,2,2,3,3,...

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 2, 5, 3, 6, 3, 7, 4, 8, 4, 9, 5, 10, 5, 11, 6, 12, 6, 13, 7, 14, 7, 15, 8, 16, 8, 17, 9, 18, 9, 19, 10, 20, 10, 21, 11, 22, 11, 23, 12, 24, 12, 25, 13, 26, 13, 27, 14, 28, 14, 29, 15, 30, 15, 31, 16, 32, 16, 33, 17, 34, 17, 35, 18, 36

Offset: 0

Paul Barry, May 03 2005

Diagonal sums of A106465.

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

With[{no=36},Riffle[Range[no],Sort[Join[Range[no/2],Range[no/2]]]]]  (* Harvey P. Dale, Feb 20 2011 *)

G.f.: (1+x+x^2)/((1+x^2)(1-x^2)^2);
a(n) = a(n-2) + a(n-4) - a(n-6);
a(n) = Sum_{k=0..floor(n/2)} (gcd(n-k+1, k+1) mod 2);
a(n) = sin(Pi*n/2)/4 + (n+2)*(-1)^n/8 + 3(n+2)/8;
a(n) = 0^n + Sum_{k=0..n-1} (if(k=floor((n-1)/2) or k=floor(n/2),1,0)*floor((k+2)/2)). - Paul Barry, Nov 22 2009
Also, a(n) = ((3 + (-1)^n)*(n+2) - i*(i^n - (-i)^n))/8, where i is the imaginary unit. - Bruno Berselli, Feb 22 2010

A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)binomial(n-j,k)((j+1) mod 2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 32, 21, 6, 1, 1, 7, 31, 65, 65, 31, 7, 1, 1, 8, 43, 116, 161, 116, 43, 8, 1, 1, 9, 57, 189, 341, 341, 189, 57, 9, 1, 1, 10, 73, 288, 645, 842, 645, 288, 73, 10, 1, 1, 11, 91, 417, 1121, 1827, 1827, 1121, 417, 91

Offset: 0

Paul Barry, May 31 2005

Or as a square array read by antidiagonals, T(n,k) = Sum_{j=0..n} binomial(k,j)*binomial(n+k-j,k)*((j+1) mod 2).
A symmetric number triangle based on 1/(1-x^2).
The construction of a symmetric triangle in this example is general. Let f(n) be a sequence, preferably with f(0)=1. Then T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*f(j) yields a symmetric triangle. When f(n)=1^n, we get Pascal's triangle. When f(n)=2^n, we get the Delannoy triangle (see A008288). In general, f(n)=k^n yields a (1,k,1)-Pascal triangle (see A081577, A081578). Row sums of triangle are A100131. Diagonal sums of the triangle are A108351. Triangle mod 2 is A106465.

			Triangle rows begin
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 13, 13,  5,  1;
  1,  6, 21, 32, 21,  6,  1;
As a square array read by antidiagonals, rows begin
  1, 1,  1,   1,   1,    1,    1, ...
  1, 2,  3,   4,   5,    6,    7, ...
  1, 3,  7,  13,  21,   31,   43, ...
  1, 4, 13,  32,  65,  116,  189, ...
  1, 5, 21,  65, 161,  341,  645, ...
  1, 6, 31, 116, 341,  842, 1827, ...
  1, 7, 43, 189, 645, 1827, 4495, ...

trgn(nn) = {for (n= 0, nn, for (k = 0, n, print1(sum(j=0, n-k, binomial(k,j)*binomial(n-j,k)*((j+1) % 2)), ", ");); print(););} \\ Michel Marcus, Sep 11 2013

Row k (and column k) has g.f. (1+C(k,2)x^2)/(1-x)^(k+1).

A106471 A number triangle with duplicated columns of the form 2^n - Sum_{j=0..2k-1} C(n,j).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 4, 1, 16, 8, 11, 4, 1, 32, 16, 26, 11, 6, 1, 64, 32, 57, 26, 22, 6, 1, 128, 64, 120, 57, 64, 22, 8, 1, 256, 128, 247, 120, 163, 64, 37, 8, 1, 512, 256, 502, 247, 382, 163, 130, 37, 10, 1, 1024, 512, 1013, 502, 848, 382, 386, 130, 56, 10, 1, 2048, 1024

Offset: 0

Paul Barry, May 03 2005

Columns include A000079, A000295, A002663, A035038, A035040.
Row sums are A106472.
Product of binomial matrix binomial(n,k) and number triangle A106465.

			Triangle begins
   1;
   2,  1;
   4,  2,  1;
   8,  4,  4,  1;
  16,  8, 11,  4,  1;
  32, 16, 26, 11,  6, 1;
  64, 32, 57, 26, 22, 6, 1;

Column 2k has g.f. x^(2*k)/((1-2*x)*(1-x)^(2*k-2)).

Column 2k+1 has g.f. x^(2*k+1)/((1-2*x)*(1-x)^(2*k)).

A359366 a(n) = (1/8)(((3n + 1) + (n - 1)(-1)^n)(n + 1)).

Original entry on oeis.org

0, 1, 3, 4, 10, 9, 21, 16, 36, 25, 55, 36, 78, 49, 105, 64, 136, 81, 171, 100, 210, 121, 253, 144, 300, 169, 351, 196, 406, 225, 465, 256, 528, 289, 595, 324, 666, 361, 741, 400, 820, 441, 903, 484, 990, 529, 1081, 576, 1176, 625, 1275, 676, 1378, 729, 1485

Offset: 0

Peter Luschny, Dec 30 2022

Daniel Mondot, Table of n, a(n) for n = 0..9999
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A014105, A106465, A056136.

a := n -> (1/8)*(((3*n + 1) + (n - 1)*(-1)^n)*(n + 1)):
# Or:
a := n -> ifelse(irem(n, 2) = 1, ((n + 1) / 2)^2, (n^2 + n)/2):
seq(a(n), n = 0..54);

a[n_] := (1/8)*(((3*n + 1) + (n - 1)*(-1)^n)*(n + 1)); Array[a,55,0] (* Stefano Spezia, Apr 28 2025 *)

a(n) = ((n + 1) / 2)^2 if n is odd, otherwise (n^2 + n) / 2.
a(n) = [x^n] -(x*(x^3 + x^2 + 3*x + 1))/(x^2 - 1)^3.
a(n) = n! * [x^n] (1/4)*((1 + x*(x + 4))*sinh(x) + x*(2*x + 3)*cosh(x)).

cp's OEIS Frontend

A106468 Absolute value of inverse of number triangle A106465.

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A106467 Inverse of number triangle A106465.

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A106466 Interleave 1,2,3,.. with 1,1,2,2,3,3,...

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A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).

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A106471 A number triangle with duplicated columns of the form 2^n - Sum_{j=0..2k-1} C(n,j).

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A359366 a(n) = (1/8)*(((3*n + 1) + (n - 1)*(-1)^n)*(n + 1)).

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Mathematica

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A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)binomial(n-j,k)((j+1) mod 2).

A359366 a(n) = (1/8)(((3n + 1) + (n - 1)(-1)^n)(n + 1)).