cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A098832 Square array read by antidiagonals: even-numbered rows of the table are of the form n*(n+m) and odd-numbered rows are of the form n*(n+m)/2.

Original entry on oeis.org

1, 3, 3, 6, 8, 2, 10, 15, 5, 5, 15, 24, 9, 12, 3, 21, 35, 14, 21, 7, 7, 28, 48, 20, 32, 12, 16, 4, 36, 63, 27, 45, 18, 27, 9, 9, 45, 80, 35, 60, 25, 40, 15, 20, 5, 55, 99, 44, 77, 33, 55, 22, 33, 11, 11, 66, 120, 54, 96, 42, 72, 30, 48, 18, 24, 6, 78, 143, 65, 117, 52, 91, 39, 65, 26, 39, 13, 13
Offset: 1

Views

Author

Eugene McDonnell (eemcd(AT)mac.com), Nov 02 2004

Keywords

Comments

The rows of this table and that in A098737 are related. Given a function f = n/( 1 + (1+n) mod(2) ), row n of A098737 can be derived from row n of T by multiplying the latter by f(n); row n of T can be derived from row n of A098737 by dividing the latter by f(n).

Examples

			Array begins as:
  1,  3,  6, 10, 15, 21,  28,  36,  45 ... A000217;
  3,  8, 15, 24, 35, 48,  63,  80,  99 ... A005563;
  2,  5,  9, 14, 20, 27,  35,  44,  54 ... A000096;
  5, 12, 21, 32, 45, 60,  77,  96, 117 ... A028347;
  3,  7, 12, 18, 25, 33,  42,  52,  63 ... A027379;
  7, 16, 27, 40, 55, 72,  91, 112, 135 ... A028560;
  4,  9, 15, 22, 30, 39,  49,  60,  72 ... A055999;
  9, 20, 33, 48, 65, 84, 105, 128, 153 ... A028566;
  5, 11, 18, 26, 35, 45,  56,  68,  81 ... A056000;
Antidiagonals begin as:
   1;
   3,  3;
   6,  8,  2;
  10, 15,  5,  5;
  15, 24,  9, 12,  3;
  21, 35, 14, 21,  7,  7;
  28, 48, 20, 32, 12, 16,  4;
  36, 63, 27, 45, 18, 27,  9,  9;
  45, 80, 35, 60, 25, 40, 15, 20,  5;
  55, 99, 44, 77, 33, 55, 22, 33, 11, 11;

Links

Crossrefs

Row m of array: A000217 (m=1), A005563 (m=2), A000096 (m=3), A028347 (m=4), A027379 (m=5), A028560 (m=6), A055999 (m=7), A028566 (m=8), A056000 (m=9), A098603 (m=10), A056115 (m=11), A098847 (m=12), A056119 (m=13), A098848 (m=14), A056121 (m=15), A098849 (m=16), A056126 (m=17), A098850 (m=18), A051942 (m=19).
Column m of array: A026741 (m=1), A022998 (m=2), A165351 (m=3).
Cf. A098737, A168509, A181900.

Programs

  • Magma
    A098832:= func< n,k | (1/4)*(3+(-1)^k)*(n+1)*(n-k+1) >;
[A098832(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 31 2022
  • Mathematica
    A098832[n_, k_]:= (1/4)*(3+(-1)^k)*(n+1)*(n-k+1);
Table[A098832[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 31 2022 *)
  • SageMath
    def A098832(n,k): return (1/4)*(3+(-1)^k)*(n+1)*(n-k+1)
flatten([[A098832(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Jul 31 2022

Formula

Item m of row n of T is given (in infix form) by: n T m = n * (n + m) / (1 + m (mod 2)). E.g. Item 4 of row 3 of T: 3 T 4 = 14.
From G. C. Greubel, Jul 31 2022: (Start)
A(n, k) = (1/4)*(3 + (-1)^n)*k*(k+n) (array).
T(n, k) = (1/4)*(3 + (-1)^k)*(n+1)*(n-k+1) (antidiagonal triangle).
Sum_{k=1..n} T(n, k) = (1/8)*(n+1)*( (3*n-1)*(n+1) + (1+(-1)^n)/2 ).
T(2*n-1, n) = A181900(n).
T(2*n+1, n) = 2*A168509(n+1). (End)

Extensions

Missing terms added by G. C. Greubel, Jul 31 2022

A079247 Number of pairs (p,q), 0 <= p < q, such that p+q divides n.

Original entry on oeis.org

1, 2, 3, 4, 4, 7, 5, 8, 8, 10, 7, 15, 8, 13, 14, 16, 10, 21, 11, 22, 18, 19, 13, 31, 17, 22, 22, 29, 16, 38, 17, 32, 26, 28, 26, 47, 20, 31, 30, 46, 22, 50, 23, 43, 42, 37, 25, 63, 30, 48, 38, 50, 28, 62, 38, 61, 42, 46, 31, 86, 32, 49, 55, 64, 44, 74, 35, 64, 50, 74, 37, 99, 38
Offset: 1

Views

Author

Vladeta Jovovic, Feb 03 2003

Keywords

Comments

Equals left border of triangle A158951. - Gary W. Adamson, Mar 31 2009
Equals row sums of triangle A168509. - Gary W. Adamson, Nov 27 2009
Let c(d_x(n)) = (d_x(n) + 1) / 2 if d_x(n) == 1 (mod 2), and d_x(n) / 2 if d_x(n) == 0 (mod 2), where d_x(n) is the x-th divisor of n, 1 <= d_x(n) <= n, and c(d_x(n)) denotes the cardinality of said divisor within the ordered set of naturals sharing its parity. Then, a(n) = Sum_{i=1..A000005(n)} c(d_i(n)). - Christopher Hohl, Apr 16 2019

Examples

			There are 7 pairs (p,q), 0 <= p < q, such that p+q divides 6: (0,1), (0,2), (0,3), (0,6), (1, 2), (1, 5), (2, 4); thus a(6) = 7.
G.f. = x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 5*x^7 + 8*x^8 + 8*x^9 + ...

Crossrefs

Cf. A069734, A008619, A158951, A168509, A001227.
Cf. A000203, A086670.

Programs

  • Maple
    with(numtheory): seq((sigma(n)+tau(2*n)-tau(n))/2,n=1 .. 80); # - Ridouane Oudra, Sep 06 2020
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, (1 + d)\2))} /* Michael Somos, Jun 11 2003 */

Formula

Inverse Moebius transform of A008619 (offset 1). - Michael Somos, Jun 11 2003
G.f.: Sum_{k>=1} x^k / ((1 - x^k) * (1 - x^(2*k))). - Michael Somos, Jun 11 2003
G.f.: Sum_{n>=1} A110654(n)*x^n/(1-x^n). - Mircea Merca, Feb 26 2014
a(n) = (1/2)*(A000203(n) + A001227(n)). - Ridouane Oudra, Sep 06 2020
a(n) = A000203(n) - A086670(n). - Ridouane Oudra, Nov 25 2022

A168508 Triangle read by rows: A101688 * A051731.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 1, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Nov 27 2009

Keywords

Comments

More precisely, form the product of the lower triangular matrix T defined in A101688 and the lower triangular matrix T defined in A051731. - N. J. A. Sloane, Dec 05 2020
Row sums = A060831: (1, 2, 4, 5, 7, 9, 11, 12, 15,...).
A168509 = A051731 * A101688

Examples

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 1, 1, 1, 1;
3, 2, 1, 1, 1, 1;
4, 2, 1, 1, 1, 1, 1;
4, 2, 1, 1, 1, 1, 1, 1;
5, 2, 2, 1, 1, 1, 1, 1, 1;
5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1;
6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
8, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
8, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
9, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...

Crossrefs

Cf. A051731, A060831, A101688, A168509.

Extensions

a(29) = 1 inserted and more terms from Georg Fischer, May 29 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.