A175009 Triangle read by rows, antidiagonals of an array formed from variants of A001318, generalized pentagonal numbers.

1, 1, 2, 1, 3, 5, 1, 4, 9, 7, 1, 5, 13, 13, 12, 1, 6, 17, 19, 23, 15, 1, 7, 21, 25, 34, 29, 22, 1, 8, 25, 31, 45, 43, 43, 26, 1, 9, 29, 37, 56, 57, 64, 51, 35, 1, 10, 33, 43, 67, 71, 85, 76, 69, 40, 1, 11, 37, 49, 78, 85, 106, 101, 103, 79, 51

Offset: 1

Author

Gary W. Adamson, Apr 03 2010

Keywords

• nonn
• tabl

Examples

			First few rows of the array:
  1, 2,  5,  7,  12,  15,  22,  26,  35,  40, ...
  1, 3,  9, 13,  23,  29,  43,  51,  69,  79, ...
  1, 4, 13, 19,  34,  43,  64,  76, 103, 118, ...
  1, 5, 17, 25,  45,  57,  85, 101, 137, 157, ...
  1, 6, 21, 31,  56,  71, 106, 126, 171, 196, ...
  ...
Example: row 3 is generated from 3 * (1, 3, 2, 5, 3, 7, ...) = (3, 9, 6, 15,...)
Preface with a 1 getting (1, 3, 9, 6, 15, ...) then take partial sums, = (1, 4, 13, 19, 34, 43, 64, ...).
...
First few rows of the triangle:
  1;
  1,  2
  1,  3,  5;
  1,  4,  9,  7;
  1,  5, 13, 13,  12;
  1,  6, 17, 29,  23,  15;
  1,  7, 21, 25,  34,  29,  22;
  1,  8, 25, 31,  45,  43,  43,  26;
  1,  9, 29, 37,  56,  57,  64,  51,  35;
  1, 10, 33, 43,  67,  71,  85,  76,  69,  40;
  1, 11, 37, 49,  78,  85, 106, 101, 103,  79,  51;
  1, 12, 41, 55,  89,  99, 127, 126, 137, 118, 101,  57;
  1, 13, 45, 61, 100, 113, 148, 151, 171, 157, 151, 113,  70;
  1, 14, 49, 67, 111, 127, 169, 176, 205, 196, 201, 169, 139, 77;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Crossrefs

Row sums are A175006.

Cf. A001318, A026741.

Programs

PARI

T(n,k)=if(k<=n, 1 + (n-k+1)*(binomial(k+1, 2) - 1 - binomial(k\2+1, 2)), 0) \\ Andrew Howroyd, Sep 08 2018

Formula

Let row 1 of the array = A001318 starting with offset 1: (1, 2, 5, 7, 12,...)

For rows k>1, begin with A026741 starting (1, 3, 2, 5, 3, 7, 4, 9, 5, 11,...)

= generator Q. Then k-th row = partial sums of (1,...(k * Q)).

T(n,k) = 1 + (n-k+1)*(binomial(k+1, 2) - 1 - binomial(floor(k/2)+1, 2)). - Andrew Howroyd, Sep 08 2018

Extensions

a(22) corrected by Andrew Howroyd, Sep 08 2018