login
A175009
Triangle read by rows, antidiagonals of an array formed from variants of A001318, generalized pentagonal numbers.
2
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
1,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
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
EXAMPLE
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;
...
PROG
(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
CROSSREFS
Row sums are A175006.
Sequence in context: A199847 A047997 A188211 * A297395 A297595 A049069
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Apr 03 2010
EXTENSIONS
a(22) corrected by Andrew Howroyd, Sep 08 2018
STATUS
approved