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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 (list; table; graph; refs; listen; history; text; internal format)
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.

Cf. A001318, A026741.

Sequence in context: A199847 A047997 A188211 * A297395 A297595 A049069

Adjacent sequences:  A175006 A175007 A175008 * A175010 A175011 A175012

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Apr 03 2010

EXTENSIONS

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

STATUS

approved

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Last modified September 27 12:51 EDT 2021. Contains 347688 sequences. (Running on oeis4.)