A127736 a(n) = n*(n^2 + 2*n - 1)/2.

1, 7, 21, 46, 85, 141, 217, 316, 441, 595, 781, 1002, 1261, 1561, 1905, 2296, 2737, 3231, 3781, 4390, 5061, 5797, 6601, 7476, 8425, 9451, 10557, 11746, 13021, 14385, 15841, 17392, 19041, 20791, 22645, 24606, 26677, 28861, 31161, 33580, 36121, 38787, 41581

Offset

1,2

Comments

Row sums of A127735.

Row sums of A162610. - Reinhard Zumkeller, Jan 19 2013

For n > 0, a(n) is the number of compositions of n+10 into n parts avoiding parts 2 and 3. - Milan Janjic, Jan 07 2016

Sum of the numbers in the top row and last column of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows (see example). - Wesley Ivan Hurt, May 18 2021

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

Row sums of triangle A134390. Also, binomial transform of [1, 6, 8, 3, 0, 0, 0, ...). - Gary W. Adamson, Oct 23 2007

a(n) = (n+1)*A000217(n) - n = A006002(n) - n. - R. J. Mathar, Jul 21 2009

From Colin Barker, Mar 12 2014: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

G.f.: -x*(x^2-3*x-1) / (x-1)^4. (End)

a(n) = A057145(n+5,n). - R. J. Mathar, Jul 28 2016

Example

From Wesley Ivan Hurt, May 18 2021: (Start)
Add all the numbers in the top row and last column.
                                                      [1   2  3  4  5]
                                      [1   2  3  4]   [6   7  8  9 10]
                            [1 2 3]   [5   6  7  8]   [11 12 13 14 15]
                   [1 2]    [4 5 6]   [9  10 11 12]   [16 17 18 19 20]
           [1]     [3 4]    [7 8 9]   [13 14 15 16]   [21 22 23 24 25]
------------------------------------------------------------------------
  n         1        2         3            4                 5
------------------------------------------------------------------------
  a(n)      1        7        21           46                85
------------------------------------------------------------------------
(End)

Maple

A127736:=n->n*(n^2+2*n-1)/2; seq(A127736(n), n=1..40); # Wesley Ivan Hurt, Mar 14 2014

Mathematica

Table[n*(n^2 + 2*n - 1)/2, {n, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 24 2011 *)
CoefficientList[Series[-(x^2 - 3 x - 1)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 7, 21, 46}, 60] (* Harvey P. Dale, Apr 22 2014 *)

Prog

(PARI) Vec(-x*(x^2-3*x-1)/(x-1)^4 + O(x^100)) \\ Colin Barker, Mar 12 2014
(PARI) a(n) = n*(n^2+2*n-1)/2; \\ Altug Alkan, Jan 07 2016

Crossrefs

Cf. A002260, A127701, A127735, A134390.

Sequence in context: A299254 A146411 A287431 * A212677 A146802 A147054

Adjacent sequences: A127733 A127734 A127735 * A127737 A127738 A127739

Keyword

nonn,easy

Author

Gary W. Adamson, Jan 26 2007

Extensions

More terms and new name from R. J. Mathar, Jul 21 2009

Status

approved