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a(n) = n*(n^2 + 2*n - 1)/2.
10

%I #48 Sep 01 2024 14:06:52

%S 1,7,21,46,85,141,217,316,441,595,781,1002,1261,1561,1905,2296,2737,

%T 3231,3781,4390,5061,5797,6601,7476,8425,9451,10557,11746,13021,14385,

%U 15841,17392,19041,20791,22645,24606,26677,28861,31161,33580,36121,38787,41581

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

%C Row sums of A127735.

%C Row sums of A162610. - _Reinhard Zumkeller_, Jan 19 2013

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A127736/b127736.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F Row sums of triangle A131416. Also, binomial transform of [1, 6, 8, 3, 0, 0, 0, ...). - _Gary W. Adamson_, Oct 23 2007

%F a(n) = (n+1)*A000217(n) - n = A006002(n) - n. - _R. J. Mathar_, Jul 21 2009

%F From _Colin Barker_, Mar 12 2014: (Start)

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

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

%F a(n) = A057145(n+5,n). - _R. J. Mathar_, Jul 28 2016

%e From _Wesley Ivan Hurt_, May 18 2021: (Start)

%e Add all the numbers in the top row and last column.

%e [1 2 3 4 5]

%e [1 2 3 4] [6 7 8 9 10]

%e [1 2 3] [5 6 7 8] [11 12 13 14 15]

%e [1 2] [4 5 6] [9 10 11 12] [16 17 18 19 20]

%e [1] [3 4] [7 8 9] [13 14 15 16] [21 22 23 24 25]

%e ------------------------------------------------------------------------

%e n 1 2 3 4 5

%e ------------------------------------------------------------------------

%e a(n) 1 7 21 46 85

%e ------------------------------------------------------------------------

%e (End)

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

%t Table[n*(n^2 + 2*n - 1)/2, {n, 60}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 24 2011 *)

%t CoefficientList[Series[-(x^2 - 3 x - 1)/(x - 1)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 14 2014 *)

%t LinearRecurrence[{4,-6,4,-1},{1,7,21,46},60] (* _Harvey P. Dale_, Apr 22 2014 *)

%o (PARI) Vec(-x*(x^2-3*x-1)/(x-1)^4 + O(x^100)) \\ _Colin Barker_, Mar 12 2014

%o (PARI) a(n) = n*(n^2+2*n-1)/2; \\ _Altug Alkan_, Jan 07 2016

%Y Cf. A002260, A127701, A127735, A131416.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Jan 26 2007

%E More terms and new name from _R. J. Mathar_, Jul 21 2009