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A162147 a(n) = n*(n+1)*(5*n+4)/6. 7
0, 3, 14, 38, 80, 145, 238, 364, 528, 735, 990, 1298, 1664, 2093, 2590, 3160, 3808, 4539, 5358, 6270, 7280, 8393, 9614, 10948, 12400, 13975, 15678, 17514, 19488, 21605, 23870, 26288, 28864, 31603, 34510, 37590, 40848, 44289, 47918, 51740, 55760 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Partial sums of A005475.

Suppose we extend the triangle in A215631 to a symmetric array by reflection about the main diagonal. The array is defined by m(i,j) = i^2+i*j+j^2: 3, 7, 13, ...; 7, 12, 19, ...; 13, 19, 27, ....  Then a(n) is the sum of the n-th antidiagonal. Examples: 3, 7 + 7, 13 + 12 + 13, 21 + 19 + 19 + 21, etc. - J. M. Bergot, Jun 25 2013

Binomial transform of [0,3,8,5,0,0,0...]. - Alois P. Heinz, Mar 10 2015

LINKS

Table of n, a(n) for n=0..40.

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

FORMULA

a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = A033994(n)+A000217(n). G.f.: x*(3+2*x)/(x-1)^4. - R. J. Mathar, Jun 27 2009

a(n) = A035005(n+1)/4. - Johannes W. Meijer, Feb 04 2010

a(n) = Sum_{i = 0..n}  i*(n + 1 + i). - Bruno Berselli, Mar 17 2016

EXAMPLE

For n=4, a(4) = 0*(5+0) + 1*(5+1) + 2*(5+2) + 3*(5+3) + 4*(5+4) = 80. - Bruno Berselli, Mar 17 2016

MATHEMATICA

f[n_]:=5*n+3; s1=s2=0; lst={}; Do[a=f[n]; s1+=a; s2+=s1; AppendTo[lst, s2], {n, 0, 6!}]; lst

Table[(n(n+1)(5n+4))/6, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 3, 14, 38}, 50] (* Harvey P. Dale, May 04 2013 *)

PROG

(PARI) a(n)=n*(n+1)*(5*n+4)/6 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A033994, A002413, A016061, A006331, A002412.

Sequence in context: A068044 A141129 A143941 * A319791 A027444 A000263

Adjacent sequences:  A162144 A162145 A162146 * A162148 A162149 A162150

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

EXTENSIONS

Definition rephrased by R. J. Mathar, Jun 27 2009

STATUS

approved

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Last modified October 17 01:03 EDT 2019. Contains 328103 sequences. (Running on oeis4.)