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Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2)=t(3)+t(6)=6+21=27.
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%I #32 Aug 24 2024 17:10:28

%S 6,27,72,150,270,441,672,972,1350,1815,2376,3042,3822,4725,5760,6936,

%T 8262,9747,11400,13230,15246,17457,19872,22500,25350,28431,31752,

%U 35322,39150,43245,47616,52272,57222,62475,68040,73926,80142,86697,93600,100860,108486

%N Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2)=t(3)+t(6)=6+21=27.

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

%F a(n) = 3/2 * n*(n+1)^2 = 3 * A006002(n).

%F a(n) = Sum_{j=1..n} (j+n+1)*(n+1). - _Zerinvary Lajos_, Sep 10 2006

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: 3*x*(x+2) / (x-1)^4. - _Colin Barker_, Mar 17 2014

%p a:=n->sum(sum(sum(j-k+1, j=1..n), k=0..n),m=0..n): seq(a(n), n=1..45); # _Zerinvary Lajos_, May 30 2007

%t LinearRecurrence[{4,-6,4,-1},{6,27,72,150},50] (* _Harvey P. Dale_, Dec 14 2017 *)

%o (PARI) v=vector(40,i,i*(i+1)/2); s=0; forstep(i=3,40,3,s+=v[i]; print1(s","))

%Y Cf. A004188, A006002.

%Y Row sums of triangle A001283.

%Y Cf. A254407. - _Bruno Berselli_, Jan 30 2015

%K nonn,easy

%O 1,1

%A _Jon Perry_, Jul 23 2003

%E Edited and more terms from _Michel Marcus_, Mar 17 2014