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Truncated triangular pyramid numbers: a(n) = (n-5)*(n^2 + 8*n - 66)/6.
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%I #41 May 05 2022 04:53:21

%S 3,13,31,58,95,143,203,276,363,465,583,718,871,1043,1235,1448,1683,

%T 1941,2223,2530,2863,3223,3611,4028,4475,4953,5463,6006,6583,7195,

%U 7843,8528,9251,10013,10815,11658,12543,13471,14443,15460,16523,17633,18791

%N Truncated triangular pyramid numbers: a(n) = (n-5)*(n^2 + 8*n - 66)/6.

%H Stefano Spezia, <a href="/A051939/b051939.txt">Table of n, a(n) for n = 6..10000</a>

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

%F a(n) = Sum_{k=6..n} (k*(k+1)/2 - 18).

%F Equals binomial transform of (3, 10, 8, 1, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 03 2008

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(6)=3, a(7)=13, a(8)=31, a(9)=58. - _Harvey P. Dale_, Oct 22 2011

%F G.f.: x^6*(-3*x^2+x+3)/(x-1)^4. - _Harvey P. Dale_, Oct 22 2011

%F Sum_{k>=6} 1/a(k) = (3/82)*((9*sqrt(82) - 82)*H(9+sqrt(82)) - (9*sqrt(82) + 82)*H(9-sqrt(82))) = 0.5039898035928909... where H(x) = Integral_{t=0..1} (1 - t^x)/(1 - t) dt is the function that interpolates the harmonic numbers. - _Stefano Spezia_, Apr 17 2022

%p A051939:=n->(n-5)*(n^2+8*n-66)/6; seq(A051939(k), k=6..100); # _Wesley Ivan Hurt_, Nov 04 2013

%t Table[(1/6)*(n - 5)*(n^2 + 8*n - 66), {n, 6, 60}] (* _Stefan Steinerberger_, Mar 31 2006 *)

%t LinearRecurrence[{4,-6,4,-1},{3,13,31,58},60] (* _Harvey P. Dale_, Oct 22 2011 *)

%o (PARI) a(n)=(n-5)*(n^2+8*n-66)/6 \\ _Charles R Greathouse IV_, Nov 10 2015

%o (Magma) [(n-5)*(n^2+8*n-66)/6 : n in [6..70]]; // _Wesley Ivan Hurt_, Apr 21 2021

%Y Cf. A000292.

%K easy,nice,nonn

%O 6,1

%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 21 1999