A051939 Truncated triangular pyramid numbers: a(n) = (n-5)*(n^2 + 8*n - 66)/6.

3, 13, 31, 58, 95, 143, 203, 276, 363, 465, 583, 718, 871, 1043, 1235, 1448, 1683, 1941, 2223, 2530, 2863, 3223, 3611, 4028, 4475, 4953, 5463, 6006, 6583, 7195, 7843, 8528, 9251, 10013, 10815, 11658, 12543, 13471, 14443, 15460, 16523, 17633, 18791

Offset

6,1

Links

Stefano Spezia, Table of n, a(n) for n = 6..10000

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

Formula

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

Equals binomial transform of (3, 10, 8, 1, 0, 0, 0, ...). - Gary W. Adamson, Jul 03 2008

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

G.f.: x^6*(-3*x^2+x+3)/(x-1)^4. - Harvey P. Dale, Oct 22 2011

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

Maple

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

Mathematica

Table[(1/6)*(n - 5)*(n^2 + 8*n - 66), {n, 6, 60}] (* Stefan Steinerberger, Mar 31 2006 *)
LinearRecurrence[{4, -6, 4, -1}, {3, 13, 31, 58}, 60] (* Harvey P. Dale, Oct 22 2011 *)

Prog

(PARI) a(n)=(n-5)*(n^2+8*n-66)/6 \\ Charles R Greathouse IV, Nov 10 2015
(Magma) [(n-5)*(n^2+8*n-66)/6 : n in [6..70]]; // Wesley Ivan Hurt, Apr 21 2021

Crossrefs

Cf. A000292.

Sequence in context: A179027 A145907 A054554 * A257764 A146728 A082709

Adjacent sequences: A051936 A051937 A051938 * A051940 A051941 A051942

Keyword

easy,nice,nonn

Author

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

Status

approved