A006597 a(n) = n^2*(5*n-3)/2.

0, 1, 14, 54, 136, 275, 486, 784, 1184, 1701, 2350, 3146, 4104, 5239, 6566, 8100, 9856, 11849, 14094, 16606, 19400, 22491, 25894, 29624, 33696, 38125, 42926, 48114, 53704, 59711, 66150, 73036, 80384, 88209, 96526, 105350, 114696, 124579, 135014, 146016, 157600

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Structured heptagonal prism numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Apart from 0, partial sums of A220083. - Bruno Berselli, Dec 11 2012

References

• W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 29.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A100177 - structured prisms; A100145 for more on structured numbers.

Cf. similar sequences, with the formula (k*n - k + 2)*n^2/2, listed in A262000.

Cf. A006592, A220083.

Programs

Magma

[n^2*(5*n-3)/2: n in [0..40]]; // Vincenzo Librandi, Jul 20 2011

Maple

A006597:=n->n^2*(5*n-3)/2; seq(A006597(n), n=0..40); # Wesley Ivan Hurt, Mar 11 2014

Mathematica

Table[n^2*(5*n-3)/2, {n, 0, 40}] (* Wesley Ivan Hurt, Mar 11 2014 *)

PARI

a(n)=n^2*(5*n-3)/2; \\ Joerg Arndt, Jul 20 2011

Formula

a(n) = (1/6)*(15*n^3 - 9*n^2). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

G.f.: x*(1+10*x+4*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012

a(n) = Sum_{i=0..n-1} n*(5*i+1) for n>0. - Bruno Berselli, Sep 08 2015

Sum_{n>=1} 1/a(n) = 1.1080093773051638036... = (sqrt(5*(5 - 2*sqrt(5)))*Pi - Pi^2 - 5*sqrt(5)*arccoth(sqrt(5)) + (25*log(5))/2)/9. - Vaclav Kotesovec, Oct 04 2016

From Elmo R. Oliveira, Aug 06 2025: (Start)

E.g.f.: exp(x)*x*(2 + 12*x + 5*x^2)/2.

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

a(n) = A006592(n)/4. (End)

Extensions

Name corrected by Arkadiusz Wesolowski, Jul 20 2011