A050509 House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.

1, 10, 36, 88, 175, 306, 490, 736, 1053, 1450, 1936, 2520, 3211, 4018, 4950, 6016, 7225, 8586, 10108, 11800, 13671, 15730, 17986, 20448, 23125, 26026, 29160, 32536, 36163, 40050, 44206, 48640, 53361, 58378, 63700, 69336, 75295, 81586, 88218, 95200, 102541

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 28 1999

Keywords

• nonn
• nice
• easy

Comments

Also as a(n) = (1/6)*(9*n^3 - 3*n^2), n>0: structured pentagonal prism numbers (Cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of inequivalent tetrahedral edge colorings using at most n+1 colors so that no color appears only once. - David Nacin, Feb 22 2017

Examples

			        *     *
a(2) = * * + * * = 10.
       * *   * *
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000217, A000578, A051662, A100145, A100177.

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

Programs

Magma

[(3*n+2)*(n+1)^2/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

Mathematica

Table[((1+n)^2*(2+3n))/2,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,10,36,88},40] (* Harvey P. Dale, Jun 26 2011 *)

PARI

a(n)=(1/2)*(3*n+2)*(n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = A000578(n+1) + (n+1)*A000217(n).

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=10, a(2)=36, a(3)=88. - Harvey P. Dale, Jun 26 2011

G.f.: (1+6*x+2*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012

a(n) = Sum_{i=0..n} (n+1)*(3*i+1). - Bruno Berselli, Sep 08 2015

Sum_{n>=0} 1/a(n) = 9*log(3) - sqrt(3)*Pi - Pi^2/3 = 1.15624437161388... . - Vaclav Kotesovec, Oct 04 2016

E.g.f.: exp(x)*(2 + 18*x + 17*x^2 + 3*x^3)/2. - Elmo R. Oliveira, Aug 06 2025