A003777 a(n) = n^3 + n^2 - 1.

1, 11, 35, 79, 149, 251, 391, 575, 809, 1099, 1451, 1871, 2365, 2939, 3599, 4351, 5201, 6155, 7219, 8399, 9701, 11131, 12695, 14399, 16249, 18251, 20411, 22735, 25229, 27899, 30751, 33791, 37025, 40459, 44099, 47951, 52021, 56315, 60839, 65599, 70601, 75851

Offset: 1

Author

N. J. A. Sloane, Jun 14 1998

Keywords

• nonn
• easy

Comments

This sequence in related to A095794 by a(n) = n*A095794(n) - Sum_{i=1..n-1} A095794(i), for n > 1. - Bruno Berselli, Dec 28 2010

a(n) is the area of a triangle with vertices at points (n-1,(n-1)^2), (n,n^2), and ((n+1)^2,n+1). - J. M. Bergot, Jun 03 2014

Old name was: "Number of stacks of n pikelets, distance 3 flips from a well-ordered stack".

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000578, A001107, A002411, A028294, A028295, A095794.

Programs

Magma

[(n^3+n^2-1): n in [1..50]]; // Vincenzo Librandi, Apr 06 2011

Maple

A003777:=n->n^3+n^2-1; seq(A003777(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014

Mathematica

Table[n^3+n^2-1,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
CoefficientList[Series[(1 + 7 x - 3 x^2 + x^3)/(1-x)^4, {x,0,50}], x] (* Vincenzo Librandi, Jun 20 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,11,35,79},50] (* Harvey P. Dale, Jul 20 2024 *)

PARI

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

SageMath

[n^3+n^2-1 for n in range(1,51)] # G. C. Greubel, Jan 03 2024

Formula

G.f.: x*(1+7*x-3*x^2+x^3)/(1-x)^4. Also, a(n) = 2*A002411(n) - 1 = A000578(n-1) + A001107(n). - Bruno Berselli, Dec 28 2010

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. - Wesley Ivan Hurt, Oct 08 2017

E.g.f.: 1 + (-1 + 2*x + 4*x^2 + x^3)*exp(x). - G. C. Greubel, Jan 03 2024

Extensions

More terms from Wesley Ivan Hurt, Jun 04 2014

Entry revised by N. J. A. Sloane, Jun 15 2014