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

1,2

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).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A233546 A092069 A103115 * A222512 A297539 A323691

Adjacent sequences: A003774 A003775 A003776 * A003778 A003779 A003780

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 14 1998

Extensions

More terms from Wesley Ivan Hurt, Jun 04 2014

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

Status

approved