A081489 a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3).

1, 3, 8, 18, 35, 61, 98, 148, 213, 295, 396, 518, 663, 833, 1030, 1256, 1513, 1803, 2128, 2490, 2891, 3333, 3818, 4348, 4925, 5551, 6228, 6958, 7743, 8585, 9486, 10448, 11473, 12563, 13720, 14946, 16243, 17613, 19058, 20580, 22181, 23863, 25628

Offset: 1

Author

Amarnath Murthy, Mar 25 2003

Keywords

• nonn
• easy

Comments

Diagonal of triangle in A081491.

First difference is given by A002522 = n^2 + 1. Second difference is odd numbers given by A005408.

With offset 0, this is the binomial transform of (0,1,1,2,0,0,0,...). - Paul Barry, Jul 03 2003

Equals row sums of triangle A144337. - Gary W. Adamson, Sep 18 2008

a(n) = sum of first (n-1) squares + n; e.g., a(5) = 35 = (1 + 4 + 9 + 16 + 5). - Gary W. Adamson, Aug 27 2010

Equals the number of functions from {1,2,...,n} to {1,2,...,n} that occur as compositions of U(x) = min{x+1,n} and D(x) = max{x-1,1}, including the identity function (the empty composition). Problem 11901 in The American Mathematical Monthly, volume 123, page 399, April 2016), proposed by Don Knuth, asked for the count of such functions (solution submitted to Monthly by Jerrold W. Grossman and Serge Kruk, August 21, 2016). - Jerrold Grossman, Aug 21 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jerrold W. Grossman and Serge Kruk, Solution to Problem 11901 in The American Mathematical Monthly, 2016
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002522, A005408, A081490, A081491, A081492, A144337.

Programs

GAP

List([1..50], n-> n*(2*n^2-3*n+7)/6); # G. C. Greubel, Aug 13 2019

Magma

I:=[1,3,8,18]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 28 2014

Maple

with(combinat):a:=n->sum(fibonacci(3,i), i=0..n): seq(a(n), n=0..42); # Zerinvary Lajos, Mar 20 2008

Mathematica

Table[n*(2*n^2-3*n+7)/6, {n,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 07 2008, modified by G. C. Greubel, Aug 13 2019 *)
LinearRecurrence[{4,-6,4,-1},{1,3,8,18},50] (* Harvey P. Dale, Aug 07 2025 *)

PARI

my(x='x+O(x^50)); Vec(serlaplace(exp(x)*(x+x^2/2+x^3/3)))

Python

def A081489(n): return n*(n*((n<<1)-3)+7)//6 # Chai Wah Wu, Nov 05 2024

Sage

[n*(2*n^2-3*n+7)/6 for n in (1..50)] # G. C. Greubel, Aug 13 2019

Formula

a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3). - Paul Barry, Jul 03 2003

E.g.f.: exp(x)*(x +x^2/2 +x^3/3).

O.g.f.: x*(1-x+2*x^2)/(1-x)^4. - Colin Barker, Jul 28 2012

a(n) = 2*n + Sum_{i=1..n} (i^2 - 2*i). - Wesley Ivan Hurt, Feb 25 2014

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003

New name, using the formulas of Paul Barry, Joerg Arndt, Feb 28 2014