A165900 a(n) = n^2 - n - 1.

-1, -1, 1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069, 2161, 2255

Offset: 0

Author

Philippe Deléham, Sep 29 2009

Keywords

• sign
• easy

Comments

Previous name was: Values of Fibonacci polynomial n^2 - n - 1.

Shifted version of the array denoted rB(0,2) in A132382, whose e.g.f. is exp(x)(1-x)^2. Taking the derivative gives the e.g.f. of this sequence. - Tom Copeland, Dec 02 2013

The Fibonacci numbers are generated by the series x/(1 - x - x^2). - T. D. Noe, Dec 04 2013

Absolute value of expression f(k)*f(k+1) - f(k-1)*f(k+2) where f(1)=1, f(2)=n. Sign is alternately +1 and -1. - Carmine Suriano, Jan 28 2014 [Can anybody clarify what is meant here? - Joerg Arndt, Nov 24 2014]

Carmine's formula is a special case related to 4 consecutive terms of a Fibonacci sequence. A generalization of this formula is |a(n)| = |f(k+i)*f(k+j) - f(k)*f(k+i+j)|/F(i)*F(j), where f denotes a Fibonacci sequence with the initial values 1 and n, and F denotes the original Fibonacci sequence A000045. The same results can be obtained with the simpler formula |a(n)| = |f(k+1)^2 - f(k)^2 - f(k+1)*f(k)|. Everything said so far is also valid for Fibonacci sequences f with the initial values f(1) = n - 2, f(2) = 2*n - 3. - Klaus Purath, Jun 27 2022

a(n) is the total number of dollars won when using the Martingale method (bet $1, if win then continue to bet $1, if lose then double next bet) for n trials of a wager with exactly one loss, n-1 wins. For the case with exactly one win, n-1 losses, see A070313. - Max Winnick, Jun 28 2022

Numbers m such that 4*m+5 is a square b^2, where b = 2*n -1, for m = a(n). - Klaus Purath, Jul 23 2022

Examples

			G.f. = -1 - x + x^2 + 5*x^3 + 11*x^4 + 19*x^5 + 29*x^6 + 41*x^7 + ... - _Michael Somos_, Mar 23 2023
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Joseph J. Heed and Lucille Kelly, An interesting sequence of Fibonacci sequence generators, Fibonacci Quarterly, 13 (1975), pp. 29-30.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

A028387 and A110331 are very similar sequences.

Cf. A000045, A005843, A015614, A070313, A132382, A152015, A188652, A214803.

Other quadratics: A002061, A002378, A005563, A008865, A014206, A014209, A028552, A034856.

Programs

Haskell

a165900 n = n * (n - 1) - 1  -- Reinhard Zumkeller, Jul 29 2012

Mathematica

Table[n^2 - n - 1, {n, 0, 50}] (* Ron Knott, Oct 27 2010 *)
LinearRecurrence[{3,-3,1},{-1,-1,1},60] (* Harvey P. Dale, Jul 05 2021 *)

PARI

a(n)=n^2-n-1 \\ Charles R Greathouse IV, Jan 12 2012

Formula

a(n+2) = (n+1)*a(n+1) - (n+2)*a(n).

G.f.: (x^2+2*x-1)/(1-x)^3.

E.g.f.: exp(x)*(x^2-1).

a(n) = - A188652(2*n) for n > 0. - Reinhard Zumkeller, Apr 13 2011

a(n) = A214803(A015614(n+1)) for n > 0. - Reinhard Zumkeller, Jul 29 2012

a(n+1) = a(n) + A005843(n) = A002378(n) - 1. - Ivan N. Ianakiev, Feb 18 2013

a(n+2) = A028387(n). - Michael B. Porter, Sep 26 2018

From Klaus Purath, Aug 25 2022: (Start)

a(2*n) = n*(a(n+1) - a(n-1)) -1.

a(2*n+1) = (2*n+1)*(a(n+1) - a(n)) - 1.

a(n+2) = a(n) + 4*n + 2.

a(n) = A014206(n-1) - 3 = A002061(n-1) - 2.

a(n) = A028552(n-2) + 1 = A014209(n-2) + 2 = 2* A034856(n-2) + 3.

a(n) = A008865(n-1) + n = A005563(n-1) - n.

a(n) = A014209(n-3) + 2*n = A028387(n-1) - 2*n.

a(n) = A152015(n)/n, n != 0.

(a(n+k) - a(n-k))/(2*k) = 2*n-1, for any k.

(End)

For n > 1, 1/a(n) = Sum_{k>=1} F(k)/n^(k+1), where F(n) = A000045(n). - Diego Rattaggi, Nov 01 2022

a(n) = a(1-n) for all n in Z. - Michael Somos, Mar 23 2023

For n > 1, 1/a(n) = Sum_{k>=1} F(2k)/((n+1)^(k+1)), where F(2n) = A001906(n). - Diego Rattaggi, Jan 20 2025

From Amiram Eldar, May 11 2025: (Start)

Sum_{n>=1} 1/a(n) = tan(sqrt(5)*Pi/2)*Pi/sqrt(5).

Product_{n>=3} 1 - 1/a(n) = -sec(sqrt(5)*Pi/2)*Pi/6.

Product_{n>=2} 1 + 1/a(n) = -sec(sqrt(5)*Pi/2)*Pi. (End)

Extensions

a(22) corrected by Reinhard Zumkeller, Apr 13 2011

Better name from Joerg Arndt, Oct 26 2024