A152015 -id:A152015 - cp's OEIS frontend

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

Philippe Deléham, Sep 29 2009

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

			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

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).

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.

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

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 *)

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

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)

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

Better name from Joerg Arndt, Oct 26 2024

A270867 a(n) = n^3 + 2n^2 + 4n + 1.

Original entry on oeis.org

1, 8, 25, 58, 113, 196, 313, 470, 673, 928, 1241, 1618, 2065, 2588, 3193, 3886, 4673, 5560, 6553, 7658, 8881, 10228, 11705, 13318, 15073, 16976, 19033, 21250, 23633, 26188, 28921, 31838, 34945, 38248, 41753, 45466, 49393, 53540, 57913, 62518, 67361, 72448

Offset: 0

Vincenzo Librandi, Apr 01 2016

Numbers of the type (m+1)^3 - (m-1)*m. Similar sequences are: A069778 with the closed form (m+1)^3 - m*(m+1), A152015 with (m+1)^3 - (m+1)*(m+2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015 (page 19, 4th row; page 21, 3rd row).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A069778, A016957, A152015, A270109.

Magma
```
[n^3+2*n^2+4*n+1: n in [0..50]];
```

A270867:=n->n^3+2*n^2+4*n+1: seq(A270867(n), n=0..100); # Wesley Ivan Hurt, Apr 01 2016

Table[n^3 + 2 n^2 + 4 n + 1, {n, 0, 40}]

x='x+O('x^99); Vec((1+4*x-x^2+2*x^3)/(1-x)^4) \\ Altug Alkan, Apr 01 2016

for i in range(0,100):print(i**3+2*i**2+4*i+1) # Soumil Mandal, Apr 02 2016

O.g.f.: (1 + 4*x - x^2 + 2*x^3)/(1 - x)^4.
E.g.f.: (1 + 7*x + 5*x^2 + x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = -A270109(-n-1). - Bruno Berselli, Apr 01 2016
a(n+2) - 2*a(n+1) + a(n) = A016957(n+1). - Wesley Ivan Hurt, Apr 02 2016

A214731 a(n) = n^3 - 2*n^2 - 1.

Original entry on oeis.org

-2, -1, 8, 31, 74, 143, 244, 383, 566, 799, 1088, 1439, 1858, 2351, 2924, 3583, 4334, 5183, 6136, 7199, 8378, 9679, 11108, 12671, 14374, 16223, 18224, 20383, 22706, 25199, 27868, 30719, 33758, 36991, 40424, 44063, 47914, 51983, 56276, 60799, 65558, 70559

Offset: 1

Marco Piazzalunga, Jul 27 2012

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

Cf. A080859, A085490, A144390 (first differences), A152619.

Similar sequences: A152015 (of the type m^3+2m^2-1), A081437 (m^3-2m^2+1).

[n^3-2*n^2-1: n in [1..50]]; // Vincenzo Librandi, Jul 29 2012

A214731:=n->n^3-2*n^2-1: seq(A214731(n), n=1..60); # Wesley Ivan Hurt, Apr 18 2017

Table[n^3 - 2 n^2 - 1, {n, 50}] (* Vincenzo Librandi, Jul 29 2012 *)

a(n)=n^3-2*n^2-1 \\ Charles R Greathouse IV, Jul 27 2012

[n^2*(n-2)-1 for n in range(1,51)] # G. C. Greubel, Dec 31 2023

From Bruno Berselli, Jul 27 2012: (Start)
G.f.: -x*(2-7*x-x^3)/(1-x)^4.
a(n) = A085490(n-1) + 2.
a(n) = A152619(n-2) - 1 for n>1.
a(n) - a(n-2) = A080859(n-2) - 1 for n>2. (End)
E.g.f.: 1 - (1-x)*(1+x)^2*exp(x). - G. C. Greubel, Dec 31 2023

a(3) corrected by Charles R Greathouse IV, Jul 27 2012

A152017 a(n) = n^5-n^4-n^3-n^2-n.

Original entry on oeis.org

0, -3, 2, 123, 684, 2345, 6222, 14007, 28088, 51669, 88890, 144947, 226212, 340353, 496454, 705135, 978672, 1331117, 1778418, 2338539, 3031580, 3879897, 4908222, 6143783, 7616424, 9358725, 11406122, 13797027, 16572948, 19778609, 23462070

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A152015, A152016.

[n^5-n^4-n^3-n^2-n: n in [0..40]]; // Vincenzo Librandi, Nov 18 2011

lst={};Do[AppendTo[lst,n^5-n^4-n^3-n^2-n],{n,0,5!}];lst
Table[n^5-Total[n^Range[4]],{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,-3,2,123,684,2345},30](* Harvey P. Dale, Sep 13 2011 *)

a(n) = 6*a(n-1)- 15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), n>5. - Harvey P. Dale, Sep 13 2011

G.f. x*(-3+20*x+66*x^2+36*x^3+x^4) / (x-1)^6. - R. J. Mathar, Nov 17 2011

Changed offset to 0 from Bruno Berselli, Nov 02 2011

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

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A270867 a(n) = n^3 + 2*n^2 + 4*n + 1.

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A214731 a(n) = n^3 - 2*n^2 - 1.

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A152017 a(n) = n^5-n^4-n^3-n^2-n.

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A270867 a(n) = n^3 + 2n^2 + 4n + 1.