A014987 -id:A014987 - cp's OEIS frontend

A014991 a(n) = (1 - (-9)^n)/10.

1, -8, 73, -656, 5905, -53144, 478297, -4304672, 38742049, -348678440, 3138105961, -28242953648, 254186582833, -2287679245496, 20589113209465, -185302018885184, 1667718169966657, -15009463529699912

Offset: 1

Olivier Gérard

q-integers for q = -9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-8,9).

Cf. A077925, A014983, A014985-A014987, A014989-A014994. - Zerinvary Lajos, Dec 16 2008

I:=[1,-8]; [n le 2 select I[n] else -8*Self(n-1)+9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-9)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

((-9)^Range[30]-1)/-10 (* or *) LinearRecurrence[{-8,9},{1,-8},30] (* Harvey P. Dale, Aug 08 2011 *)
CoefficientList[Series[1/((1 - x)*(1 + 9*x)), {x, 0, 30}], x]; (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1,30, print1((1-(-9)^n)/10, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-9) for n in range(1,19)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).
a(0)=1, a(1)=-8, a(n) = -8*a(n-1) + 9*a(n-2). - Harvey P. Dale, Aug 08 2011
G.f.: x/((1 - x)*(1 + 9*x)). - Vincenzo Librandi, Oct 22 2012
E.g.f.: (exp(x) - exp(-9*x))/10. - G. C. Greubel, May 26 2018

Better name from Ralf Stephan, Jul 14 2013

A014993 a(n) = (1 - (-11)^n)/12.

Original entry on oeis.org

1, -10, 111, -1220, 13421, -147630, 1623931, -17863240, 196495641, -2161452050, 23775972551, -261535698060, 2876892678661, -31645819465270, 348104014117971, -3829144155297680, 42120585708274481

Offset: 1

Olivier Gérard

q-integers for q = -11.

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Index entries for linear recurrences with constant coefficients, signature (-10,11).

Cf. A077925, A014983, A014985, A014986, A014987, A014989, A014990, A014991, A014992, A014994. - Zerinvary Lajos, Dec 16 2008

I:=[1, -10]; [n le 2 select I[n] else -10*Self(n-1) +11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-11)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

LinearRecurrence[{-10, 11}, {1, -10}, 40] (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1,30, print1((1-(-11)^n)/12, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-11) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^{(n-1)} = {(q^n - 1) / (q - 1)}.
G.f.: x/((1 - x)*(1 + 11*x)). - Vincenzo Librandi, Oct 22 2012
a(n) = -10*a(n-1) + 11*a(n-2). - Vincenzo Librandi, Oct 22 2012
E.g.f.: (exp(x) - exp(-11*x))/12. - G. C. Greubel, May 26 2018

Better name from Ralf Stephan, Jul 14 2013

A015000 q-integers for q=-13.

Original entry on oeis.org

1, -12, 157, -2040, 26521, -344772, 4482037, -58266480, 757464241, -9847035132, 128011456717, -1664148937320, 21633936185161, -281241170407092, 3656135215292197, -47529757798798560, 617886851384381281

Offset: 1

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (-12,13).

Cf. A077925, A014983, A014985-A014987, A014989-A014994.

I:=[1,-12]; [n le 2 select I[n] else -12*Self(n-1)+13*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-13)^j, j=0..n-1): seq(a(n), n=0..20); # Zerinvary Lajos, Dec 16 2008

QBinomial[Range[20],1,-13] (* Harvey P. Dale, May 02 2012 *)
LinearRecurrence[{-12, 13}, {1, -12}, 30] (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1, 30, print1((1-(-13)^n)/14, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-13) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1), with q=-13.
a(n) = Sum_{j=0..n-1} (-13)^j. - Zerinvary Lajos, Dec 16 2008
G.f.: x/((1 - x)*(1 + 13*x)). - Vincenzo Librandi, Oct 22 2012
a(n) = -12*a(n-1) + 13*a(n-2). - Vincenzo Librandi, Oct 22 2012
From G. C. Greubel, May 26 2018: (Start)
a(n) = (1 - (-13)^n)/14.
E.g.f.: (exp(x) - exp(-13*x))/14. (End)

Edited by N. J. A. Sloane, Jan 13 2009 at the suggestion of R. J. Mathar

A179897 a(n) = (n^(2*n+1) + 1) / (n+1).

Original entry on oeis.org

1, 1, 11, 547, 52429, 8138021, 1865813431, 593445188743, 250199979298361, 135085171767299209, 90909090909090909091, 74619186937936447687211, 73381705110822317661638341, 85180949465178001182799643437, 115244915978498073437814463065839, 179766618030828831251710653305053711

Offset: 0

Martin Saturka (martin(AT)saturka.net), Jul 31 2010

a(n) is the arithmetic mean of the multiset consisting of n lots of 1/n and one lot of n^(2*n+1). This multiset also has an integer valued geometric mean which is equal to n for n > 0.

According to search at OEIS for particular sequence members, a(n) is also: (1+2*n)-th q-integer for q=-n, (2*(n+1))-th cyclotomic polynomial at q=-n, Gaussian binomial coefficient [2*n+1, 2*n] for q=-n, number of walks of length 1+2*n between any two distinct vertices of the complete graph K_(n+1).

			For n = 2, a(2) = 11 which is the arithmetic mean of {1/2, 1/2, 2^5} = 33 / 3 = 11. The geometric mean is 8^(1/3) = 2, i.e. both are integral.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Google Groups, Integer-valued arithmetic and geometric means of sequences with non-integer numbers

Main diagonal of A362783.

Values for n = 5, 6 via other ways. Q-integers: A014986, A014987, K_n paths: A015531, A015540, Cyclotomic polynomials: A020504, A020505, Gaussian binomial coefficients: A015391, A015429.

a(n) = (n^(2*n + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023

[(n**(2*n+1)+1)//(n+1) for n in range(1,11)]

a(n) = Sum_{i=0..2*n} (-n)^i.

Edited, a(0)=1 prepended and more terms from Andrew Howroyd, May 03 2023

A268413 a(n) = Sum_{k = 0..n} (-1)^k*14^k.

Original entry on oeis.org

1, -13, 183, -2561, 35855, -501969, 7027567, -98385937, 1377403119, -19283643665, 269971011311, -3779594158353, 52914318216943, -740800455037201, 10371206370520815, -145196889187291409, 2032756448622079727, -28458590280709116177, 398420263929927626479

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

Alternating sum of powers of 14.

More generally, the ordinary generating function for the Sum_{k = 0..n} (-1)^k*m^k is 1/(1 + (m - 1)*x - m*x^2). Also, Sum_{k = 0..n} (-1)^k*m^k = ((-1)^n*m^(n + 1) + 1)/(m + 1).

G. C. Greubel, Table of n, a(n) for n = 0..870
Index entries for linear recurrences with constant coefficients, signature (-13,14).

Cf. A001023, A033999.

Cf. similar sequences of the type Sum_{k=0..n} (-1)^k*m^k: A059841 (m=1), A077925 (m=2), A014983 (m=3), A014985 (m=4), A014986 (m=5), A014987 (m=6), A014989 (m=7), A014990 (m=8), A014991 (m=9), A014992 (m=10), A014993 (m=11), A014994 (m=12), A015000 (m=13), this sequence (m=14), A239284 (m=15).

I:=[1,-19]; [n le 2 select I[n] else -13*Self(n-1) +14*Self(n-2): n in [1..30]]; // G. C. Greubel, May 26 2018

Table[((-1)^n 14^(n + 1) + 1)/15, {n, 0, 18}]
LinearRecurrence[{-13, 14}, {1, -13}, 19]
Table[Sum[(-1)^k*14^k, {k, 0, n}], {n, 0, 18}]

x='x+O('x^30); Vec(1/(1 + 13*x - 14*x^2)) \\ G. C. Greubel, May 26 2018

G.f.: 1/(1 + 13*x - 14*x^2).
a(n) = ((-1)^n*14^(n + 1) + 1)/15.
a(n) = 1 - 14*a(n - 1) for n>0 and a(0)=1.
a(n) = Sum_{k = 0..n} A033999(k)*A001023(k).
Lim_{n -> infinity} a(n)/a(n + 1) = - 1/14.

cp's OEIS Frontend

A014991 a(n) = (1 - (-9)^n)/10.

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A014993 a(n) = (1 - (-11)^n)/12.

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A015000 q-integers for q=-13.

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A179897 a(n) = (n^(2*n+1) + 1) / (n+1).

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A268413 a(n) = Sum_{k = 0..n} (-1)^k*14^k.

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