A014990 -id:A014990 - cp's OEIS frontend

A014983 a(n) = (1 - (-3)^n)/4.

0, 1, -2, 7, -20, 61, -182, 547, -1640, 4921, -14762, 44287, -132860, 398581, -1195742, 3587227, -10761680, 32285041, -96855122, 290565367, -871696100, 2615088301, -7845264902, 23535794707, -70607384120, 211822152361, -635466457082, 1906399371247

Offset: 0

Olivier Gérard

q-integers for q=-3.
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^n*charpoly(A,0). - Milan Janjic, Jan 27 2010
Pisano period lengths:  1, 2, 1, 4, 4, 2, 3, 8, 1, 4, 10, 4, 6, 6, 4, 16, 16, 2, 9, 4, ... - R. J. Mathar, Aug 10 2012

T. D. Noe, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 927
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
Index entries for linear recurrences with constant coefficients, signature (-2,3)

Cf. A077925, A014985, A014986, A014987, A014989, A014990, A014991, A014992, A014993, A014994. - Zerinvary Lajos, Dec 16 2008

[(1-(-3)^n)/4: n in [0..30]]; // G. C. Greubel, May 26 2018

a:=n->sum ((-3)^j, j=0..n): seq(a(n), n=-1..25); # Zerinvary Lajos, Dec 16 2008

nn = 25; CoefficientList[Series[x/((1 - x)*(1 + 3*x)), {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
Table[(1 - (-3)^n)/4, {n, 0, 27}] (* Michael De Vlieger, Nov 23 2016 *)

PARI
```
a(n)=(1-(-3)^n)/4
```

[gaussian_binomial(n,1,-3) for n in range(0,27)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + (-3)^(n-1).
G.f.: x/((1-x)*(1+3*x)).
a(n) = -(-1)^n*A015518(n).
a(n) = the (1, 2)-th element of M^n, where M = ((1, 1, 1, -2), (1, 1, -2, 1), (1, -2, 1, 1), (-2, 1, 1, 1)). - Simone Severini, Nov 25 2004
a(0)=0, a(1)=1, a(n) = -2*a(n-1) + 3*a(n-2) for n>1. - Philippe Deléham, Sep 19 2009
From Sergei N. Gladkovskii, Apr 29 2012: (Start)
G.f. A(x) = G(0)/4; G(k) = 1 - 1/(3^(2*k) - 3*x*3^(4*k)/(3*x*3^(2*k) + 1/(1 + 1/(3*3^(2*k) - 3^(3)*x*3^(4*k)/(3^2*x*3^(2*k) - 1/G(k+1)))))); (continued fraction, 3rd kind, 6-step).
E.g.f. E(x) = G(0)/4; G(k) = 1 - 1/(9^k - 3*x*81^k/(3*x*9^k + (2*k+1)/(1 + 1/(3*9^k - 27*x*81^k/(9*x*9^k - (2*k+2)/G(k+1)))))); (continued fraction, 3rd kind, 6-step). (End)
a(n) = A084222(n) - 1. - Filip Zaludek, Nov 19 2016
E.g.f.: sinh(x)*cosh(x)*exp(-x). - Ilya Gutkovskiy, Nov 20 2016

A015565 a(n) = 7a(n-1) + 8a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 7, 57, 455, 3641, 29127, 233017, 1864135, 14913081, 119304647, 954437177, 7635497415, 61083979321, 488671834567, 3909374676537, 31274997412295, 250199979298361, 2001599834386887, 16012798675095097, 128102389400760775, 1024819115206086201, 8198552921648689607

Offset: 0

Olivier Gérard

A linear 2nd order recurrence. A Jacobsthal number sequence.
Binomial transform of A053573 (preceded by zero). - Paul Barry, Apr 09 2003
Second binomial transform of A080424. Binomial transform of A053573, with leading zero. Binomial transform is 0,1,9,81,729,....(9^n - 0^n)/9. Second binomial transform is 0,1,11,111,1111,... (A002275: repunits). - Paul Barry, Mar 14 2004
Number of walks of length n between any two distinct nodes of the complete graph K_9. Example: a(2)=7 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHI are: ACB, ADB, AEB, AFB, AGB, AHB and AIB. - Emeric Deutsch, Apr 01 2004
Unsigned version of A014990. - Philippe Deléham, Feb 13 2007
General form: k=8^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
The ratio a(n+1)/a(n) converges to 8 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

			G.f. = x + 7*x^2 + 57*x^3 + 455*x^4 + 3641*x^5 + 29127*x^6 + 233017*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Dale Gerdemann, Fractal generated from (7,8) recursion, YouTube Video, Dec 5, 2014.
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A002275, A014990, A053573, A080424, A082311, A082365.
Cf. A062160, A099322, A106566.
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[Round(8^n/9): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(8^n/9),n=0..25); # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=8^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{7,8},{0,1},30] (* Harvey P. Dale, Mar 04 2016 *)

x='x+O('x^30); concat([0], Vec(x/(1-7*x-8*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,7,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

From Paul Barry, Apr 09 2003: (Start)
a(n) = (8^n - (-1)^n)/9.
a(n) = J(3*n)/3 = A001045(3*n)/3. (End)
From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 8^(n-1) - a(n-1).
G.f.: x/(1-7*x-8*x^2). (End)
a(n) = Sum_{k = 0..n} A106566(n,k)*A099322(k). - Philippe Deléham, Oct 30 2008
a(n) = round(8^n/9). - Mircea Merca, Dec 28 2010
From Peter Bala, May 31 2024: (Start)
G.f: A(x) = x/(1 - x^2) o x/(1 - x^2), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A054878.
The black diamond product A(x) o A(x) is the g.f. for the number of walks of length n between any two distinct nodes of the complete graph K_81.
Row 8 of A062160. (End)
E.g.f.: exp(-x)*(exp(9*x) - 1)/9. - Elmo R. Oliveira, Aug 17 2024

A015118 Triangle of q-binomial coefficients for q=-8.

Original entry on oeis.org

1, 1, 1, 1, -7, 1, 1, 57, 57, 1, 1, -455, 3705, -455, 1, 1, 3641, 236665, 236665, 3641, 1, 1, -29127, 15150201, -120935815, 15150201, -29127, 1, 1, 233017, 969583737, 61934287481, 61934287481, 969583737, 233017, 1, 1, -1864135, 62053592185

Offset: 0

Olivier Gérard

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals of the former, or rows/columns of the latter, are: A000012 (k=0), A014990 (k=1), A015259 (k=2), A015276 (k=3), A015294 (k=4), A015313 (k=5), A015331 (k=6), A015347 (k=7), A015364 (k=8), A015380 (k=9), A015394 (k=10), A015413 (k=11), A015431 (k=12). - M. F. Hasler, Nov 04 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 05 2012

Table[QBinomial[n, k, -8], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015118(n, k, q=-8)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A014986 a(n) = (1 - (-5)^n)/6.

Original entry on oeis.org

1, -4, 21, -104, 521, -2604, 13021, -65104, 325521, -1627604, 8138021, -40690104, 203450521, -1017252604, 5086263021, -25431315104, 127156575521, -635782877604, 3178914388021, -15894571940104, 79472859700521

Offset: 1

Olivier Gérard

q-integers for q = -5.

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-4,5).

Cf. A077925, A014983, A014985, A014987, A014989, A014990, A014991, A014992, A014993, A014994.

I:=[1, -4]; [n le 2 select I[n] else -4*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2012

a:=n->sum ((-5)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

LinearRecurrence[{-4,5},{1,-4},30] (* Vincenzo Librandi, Jun 19 2012 *)

a(n)=(1-(-5)^n)/6 \\ Charles R Greathouse IV, Dec 07 2011

[gaussian_binomial(n,1,-5) for n in range(1,22)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).
G.f.: x/((1-x)*(1+5*x)). - Bruno Berselli, Dec 07 2011
a(n) = -4*a(n-1) + 5*a(n-2). - Vincenzo Librandi, Jun 19 2012
E.g.f.: (exp(x) - exp(-5*x))/6. - G. C. Greubel, May 26 2018

Better name from Ralf Stephan, Jul 14 2013

A014989 a(n) = (1 - (-7)^n)/8.

Original entry on oeis.org

1, -6, 43, -300, 2101, -14706, 102943, -720600, 5044201, -35309406, 247165843, -1730160900, 12111126301, -84777884106, 593445188743, -4154116321200, 29078814248401, -203551699738806, 1424861898171643, -9974033287201500

Offset: 1

Olivier Gérard

q-integers for q = -7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (-6,7).

Cf. A077925, A014983, A014985-A014987, A014990-A014994.

I:=[1,-6]; [n le 2 select I[n] else -6*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-7)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

LinearRecurrence[{-6, 7}, {1, -6}, 30] (* Vincenzo Librandi, Oct 22 2012 *)

x='x+O('x^30); Vec(x/((1-x)*(1+7*x))) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-7) for n in range(1,21)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).
a(n) = -6*a(n-1) + 7*a(n-2). - Vincenzo Librandi, Oct 22 2012
From G. C. Greubel, May 26 2018: (Start)
G.f.: x/((1-x)*(1+7*x)).
E.g.f.: (exp(x) - exp(-7*x))/8. (End)

Better name from Ralf Stephan, Jul 14 2013

A014992 a(n) = (1 - (-10)^n)/11.

Original entry on oeis.org

1, -9, 91, -909, 9091, -90909, 909091, -9090909, 90909091, -909090909, 9090909091, -90909090909, 909090909091, -9090909090909, 90909090909091, -909090909090909, 9090909090909091, -90909090909090909

Offset: 1

Olivier Gérard

q-integers for q = -10.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-9,10).

Cf. A077925, A014983, A014985, A014986, A014987, A014989, A014990, A014991, A014993, A014994. - Zerinvary Lajos, Dec 16 2008

I:=[1, -9]; [n le 2 select I[n] else -9*Self(n-1) +10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

a:=n->sum ((-10)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

CoefficientList[Series[1/((1 - x)*(1 + 10*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2012 *)

for(n=1, 30, print1((1-(-10)^n)/11, ", ")) \\ G. C. Greubel, May 26 2018

[gaussian_binomial(n,1,-10) for n in range(1,19)] # Zerinvary Lajos, May 28 2009

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).
G.f.: x/((1 - x)*(1 + 10*x)). - Vincenzo Librandi, Oct 22 2012
a(n) = -9*a(n-1) + 10*a(n-2). - Vincenzo Librandi, Oct 22 2012
a(n) = (-1)^(n+1)*A015585(n). - R. J. Mathar, Oct 26 2015
E.g.f.: (exp(x) - exp(-10*x))/11. - 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

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.

A328824 Numerators of A113405(-n) (see the comment for details).

Original entry on oeis.org

0, 1, 1, 1, -7, -7, -7, 57, 57, 57, -455, -455, -455, 3641, 3641, 3641, -29127, -29127, -29127, 233017, 233017, 233017, -1864135, -1864135, -1864135, 14913081, 14913081, 14913081, -119304647, -119304647, -119304647

Offset: 0

Paul Curtz, Oct 28 2019

Let A(n) = (2^n + (-1)^(n+1) - 2*sqrt(3)*sin((Pi*n)/3))/9. Then A(n) = A113405(n) and a(n) = numerator(A(-n)).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,-8,8).

Cf. A000079, A014990, A015565, A113405, A139818.

gf := x / ((1 - x)*(1 + 2*x)*(1 - 2*x + 4*x^2)): ser := series(gf, x, 36):
seq(coeff(ser,x,n),n=0..30); # Peter Luschny, Nov 11 2019

LinearRecurrence[{1,0,-8,8},{0,1,1,1},50] (* Paolo Xausa, Nov 13 2023 *)

concat(0, Vec(x / ((1 - x)*(1 + 2*x)*(1 - 2*x + 4*x^2)) + O(x^40))) \\ Colin Barker, Nov 11 2019

From Colin Barker, Nov 11 2019: (Start)
G.f.: x / ((1 - x)*(1 + 2*x)*(1 - 2*x + 4*x^2)).
a(n) = a(n-1) - 8*a(n-3) + 8*a(n-4) for n>3. (End)

cp's OEIS Frontend

A014983 a(n) = (1 - (-3)^n)/4.

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A015565 a(n) = 7*a(n-1) + 8*a(n-2), a(0) = 0, a(1) = 1.

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A015118 Triangle of q-binomial coefficients for q=-8.

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A014986 a(n) = (1 - (-5)^n)/6.

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A014989 a(n) = (1 - (-7)^n)/8.

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A014992 a(n) = (1 - (-10)^n)/11.

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A015565 a(n) = 7a(n-1) + 8a(n-2), a(0) = 0, a(1) = 1.