A062160 -id:A062160 - cp's OEIS frontend

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

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

A062157 a(n) = 0^n - (-1)^n.

Original entry on oeis.org

0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1

Offset: 0

Henry Bottomley, Jun 08 2001

Also the numerators of the series expansion of log(1+x). Denominators are A028310. - Robert G. Wilson v, Aug 14 2015

			G.f. = x - x^2 + x^3 - x^4 + x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + ... - _Michael Somos_, Feb 20 2024

Antti Karttunen, Table of n, a(n) for n = 0..10000
Wikipedia, Dirichlet eta function
Index entries for linear recurrences with constant coefficients, signature (-1).

Convolution inverse of A019590.

Cf. A000007, A028310, A033999, A062160.

[0^n-(-1)^n: n in [0..100]]; // Vincenzo Librandi, Aug 15 2015

[0] cat &cat[ [1, -1]: n in [1..80] ]; // Vincenzo Librandi, Aug 15 2015

PadRight[{0},120,{-1,1}] (* Harvey P. Dale, Aug 20 2012 *)
Join[{0},LinearRecurrence[{-1},{1},101]] (* Ray Chandler, Aug 12 2015 *)
f[n_] := 0^n - (-1)^n; f[0] = 0; Array[f, 105, 0] (* or *)
CoefficientList[ Series[ x/(1 + x), {x, 0, 80}], x] (* or *)
Numerator@ CoefficientList[ Series[ Log[1 + x], {x, 0, 80}], x] (* Robert G. Wilson v, Aug 14 2015 *)

{a(n) = if( n<1, 0, -(-1)^n )}; /* Michael Somos, Jul 05 2009 */

a(n) = A000007(n) - A033999(n) = A062160(0, n).
G.f.: x/(1+x).
Euler transform of length 2 sequence [-1, 1]. - Michael Somos, Jul 05 2009
Moebius transform is length 2 sequence [1, -2]. - Michael Somos, Jul 05 2009
a(n) is multiplicative with a(2^e) = -1 if e > 0, a(p^e) = 1 if p > 2. - Michael Somos, Jul 05 2009
Dirichlet g.f.: zeta(s) * (1 - 2^(1-s)). - Michael Somos, Jul 05 2009
Also, Dirichlet g.f.: eta(s). - Ralf Stephan, Mar 25 2015
E.g.f.: 1 - exp(-x). - Alejandro J. Becerra Jr., Feb 16 2021

A062158 a(n) = n^3 - n^2 + n - 1 = (n-1) * (n^2 + 1).

Original entry on oeis.org

-1, 0, 5, 20, 51, 104, 185, 300, 455, 656, 909, 1220, 1595, 2040, 2561, 3164, 3855, 4640, 5525, 6516, 7619, 8840, 10185, 11660, 13271, 15024, 16925, 18980, 21195, 23576, 26129, 28860, 31775, 34880, 38181, 41684, 45395, 49320, 53465, 57836, 62439, 67280, 72365, 77700, 83291, 89144, 95265, 101660

Offset: 0

Henry Bottomley, Jun 08 2001

Number of walks of length 4 between any two distinct vertices of the complete graph K_{n+1} (n >= 1). Example: a(2) = 5 because in the complete graph ABC we have the following walks of length 4 between A and B: ABACB, ABCAB, ACACB, ACBAB and ACBCB. - Emeric Deutsch, Apr 01 2004
1/a(n) for n >= 2, is in base n given by 0.repeat(0,0,1,1), due to (1/n^3 + 1/n^4)*(1/(1-1/n^4)) = 1/((n-1)*(n^2+1)). - Wolfdieter Lang, Jun 20 2014
For n>3, a(n) is 1220 in base n-1. - Bruno Berselli, Jan 26 2016
For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^4 consecutive integers that equals n^8. - Patrick J. McNab, Dec 26 2016

			a(4) = 4^3 - 4^2 + 4 - 1 = 64 - 16 + 4 - 1 = 51.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002061, A023443, A053698, A060884, A060888, A062159, A062160, A268086.

[n^3 - n^2 + n - 1 : n in [0..50]]; // Wesley Ivan Hurt, Dec 26 2016

[seq(n^3-n^2+n-1,n=0..49)]; # Zerinvary Lajos, Jun 29 2006
a:=n->sum(1+sum(n, k=1..n), k=2..n):seq(a(n), n=0...43); # Zerinvary Lajos, Aug 24 2008

Table[n^3 - n^2 + n - 1, {n, 0, 49}] (* Alonso del Arte, Apr 30 2014 *)

a(n) = { n*(n*(n - 1) + 1) - 1 } \\ Harry J. Smith, Aug 02 2009

a(n) = round(n^4/(n+1)) for n >= 2.
a(n) = A062160(n, 4), for n > 2.
G.f.: (4*x-1)*(1+x^2)/(1-x)^4 (for the signed sequence). - Emeric Deutsch, Apr 01 2004
a(n) = floor(n^5/(n^2+n)) for n > 0. - Gary Detlefs, May 27 2010
a(n) = -A053698(-n). - Bruno Berselli, Jan 26 2016
Sum_{n>=2} 1/a(n) = A268086. - Amiram Eldar, Nov 18 2020
E.g.f.: exp(x)*(x^3 + 2*x^2 + x - 1). - Stefano Spezia, Apr 22 2023

More terms from Emeric Deutsch, Apr 01 2004

A081216 a(n) = (n^n-(-1)^n)/(n+1).

Original entry on oeis.org

0, 1, 1, 7, 51, 521, 6665, 102943, 1864135, 38742049, 909090909, 23775972551, 685853880635, 21633936185161, 740800455037201, 27368368148803711, 1085102592571150095, 45957792327018709121, 2070863582910344082917, 98920982783015679456199

Offset: 0

Vladeta Jovovic, Apr 17 2003

a(n) is prime for n = {3, 5, 17, 157} = A056826(n) Primes p such that (p^p + 1)/(p + 1) is a prime. Prime a(n) are {7, 521, 45957792327018709121, ...}. Bisection of a(n) is Sierpinski quotient a(2n-1) = A124899(n) = ((2n-1)^(2n-1) + 1)/(2n) = A014566(2n-1)/(2n). - Alexander Adamchuk, Nov 12 2006

This is related to the dimension of the primitive middle cohomology of Dwork hypersurfaces x1**n+x2**n+...+xn**n=n*psi*x1*x2*...*xn. [F. Chapoton, Dec 11 2009]

Seiichi Manyama, Table of n, a(n) for n = 0..387
Philippe Goutet, Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions of Dwork Hypersurfaces, arXiv:0912.2075 [math.NT], 2009.

Main diagonal of A062160.

Cf. A056826, A124899, A014566 (Sierpinski numbers of the first kind: n^n + 1).

a:= n-> (n^n-(-1)^n)/(n+1):
seq(a(n), n=0..20);  # Alois P. Heinz, May 11 2023

a(n) = (n^n-(-1)^n)/(n+1); \\ Michel Marcus, Jul 29 2017

[((n - 1)**(n - 1) + (-1)**n) // n for n in range(1, 16)]

Edited by F. Chapoton, Feb 03 2011

A109502 Array read by antidiagonals: T(m,n) is the number of closed walks of length n on the complete graph on m nodes, m >= 1, n >= 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 3, 2, 1, 0, 1, 0, 4, 6, 6, 0, 0, 1, 0, 5, 12, 21, 10, 1, 0, 1, 0, 6, 20, 52, 60, 22, 0, 0, 1, 0, 7, 30, 105, 204, 183, 42, 1, 0, 1, 0, 8, 42, 186, 520, 820, 546, 86, 0, 0, 1, 0, 9, 56, 301, 1110, 2605, 3276, 1641, 170, 1, 0

Offset: 1

Mitch Harris, Jun 30 2005

			Array begins:
  m\n| 0  1  2  3   4    5     6      7       8        9        10
  ---+------------------------------------------------------------
   1 | 1  0  0  0   0    0     0      0       0        0         0
   2 | 1  0  1  0   1    0     1      0       1        0         1
   3 | 1  0  2  2   6   10    22     42      86      170       342
   4 | 1  0  3  6  21   60   183    546    1641     4920     14763
   5 | 1  0  4 12  52  204   820   3276   13108    52428    209716
   6 | 1  0  5 20 105  520  2605  13020   65105   325520   1627605
   7 | 1  0  6 30 186 1110  6666  39990  239946  1439670   8638026
   8 | 1  0  7 42 301 2100 14707 102942  720601  5044200  35309407
   9 | 1  0  8 56 456 3640 29128 233016 1864136 14913080 119304648
  10 | 1  0  9 72 657 5904 53145 478296 4304673 38742048 348678441

M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Rows are A078008, A054878, A109499, A109500, A109501.

Cf. A062160.

T := proc(m, n); ((m-1)^n + (m-1)*(-1)^n)/m end:
seq(print(seq(T(m, n), n = 0..10)), m = 1..10); # Peter Bala, May 30 2024

T(m,n) = ((m-1)^n + (m-1)(-1)^n)/m.
G.f.: T(m, n) = [z^n](1 - (m-2)z)/(1 - (m-2)z - (m-1)z^2).
From Peter Bala, May 29 2024: (Start)
Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = m^(n-1) for n >= 1.
Let R(m, x) denote the g.f. of the m-th row of the square array. Then R(m_1, x) o R(m_2, x) = R(m_1*m_2, x), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A062160.
T(m_1*m_2, n) = Sum_{k = 0..n} Sum_{i = k..n} binomial(n, k)*binomial(n-k, i-k)*T(m_1, i)*T(m_2, n-k). (End)

Corrected by Franklin T. Adams-Watters, Sep 18 2006

A062159 a(n) = n^5 - n^4 + n^3 - n^2 + n - 1.

Original entry on oeis.org

-1, 0, 21, 182, 819, 2604, 6665, 14706, 29127, 53144, 90909, 147630, 229691, 344772, 501969, 711914, 986895, 1340976, 1790117, 2352294, 3047619, 3898460, 4929561, 6168162, 7644119, 9390024, 11441325, 13836446, 16616907, 19827444, 23516129, 27734490, 32537631, 37984352, 44137269

Offset: 0

Henry Bottomley, Jun 08 2001

Number of walks of length 6 between any two distinct nodes of the complete graph K_{n+1} (n>=1). - Emeric Deutsch, Apr 01 2004

For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^6 consecutive integers that equals n^12. - Patrick J. McNab, Dec 26 2016

			a(4) = 4^5 - 4^4 + 4^3 - 4^2 + 4 - 1 = 1024 - 256 + 64 - 16 + 4 - 1 = 819.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A023443, A002061, A062158, A060884, A060888.

Cf. A002061.

A062159:=n->n^5-n^4+n^3-n^2+n-1; seq(A062159(k), k=0..100); # Wesley Ivan Hurt, Nov 06 2013

Table[n^5-n^4+n^3-n^2+n-1, {n,0,100}] (* Wesley Ivan Hurt, Nov 06 2013 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{-1,0,21,182,819,2604},40] (* Harvey P. Dale, Dec 20 2015 *)

a(n) = { n*(n*(n*(n*(n - 1) + 1) - 1) + 1) - 1 } \\ Harry J. Smith, Aug 02 2009

a(n) = round(n^6/(n+1)) for n>2 = A062160(n,6).
G.f.: (76x^3 + 6x^2 + 27x^4 + 6x^5 + 6x - 1)/(1-x)^6 (for the signed sequence). - Emeric Deutsch, Apr 01 2004
a(n) = (n^6 - 1)/(n+1). a(n) = (n-1)(n^2 - n + 1)(n^2 + n + 1) = (n-1)*A002061(n)*A002061(n+1). - Alexander Adamchuk, Apr 12 2006
a(0)=-1, a(1)=0, a(2)=21, a(3)=182, a(4)=819, a(5)=2604, 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). - Harvey P. Dale, Dec 20 2015
E.g.f.: exp(x)*(x^5 + 9*x^4 + 20*x^3 + 10*x^2 + x - 1). - Stefano Spezia, Apr 22 2023

More terms from Emeric Deutsch, Apr 01 2004

A239284 a(n) = (15^n - (-1)^n)/16.

Original entry on oeis.org

0, 1, 14, 211, 3164, 47461, 711914, 10678711, 160180664, 2402709961, 36040649414, 540609741211, 8109146118164, 121637191772461, 1824557876586914, 27368368148803711, 410525522232055664, 6157882833480834961, 92368242502212524414, 1385523637533187866211

Offset: 0

Felix P. Muga II, Mar 14 2014

Let k and t be positive integers and consider a(n) = k*a(n-1)+t*a(n-2) for n>=2, with a(0)=0, a(1)=1.
The roots of its characteristic equation are r1 = (k+sqrt(k^2+4t))/2 and r2 =(k-sqrt(k^2+4t))/2. Hence, the solution to the recurrence relation is the sequence {a(n)} where a(n) = alpha1*r1^n + alpha2*r2^n. It can be shown that alpha1 = 1/sqrt(k^2+4t) and alpha2 = -alpha1. It can be shown also that |r2/r1|< 1. Thus, the ratio a(n+1)/a(n) converges to r as n approaches infinity.
Note that limit a(n+1)/a(n) = 15 as n approaches infinity with k=14 and t=15.
If n > 15 then | a(n+1)/a(n) - 15 | < 10^(-16).
The number of walks of length n between any two distinct vertices of the complete graph K_16. - Peter Bala, May 30 2024

G. C. Greubel, Table of n, a(n) for n = 0..849
Index entries for linear recurrences with constant coefficients, signature (14,15).

Cf. A062160 (row 15).

[(15^n - (-1)^n)/16: n in [0..30]]; // G. C. Greubel, May 26 2018

CoefficientList[Series[x/(1-14*x-15*x^2), {x,0,50}], x] (* or *) Table[ (15^n - (-1)^n)/16, {n,0,30}] (* or *) LinearRecurrence[{14,15}, {0,1}, 30] (* G. C. Greubel, May 26 2018 *)

a(n) = (15^n - (-1)^n)/16; \\ Michel Marcus, Mar 16 2014

my(x='x+O('x^30)); concat([0], Vec(x/(1 -14*x - 15*x^2))) \\ G. C. Greubel, May 26 2018

G.f.: x/(1 - 14*x - 15*x^2).
a(n) = 14*a(n-1) + 15*a(n-2) for n > 1, a(0) = 0, a(1) = 1.
a(n) = (1/16)*(15^n - (-1)^n).
a(n) = (1/16)*( A001024(n) - A033999(n) ).
E.g.f.: (exp(15*x) - exp(-x))/16. - G. C. Greubel, May 26 2018

cp's OEIS Frontend

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

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A062157 a(n) = 0^n - (-1)^n.

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A062158 a(n) = n^3 - n^2 + n - 1 = (n-1) * (n^2 + 1).

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A081216 a(n) = (n^n-(-1)^n)/(n+1).

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A109502 Array read by antidiagonals: T(m,n) is the number of closed walks of length n on the complete graph on m nodes, m >= 1, n >= 0.

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Maple

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A062159 a(n) = n^5 - n^4 + n^3 - n^2 + n - 1.

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