A109499 -id:A109499 - cp's OEIS frontend

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.

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

A015592 a(n) = 10a(n-1) + 11a(n-2).

Original entry on oeis.org

0, 1, 10, 111, 1220, 13421, 147630, 1623931, 17863240, 196495641, 2161452050, 23775972551, 261535698060, 2876892678661, 31645819465270, 348104014117971, 3829144155297680, 42120585708274481, 463326442791019290, 5096590870701212191, 56062499577713334100

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_12. Example: a(2)=10 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJKL are ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB, AKB and ALB. - Emeric Deutsch, Apr 01 2004

General form: k=11^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,11).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[-(1/12)*(-1)^n+(1/12)*11^n: n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

k=0;lst={k};Do[k=11^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)

[lucas_number1(n,10,-11) for n in range(0, 18)] # Zerinvary Lajos, Apr 26 2009

a(n) = 11^(n-1) - a(n-1). G.f.: x/(1 - 10x - 11x^2). - Emeric Deutsch, Apr 01 2004
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(5*x)*sinh(6*x)/6.
a(n) = (11^n - (-1)^n)/12. (End)

A015609 a(n) = 11a(n-1) + 12a(n-2).

Original entry on oeis.org

0, 1, 11, 133, 1595, 19141, 229691, 2756293, 33075515, 396906181, 4762874171, 57154490053, 685853880635, 8230246567621, 98762958811451, 1185155505737413, 14221866068848955, 170662392826187461

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_13. Example: a(2)=11 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJKLM are ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB, AKB, ALB and AMB. - Emeric Deutsch, Apr 01 2004

General form: k=12^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (11,12).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585, A015592. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(1/13)*(12^n-(-1)^n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

CoefficientList[Series[x/(1-11*x-12*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{11,12}, {0,1}, 30] (* G. C. Greubel, Dec 30 2017 *)

x='x+O('x^30); concat([0], Vec(x/(1-11*x-12*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,11,-12) for n in range(0, 18)] # Zerinvary Lajos, Apr 27 2009

[abs(gaussian_binomial(n,1,-12)) for n in range(0,18)] # Zerinvary Lajos, May 28 2009

From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 12^(n-1) - a(n-1).
G.f.: x/(1 - 11*x - 12*x^2). (End)
E.g.f.: exp(-x)*(exp(13*x) - 1)/13. - Stefano Spezia, Mar 11 2020

A137505 Inverse binomial transform of A007910.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 4, -4, 4, 4, -12, 20, -12, -12, 52, -76, 52, 52, -204, 308, -204, -204, 820, -1228, 820, 820, -3276, 4916, -3276, -3276, 13108, -19660, 13108, 13108, -52428, 78644, -52428, -52428, 209716, -314572, 209716, 209716, -838860, 1258292, -838860, -838860, 3355444, -5033164, 3355444

Offset: 0

Paul Curtz, Apr 23 2008

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,2).

LinearRecurrence[{-1,0,2},{1,1,0},50] (* Harvey P. Dale, Sep 17 2012 *)

Recurrence: a(n) = -a(n-1) + 2a(n-3), starting 1,1,0.
O.g.f.: (1+x)^2/((1-x)(1+2x+2x^2)). - R. J. Mathar, Jun 12 2008
a(4n) = a(4n+1) = (-1)^n*A109499(n). - Paul Curtz, Nov 01 2009
a(n) = (1/5) * (A137429(n-1) + 4) = A077973(n-2) + 2*A077973(n-1) + A077973(n). - Ralf Stephan, Aug 18 2013

More terms from R. J. Mathar, Jun 12 2008

A167291 a(n) = A137505(2n) + A137505(2n+1).

Original entry on oeis.org

2, 2, 0, 0, 8, 8, -24, -24, 104, 104, -408, -408, 1640, 1640, -6552, -6552, 26216, 26216, -104856, -104856, 419432, 419432, -1677720, -1677720, 6710888, 6710888, -26843544, -26843544, 107374184, 107374184, -429496728, -429496728, 1717986920, 1717986920

Offset: 0

Paul Curtz, Nov 01 2009

The formula for the singular sequence, i.e., just each unique term of the sequence (without duplication), is: a(n) = 1/10 (16-(-4)^n). - Harvey P. Dale, Jun 12 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-4,4)

CoefficientList[Series[(-2-6x^2)/((x-1)(4x^2+1)),{x,0,50}],x] (* or *) LinearRecurrence[{1,-4,4},{2,2,0},50] (* Harvey P. Dale, Jun 12 2013 *)

a(2n) = a(2n+1) = 2*(-1)^n*A109499(n).
G.f. ( -2-6*x^2 ) / ( (x-1)*(4*x^2+1) ). - R. J. Mathar, Feb 06 2011
a(0)=2, a(1)=2, a(2)=0, a(n) = a(n-1)-4*a(n-2)+4*a(n-3). - Harvey P. Dale, Jun 12 2013

A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 7, 42, 301, 2100, 14707, 102942, 720601, 5044200, 35309407, 247165842, 1730160901, 12111126300, 84777884107, 593445188742, 4154116321201, 29078814248400, 203551699738807, 1424861898171642, 9974033287201501, 69818233010410500, 488727631072873507, 3421093417510114542

Offset: 0

Ilya Gutkovskiy, Apr 13 2016

In general, the ordinary generating function for the recurrence b(n) = k^n - b(n-1), where n>0 and b(0)=0, is k*x/((1 + x)*(1 - k*x)). This recurrence gives the closed form b(n) = k*(k^n - (-1)^n)/(k + 1).

			a(2) = 7^2 - a(2-1) = 49 - 7 = 42.
a(4) = 7^4 - a(4-1) = 2401 - 301 = 2100.

Index entries for linear recurrences with constant coefficients, signature (6,7).

Cf. A000420, A015552.

Cf. similar sequences with the recurrence b(n) = k^n - b(n-1): A125122 (k=1), A078008 (k=2), A054878 (k=3), A109499 (k=4), A109500 (k=5), A109501 (k=6), this sequence (k=7), A093134 (k=8), A001099 (k=n).

LinearRecurrence[{6, 7}, {0, 7}, 30]
Table[7 (7^n - (-1)^n)/8, {n, 0, 30}]

vector(50, n, n--; 7*(7^n-(-1)^n)/8) \\ Altug Alkan, Apr 13 2016

for n in range(0,10**2):print((int)((7*(7**n-(-1)**n))/8))
# Soumil Mandal, Apr 14 2016

O.g.f.: 7*x/(1 - 6*x - 7*x^2).
E.g.f.: (7/8)*(exp(7*x) - exp(-x)).
a(n) = 6*a(n-1) + 7*a(n-2).
a(n) = 7*(7^n - (-1)^n)/8.
a(n) = 7*A015552(n).
Sum_{n>0} 1/(a(n) + a(n-1)) = 1/6 = A020793.
Limit_{n->oo} a(n-1)/a(n) = 1/7 = A020806.

cp's OEIS Frontend

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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A015592 a(n) = 10*a(n-1) + 11*a(n-2).

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A015609 a(n) = 11*a(n-1) + 12*a(n-2).

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A137505 Inverse binomial transform of A007910.

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A167291 a(n) = A137505(2n) + A137505(2n+1).

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Mathematica

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A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

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A015592 a(n) = 10a(n-1) + 11a(n-2).

A015609 a(n) = 11a(n-1) + 12a(n-2).