cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A033115 Base-5 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.

Original entry on oeis.org

1, 5, 26, 130, 651, 3255, 16276, 81380, 406901, 2034505, 10172526, 50862630, 254313151, 1271565755, 6357828776, 31789143880, 158945719401, 794728597005, 3973642985026, 19868214925130, 99341074625651, 496705373128255
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Partial sums of A015531. - Mircea Merca, Dec 28 2010

Links

Crossrefs

Cf. A015531.

Programs

  • Magma
    [Round((5*5^n-3)/24): n in [1..30]]; // Vincenzo Librandi, Jun 25 2011
  • Maple
    seq(1/3*floor(5^(n+1)/8),n=1..32); # Mircea Merca, Dec 26 2010
  • Mathematica
    Table[FromDigits[PadRight[{},n,{1,0}],5],{n,30}] (* or *) LinearRecurrence[ {5,1,-5},{1,5,26},30] (* Harvey P. Dale, Jan 28 2017 *)

Formula

a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3). - Joerg Arndt, Jan 08 2011
From Paul Barry, Nov 12 2003: (Start)
a(n) = floor(5^(n+2)/24);
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k);
a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^(j+k)*5^j.
Partial sums of A083425.
G.f.: 1/((1-x)*(1+x)*(1-5*x));
a(n) = 4*a(n-1) + 5*a(n-2) + 1. (End)
From Mircea Merca, Dec 28 2010: (Start)
a(n) = (1/3)*floor(5^(n+1)/8) = floor((5*5^n - 1)/24) = round((5*5^n - 3)/24) = round((5*5^n - 5)/24) = ceiling((5*5^n - 5)/24);
a(n) = a(n-2) + 5^(n-1), n > 1. (End)

A213128 Polylogarithm li(-n,-1/5) multiplied by (6^(n+1))/5.

Original entry on oeis.org

1, -1, -4, -6, 96, 1104, 2016, -112176, -1718784, -642816, 437031936, 7656021504, -24274059264, -3939918299136, -72733516959744, 699443277686784, 67781787782086656, 1236409075147014144, -25430445045847425024
Offset: 0

Views

Author

Stanislav Sykora, Jun 06 2012

Keywords

Comments

See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=5.

Examples

			polylog(-5,-1/5)*6^6/5 = 1104.

Links

Crossrefs

Cf. A212846, A210246, A212847, A213127
Cf. A213129 through A213157.
Cf. A015531.

Programs

  • Maple
    seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*5^k, k=0..n),n=0..18); # Peter Luschny, Apr 21 2013
  • Mathematica
    Table[If[n == 0, 1, PolyLog[-n, -1/5] 6^(n+1)/5], {n, 0, 18}] (* Jean-François Alcover, Jun 29 2019 *)
  • PARI
    /*See A212846; run limnpq(nmax,1,5) */
  • PARI
    x='x+O('x^66); Vec(serlaplace( 6/(5+exp(6*x)) )) \\ Joerg Arndt, Apr 21 2013
  • PARI
    a(n) = sum(k=0, n, k!*(-1)^k*6^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

Formula

See formula in A212846, setting p=1,q=5
From Peter Bala, Jun 24 2012: (Start)
E.g.f.: A(x) = 6/(5 + exp(6*x)) = 1 - x - 4*x^2/2! - 6 x^3/3! + 96*x^4/4! + ....
The compositional inverse (A(-x) - 1)^(-1) = x + 4*x^2/2 + 21*x^3/3 + 104*x^4/4 + 521*x^5/5 + ... is the logarithmic generating function for A015531.
(End)
G.f.: 1/Q(0), where Q(k) = 1 + x*(k+1)/( 1 - 5*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 17 2013
a(n) = Sum_{k=0..n} k! * (-1)^k * 6^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022

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

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Crossrefs

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

Programs

Formula

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)

A083425 a(n) = (5*5^n + (-1)^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, 397364298502604, 1986821492513021, 9934107462565104
Offset: 0

Views

Author

Paul Barry, Apr 30 2003

Keywords

Comments

Binomial transform of A083424. Inverse binomial transform of A052934.
Primes occur at indices n = 4, 66, 100, 102, 228, 346, ..., see A138647. - R. J. Mathar, Jan 19 2011
Sum_{i=0..m} (-1)^(m+i)*5^i, for m >= 0, gives all terms of the sequence. - Bruno Berselli, Aug 28 2013

Links

Programs

  • GAP
    List([0..25],n->(5*5^n+(-1)^n)/6); # Muniru A Asiru, Sep 21 2018
  • Magma
    [n le 2 select n^2 else 4*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012
  • Maple
    seq(coeff(series(factorial(n)*(5*exp(5*x)+exp(-x))/6,x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Sep 21 2018
  • Mathematica
    LinearRecurrence[{4,5},{1,4},40] (* Vincenzo Librandi, Jun 23 2012 *)
  • PARI
    a(n)=(5*5^n+(-1)^n)/6 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (5*5^n + (-1)^n)/6.
G.f.: 1/((1+x)*(1-5x)).
E.g.f.: (5*exp(5x) + exp(-x))/6.
a(n) = Sum_{k=0..n} C(n-k,k)*4^(n-2k)*5^k. - Paul Barry, Jul 29 2004
a(n) = A015531(n+1). - R. J. Mathar, Sep 17 2008
a(n) = 4*a(n-1) + 5*a(n-2). - Vincenzo Librandi, Jun 23 2012

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

Links

Crossrefs

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
1..A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
2..A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
3..A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
4..A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
5..A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
6..A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
7..A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
8..A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
9..A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
10.A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
  • Mathematica
    LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)
  • PARI
    a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A015541 Expansion of x/(1 - 5*x - 7*x^2).

Original entry on oeis.org

0, 1, 5, 32, 195, 1199, 7360, 45193, 277485, 1703776, 10461275, 64232807, 394392960, 2421594449, 14868722965, 91294775968, 560554940595, 3441838134751, 21133075257920, 129758243232857, 796722742969725, 4891921417478624, 30036666288181195
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Pisano period lengths: 1, 3, 8, 6, 8, 24, 6, 6, 24, 24, 5, 24, 12, 6, 8, 12, 16, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Links

Crossrefs

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

Programs

Formula

a(n) = 5*a(n-1) + 7*a(n-2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

Programs

  • Magma
    [n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012
  • Mathematica
    a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)
  • PARI
    A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009
  • PARI
    x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018
  • Sage
    [lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

Formula

a(n) = 5*a(n-1) + 8*a(n-2).
G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

Extensions

More precise definition by M. F. Hasler, Mar 06 2009

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

Views

Author

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

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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.

Programs

  • PARI
    a(n) = (n^(2*n + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023
  • Python
    [(n**(2*n+1)+1)//(n+1) for n in range(1,11)]

Formula

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

Extensions

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

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

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [(1/13)*(12^n-(-1)^n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011
  • Mathematica
    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 *)
  • PARI
    x='x+O('x^30); concat([0], Vec(x/(1-11*x-12*x^2))) \\ G. C. Greubel, Dec 30 2017
  • Sage
    [lucas_number1(n,11,-12) for n in range(0, 18)] # Zerinvary Lajos, Apr 27 2009
  • Sage
    [abs(gaussian_binomial(n,1,-12)) for n in range(0,18)] # Zerinvary Lajos, May 28 2009

Formula

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

A097139 Convolution of 5^n and floor(n/2).

Original entry on oeis.org

0, 0, 1, 6, 32, 162, 813, 4068, 20344, 101724, 508625, 2543130, 12715656, 63578286, 317891437, 1589457192, 7947285968, 39736429848, 198682149249, 993410746254, 4967053731280, 24835268656410, 124176343282061, 620881716410316
Offset: 0

Views

Author

Paul Barry, Jul 29 2004

Keywords

Comments

a(n+1) gives partial sums of A033115 and second partial sums of A015531.
Partial sums of (1/4)*floor(5^n/6) = (1/3)*floor(5^n/8). - Mircea Merca, Dec 27 2010

Links

Crossrefs

Column k=5 of A368296.

Programs

  • Magma
    [5^(n+1)/96 -n/8 -3/32 +(-1)^n/24: n in [0..30]]; // Vincenzo Librandi, Jun 25 2011
  • Maple
    A097139 := proc(n) 5^(n+1)/96 -n/8 -3/32 +(-1)^n/24 ; end proc: # R. J. Mathar, Jan 08 2011
  • Mathematica
    f[n_] := Floor[5^n/6]/4; Accumulate@ Array[f, 24, 0]
a[n_] := a[n] = 6 a[n - 1] - 4 a[n - 2] - 6 a[n - 3] + 5 a[n - 4]; a[0] = a[1] = 0; a[2] = 1; a[3] = 6; Array[a, 24, 0]
CoefficientList[ Series[x^2/((1 - x) (1 - 5 x) (1 - x^2)), {x, 0, 23}], x] (* Robert G. Wilson v, Jan 02 2011 *)
LinearRecurrence[{6,-4,-6,5},{0,0,1,6},30] (* Harvey P. Dale, Mar 16 2019 *)

Formula

a(n) = 5^(n+1)/96 -n/8 -3/32 +(-1)^n/24. - R. J. Mathar, Jan 08 2011
G.f.: x^2/((1-x)*(1-5*x)*(1-x^2)).
a(n) = 6*a(n-1) - 4*a(n-2) - 6*a(n-3) + 5*a(n-4).
a(n) = Sum_{k=0..n} floor((n-k)/2)*4^k = Sum_{k=0..n} floor(k/2)*4^(n-k).
From Mircea Merca, Dec 27 2010: (Start)
4*a(n) = round((5*5^n-12*n-9)/24) = floor((5*5^n-12*n-5)/24) = ceiling((5*5^n-12*n-13)/24) = round((5*5^n-12*n-5)/24).
a(n) = a(n-2) + (5^(n-1)-1)/4, n>1. (End)
a(n) = (floor(5^(n+1)/24) - floor((n+1)/2))/4. - Seiichi Manyama, Dec 22 2023
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