A051958 -id:A051958 - cp's OEIS frontend

A015540 a(n) = 5a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

0, 1, 5, 31, 185, 1111, 6665, 39991, 239945, 1439671, 8638025, 51828151, 310968905, 1865813431, 11194880585, 67169283511, 403015701065, 2418094206391, 14508565238345, 87051391430071, 522308348580425, 3133850091482551, 18803100548895305, 112818603293371831

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct vertices of the complete graph K_7. Example: a(2)=5 because the walks of length 2 between the vertices A and B of the complete graph ABCDEFG are ACB, ADB, AEB, AFB and AGB. - Emeric Deutsch, Apr 01 2004
General form: k=6^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500. -Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Pisano period lengths: 1, 1, 2, 2, 2, 2, 14, 2, 2, 2, 10, 2, 12, 14, 2, 2, 16, 2, 18, 2, ... - R. J. Mathar, Aug 10 2012
Sum_{i=0..m} (-1)^(m+i)*6^i, for m >= 0, gives all terms after 0. - Bruno Berselli, Aug 28 2013
The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. Also A053524, A080424, A051958. - Felix P. Muga II, Mar 09 2014

			G.f. = x + 5*x^2 + 31*x^3 + 185*x^4 + 1111*x^5 + 6665*x^6 + 39991*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (5,6).

Partial sums are in A033116. Cf. A014987.

[Floor(6^n/7-(-1)^n/7): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(6^n/7),n=0..25); # Mircea Merca, Dec 28 2010

k=0; lst={k}; Do[k = 6^n-k; AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[x / ((1 - 6 x) (1 + x)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
LinearRecurrence[{5,6},{0,1},30] (* Harvey P. Dale, May 12 2015 *)

my(x='x+O('x^30)); concat([0], Vec(x/((1-6*x)*(1+x)))) \\ G. C. Greubel, Jan 24 2018

a(n) = round(6^n/7); \\ Altug Alkan, Sep 05 2018

[lucas_number1(n,5,-6) for n in range(21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 6*a(n-2).
From Paul Barry, Apr 20 2003: (Start)
a(n) = (6^n - (-1)^n)/7.
G.f.: x/((1-6*x)*(1+x)).
E.g.f.: (exp(6*x) - exp(-x))/7. (End)
a(n) = 6^(n-1) - a(n-1). - Emeric Deutsch, Apr 01 2004
a(n+1) = Sum_{k=0..n} binomial(n-k, k)*5^(n-2*k)*6^k. - Paul Barry, Jul 29 2004
a(n) = round(6^n/7). - Mircea Merca, Dec 28 2010
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} A135278(n-1,k)*(-7)^k = (6^n - (-1)^n)/7 = (-1)^(n-1)*Sum_{k=0..n-1} (-6)^k. Equals (-1)^(n-1)*Phi(n,-6), where Phi is the cyclotomic polynomial when n is an odd prime. (For n > 0.) - Tom Copeland, Apr 14 2014

A079773 a(n) = 2a(n-1)+15a(n-2) with n>0, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 19, 68, 421, 1862, 10039, 48008, 246601, 1213322, 6125659, 30451148, 152787181, 762341582, 3816490879, 19068105488, 95383574161, 476788730642, 2384331073699, 11920493107028, 59605952319541, 298019301244502

Offset: 0

Paul Barry, Feb 20 2003

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,15)

Cf. A051958, A015441.

[(5^n-(-3)^n)/8: n in [0..25]]; // Vincenzo Librandi, Aug 05 2013

Join[{a=0,b=1},Table[c=2*b+15*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[x / ((1 + 3 x) (1 - 5 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)

[lucas_number1(n,2,-15) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009

G.f.: x/((1+3*x)*(1-5*x)).
a(n) = (5^n-(-3)^n)/8.
a(n) = sum(k=1..n, binomial(n, 2*k-1)*4^(2*(k-1))).
E.g.f.: exp(x)*sinh(4*x)/4. - Paul Barry, Jul 09 2003
a(n+1) = Sum_{k = 0..n} A238801(n,k)*4^k. - Philippe Deléham, Mar 07 2014

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0

Offset: 0

Philippe Deléham, Mar 05 2014

Row sums are powers of 2.

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.

Cf. A095704, A178616.

Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A080424 a(n) = 3a(n-1) + 18a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 27, 135, 891, 5103, 31347, 185895, 1121931, 6711903, 40330467, 241805655, 1451365371, 8706597903, 52244370387, 313451873415, 1880754287211, 11284396583103, 67706766919107, 406239439253175, 2437440122303451, 14624630273467503, 87747813021864627, 526486783988008935

Offset: 0

Paul Barry, Feb 24 2003

The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (3,18).

Cf. A001045, A015441, A051958, A079773.

[(6^n-(-3)^n)/9: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

a[n_]:=(6^n - (-3)^n)/9; Array[a, 22, 0] (* Robert G. Wilson v, Aug 13 2011 *)
LinearRecurrence[{3,18}, {0,1}, 31] (* G. C. Greubel, Dec 22 2023 *)

a(n)=(6^n-(-3)^n)/9 \\ Charles R Greathouse IV, Jun 10 2011

[3^(n-1)*lucas_number1(n,1,-2) for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: x/((1+3*x)*(1-6*x)).
a(n) = (6^n - (-3)^n)/9.
a(n+1) = 6*a(n) + (-3)^n. - Paul Curtz, Jun 07 2011
a(n) = 3^(n-1)*A001045(n). - R. J. Mathar, Mar 08 2021

A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 2, 6, 4, 12, 36, 8, 24, 72, 216, 16, 48, 144, 432, 1296, 32, 96, 288, 864, 2592, 7776, 64, 192, 576, 1728, 5184, 15552, 46656, 128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616, 512, 1536, 4608, 13824, 41472, 124416, 373248, 1119744, 3359232, 10077696

Offset: 0

Reinhard Zumkeller, Nov 20 2004

			From _Stefano Spezia_, Apr 28 2024: (Start)
Triangle begins:
   1;
   2,  6;
   4, 12,  36;
   8, 24,  72, 216;
  16, 48, 144, 432, 1296;
  32, 96, 288, 864, 2592, 7776;
  ...
(End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A000400, A001021, A003586, A005010, A007283, A016129.
Cf. A016149, A051958, A053524, A100852, A248337.
Cf. A065333(T(n, k))=1.

A100851:= func< n,k | 2^n*3^k >;
[A100851(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 11 2024

A100851[n_, k_]= 2^n*3^k;
Table[A100851[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 11 2024 *)

def A100851(n,k): return 2^n*3^k
flatten([[A100851(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 11 2024

T(n,0) = A000079(n).
T(n,1) = A007283(n) for n>0.
T(n,2) = A005010(n) for n>1.
T(n,n) = A000400(n) = A100852(n,n).
Sum_{k=0..n} T(n, k) = A016129(n).
T(2*n, n) = A001021(n). - Reinhard Zumkeller, Mar 04 2006
G.f.: 1/((1 - 2*x)*(1 - 6*x*y)). - Stefano Spezia, Apr 28 2024
From G. C. Greubel, Nov 11 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A053524(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*((1-(-1)^n)*A248337((n+1)/2) + (1 + (-1)^n)*A016149(n/2)).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n) *A051958((n+2)/2) + 2*(1-(-1)^n)*A051958((n+1)/2)). (End)
Sum_{n>=0, k=0..n} 1/T(n,k) = 12/5. - Amiram Eldar, May 12 2025

A053455 a(n) = ((8^n) - (-6)^n)/14.

Original entry on oeis.org

0, 1, 2, 52, 200, 2896, 15392, 169792, 1078400, 10306816, 72376832, 639480832, 4753049600, 40201179136, 308548739072, 2546754076672, 19903847628800, 162051890937856, 1279488468058112, 10337467701133312, 82090381869056000, 660379213392510976, 5261096756499709952, 42220395755839946752

Offset: 0

Barry E. Williams, Jan 13 2000

Previous name was: A linear recursive sequence.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,48).

Cf. A053404, A051958, A015441, A080921.

A053455:=n->((8^n)-(-6)^n)/14; seq(A053455(n), n=0..30); # Wesley Ivan Hurt, Mar 07 2014

LinearRecurrence[{2,48},{1,2},30] (* Harvey P. Dale, Nov 28 2011 *)
CoefficientList[Series[x / (1 - 2 x - 48 x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Mar 08 2014 *)

a(n) = ((8^n)-(-6)^n)/14; \\ Joerg Arndt, Mar 08 2014

a(n) = 2*a(n-1) + 48*a(n-2), n>=2; a(0)=0, a(1)=1.
a(n) = ((8^n)-(-6)^n)/14 = (2^(n-1))*((4^n) - (-3)^n)/7 = 2^(n-1)*A053404(n).
G.f.: x/((1+6*x)*(1-8*x)). - Harvey P. Dale, Nov 28 2011
a(n) = A080921(n). - Philippe Deléham, Mar 05 2014
a(n+1) = Sum_{k=0..n} A238801(n,k)*7^k. - Philippe Deléham, Mar 07 2014

More terms from James Sellers, Feb 02 2000

New name (from formula), Joerg Arndt, Mar 05 2014

A080920 a(n) = 2a(n-1) + 35a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 39, 148, 1661, 8502, 75139, 447848, 3525561, 22725802, 168846239, 1133095548, 8175809461, 56009963102, 398173257339, 2756695223248, 19449454453361, 135383241720402, 951497389308439, 6641408238830948

Offset: 0

Paul Barry, Feb 24 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,35).

Cf. A079773, A051958, A015441.

Join[{a=0,b=1},Table[c=2*b+35*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1 / ((1 + 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{2,35},{0,1},30] (* Harvey P. Dale, Aug 24 2017 *)

a(n) = 7^n/12 - (-5)^n/12.
a(n) = Sum{k=1..n, binomial(n, 2k-1)*6^(2(k-1))}.
G.f.: 1/((1+5x)(1-7x)).
a(n+1) = Sum_{k = 0..n} A238801(n,k)*6^k. - Philippe Deléham, Mar 07 2014

A080921 a(n) = 2a(n-1) + 48a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 52, 200, 2896, 15392, 169792, 1078400, 10306816, 72376832, 639480832, 4753049600, 40201179136, 308548739072, 2546754076672, 19903847628800, 162051890937856, 1279488468058112, 10337467701133312, 82090381869056000

Offset: 0

Paul Barry, Feb 24 2003

Essentially the same as A053455: a(n) = A053455(n-1), n>=1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,48).

Cf. A079773, A051958, A015441, A080920.

CoefficientList[Series[x / ((1 + 6 x) (1 - 8 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{2,48},{0,1},30] (* Harvey P. Dale, Jan 20 2016 *)

a(n) = (8^n - (-6)^n)/14.
a(n) = Sum{k=1..n, binomial(n, 2k-1) * 7^(2(k-1)) }
G.f.: x/((1+6*x)*(1-8*x)).
a(n) = A053455(n-1), n>=1. [R. J. Mathar, Sep 18 2008]

A083578 a(n) = (6^n + (-4)^n)/2.

Original entry on oeis.org

1, 1, 26, 76, 776, 3376, 25376, 131776, 872576, 4907776, 30757376, 179301376, 1096779776, 6496792576, 39316299776, 234555621376, 1412702437376, 8454739787776, 50814338072576, 304542431051776, 1828628975845376

Offset: 0

Paul Barry, Apr 30 2003

Index entries for linear recurrences with constant coefficients, signature (2,24).

Cf. A083579. First differences of A051958.

LinearRecurrence[{2,24},{1,1},30] (* Harvey P. Dale, Nov 29 2017 *)

a(n) = (6^n + (-4)^n)/2.
G.f.: (1-x)/((1+4x)(1-6x)).
E.g.f.: (exp(6x) + exp(-4x))/2.

A383651 Expansion of 1/((1-x) * (1+4x) (1-6*x)).

Original entry on oeis.org

1, 3, 31, 135, 1015, 5271, 34903, 196311, 1230295, 7172055, 43871191, 259871703, 1572651991, 9382224855, 56508097495, 338189591511, 2032573522903, 12181697242071, 73145159033815, 438651051877335, 2632785920566231, 15793197086188503, 94773256265966551

Offset: 0

Seiichi Manyama, May 04 2025

Index entries for linear recurrences with constant coefficients, signature (3,22,-24).

Cf. A051958, A083578.

PARI
```
a(n) = (6^(n+2)-2+(-4)^(n+2))/50;
```

a(n) = Sum_{k=0..n} 5^k * (-4)^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-5)^k * 6^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = (6^(n+2) - 2 + (-4)^(n+2))/50 = (A083578(n+2) - 1)/25.
a(n) = 3*a(n-1) + 22*a(n-2) - 24*a(n-3).
E.g.f.: exp(-4*x)*(8 - exp(5*x) + 18*exp(10*x))/25. - Stefano Spezia, May 04 2025

cp's OEIS Frontend

A015540 a(n) = 5*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.

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A079773 a(n) = 2*a(n-1)+15*a(n-2) with n>0, a(0)=0, a(1)=1.

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A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

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A080424 a(n) = 3*a(n-1) + 18*a(n-2), a(0)=0, a(1)=1.

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A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

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A053455 a(n) = ((8^n) - (-6)^n)/14.

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A015540 a(n) = 5a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

A079773 a(n) = 2a(n-1)+15a(n-2) with n>0, a(0)=0, a(1)=1.

A080424 a(n) = 3a(n-1) + 18a(n-2), a(0)=0, a(1)=1.

A080921 a(n) = 2a(n-1) + 48a(n-2), a(0)=0, a(1)=1.

A383651 Expansion of 1/((1-x) * (1+4x) (1-6*x)).