A005057 -id:A005057 - cp's OEIS frontend

A024036 a(n) = 4^n - 1.

0, 3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663, 281474976710655

Offset: 0

N. J. A. Sloane

This sequence is the normalized length per iteration of the space-filling Peano-Hilbert curve. The curve remains in a square, but its length increases without bound. The length of the curve, after n iterations in a unit square, is a(n)*2^(-n) where a(n) = 4*a(n-1)+3. This is the sequence of a(n) values. a(n)*(2^(-n)*2^(-n)) tends to 1, the area of the square where the curve is generated, as n increases. The ratio between the number of segments of the curve at the n-th iteration (A015521) and a(n) tends to 4/5 as n increases. - Giorgio Balzarotti, Mar 16 2006
Numbers whose base-4 representation is 333....3. - Zerinvary Lajos, Feb 03 2007
From Eric Desbiaux, Jun 28 2009: (Start)
It appears that for a given area, a square n^2 can be divided into n^2+1 other squares.
It's a rotation and zoom out of a Cartesian plan, which creates squares with side
= sqrt( (n^2) / (n^2+1) ) --> A010503|A010532|A010541... --> limit 1,
and diagonal sqrt(2*sqrt((n^2)/(n^2+1))) --> A010767|... --> limit A002193.
(End)
Also the total number of line segments after the n-th stage in the H tree, if 4^(n-1) H's are added at the n-th stage to the structure in which every "H" is formed by 3 line segments. A164346 (the first differences of this sequence) gives the number of line segments added at the n-th stage. - Omar E. Pol, Feb 16 2013
a(n) is the cumulative number of segment deletions in a Koch snowflake after (n+1) iterations. - Ivan N. Ianakiev, Nov 22 2013
Inverse binomial transform of A005057. - Wesley Ivan Hurt, Apr 04 2014
For n > 0, a(n) is one-third the partial sums of A002063(n-1). - J. M. Bergot, May 23 2014
Also the cyclomatic number of the n-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 18 2017

			G.f. = 3*x + 15*x^2 + 63*x^3 + 255*x^4 + 1023*x^5 + 4095*x^6 + ...

Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Felix Fröhlich, Table of n, a(n) for n = 0..99
Alexander V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 15 (2019), 046, 53 pages; arXiv preprint, arXiv:1809.00122 [math.CA], 2018-2019.
Eric Weisstein's World of Mathematics, Cyclomatic Number.
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000051, A000120, A000225, A000302, A002001, A002063, A002193, A002450, A005057, A010503, A010532, A010541, A010767, A015521, A020988, A027637 (partial products), A078904 (partial sums), A079978, A080674, A164346 (first differences), A178789, A179857, A248721.

a024036 = (subtract 1) . a000302
a024036_list = iterate ((+ 3) . (* 4)) 0
-- Reinhard Zumkeller, Oct 03 2012

A024036:=n->4^n-1; seq(A024036(n), n=0..30); # Wesley Ivan Hurt, Apr 04 2014

Array[4^# - 1 &, 50, 0] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
(* Start from Eric W. Weisstein, Sep 19 2017 *)
Table[4^n - 1, {n, 0, 20}]
4^Range[0, 20] - 1
LinearRecurrence[{5, -4}, {0, 3}, 20]
CoefficientList[Series[3 x/(1 - 5 x + 4 x^2), {x, 0, 20}], x]
(* End *)

for(n=0, 100, print1(4^n-1, ", ")) \\ Felix Fröhlich, Jul 04 2014

[gaussian_binomial(2*n,1, 2) for n in range(21)] # Zerinvary Lajos, May 28 2009

[stirling_number2(2*n+1, 2) for n in range(21)] # Zerinvary Lajos, Nov 26 2009

a(n) = 3*A002450(n). - N. J. A. Sloane, Feb 19 2004
G.f.: 3*x/((-1+x)*(-1+4*x)) = 1/(-1+x) - 1/(-1+4*x). - R. J. Mathar, Nov 23 2007
E.g.f.: exp(4*x) - exp(x). - Mohammad K. Azarian, Jan 14 2009
a(n) = A000051(n)*A000225(n). - Reinhard Zumkeller, Feb 14 2009
A079978(a(n)) = 1. - Reinhard Zumkeller, Nov 22 2009
a(n) = A179857(A000225(n)), for n > 0; a(n) > A179857(m), for m < A000225(n). - Reinhard Zumkeller, Jul 31 2010
a(n) = 4*a(n-1) + 3, with a(0) = 0. - Vincenzo Librandi, Aug 01 2010
A000120(a(n)) = 2*n. - Reinhard Zumkeller, Feb 07 2011
a(n) = (3/2)*A020988(n). - Omar E. Pol, Mar 15 2012
a(n) = (Sum_{i=0..n} A002001(i)) - 1 = A178789(n+1) - 3. - Ivan N. Ianakiev, Nov 22 2013
a(n) = n*E(2*n-1,1)/B(2*n,1), for n > 0, where E(n,x) denotes the Euler polynomials and B(n,x) the Bernoulli polynomials. - Peter Luschny, Apr 04 2014
a(n) = A000302(n) - 1. - Sean A. Irvine, Jun 18 2019
Sum_{n>=1} 1/a(n) = A248721. - Amiram Eldar, Nov 13 2020
a(n) = A080674(n) - A002450(n). - Elmo R. Oliveira, Dec 02 2023

More terms Wesley Ivan Hurt, Apr 04 2014

A155588 a(n) = 5^n + 2^n - 1^n.

Original entry on oeis.org

1, 6, 28, 132, 640, 3156, 15688, 78252, 390880, 1953636, 9766648, 48830172, 244144720, 1220711316, 6103532008, 30517610892, 152587956160, 762939584196, 3814697527768, 19073486852412, 95367432689200, 476837160300276

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Michael De Vlieger, Table of n, a(n) for n = 0..1430
Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A074501, A074600, A005057, A005056, A099393.

CoefficientList[Series[1/(1 - 5 x) + 1/(1 - 2 x) - 1/(1 - x), {x, 0, 21}], x] (* Michael De Vlieger, Mar 02 2022 *)

a(n)=5^n+2^n-1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-5*x)+1/(1-2*x)-1/(1-x).
E.g.f.: e^(5*x)+e^(2*x)-e^x.
a(n) = 7*a(n-1)-10*a(n-2)-4 with a(0)=1, a(1)=6. - Vincenzo Librandi, Jul 21 2010
a(n) = A074600(n)-1. - R. J. Mathar, Mar 10 2022

A155596 a(n) = 5^n - 2^n + 1^n.

Original entry on oeis.org

1, 4, 22, 118, 610, 3094, 15562, 77998, 390370, 1952614, 9764602, 48826078, 244136530, 1220694934, 6103499242, 30517545358, 152587825090, 762939322054, 3814697003482, 19073485803838, 95367430592050, 476837156105974

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594.

Table[5^n-2^n+1,{n,0,30}] (* or *) LinearRecurrence[{8,-17,10},{1,4,22},30] (* Harvey P. Dale, Sep 11 2019 *)

a(n)=5^n-2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-5*x)-1/(1-2*x)+1/(1-x).
E.g.f.: e^(5*x) - e^(2*x) + e^x.
a(n) = 7*a(n-1)-10*a(n-2)+4 with a(0)=1, a(1)=4. - Vincenzo Librandi, Jul 21 2010
a(n) = A005057(n)+1. - R. J. Mathar, Mar 10 2022

A016127 Expansion of g.f. 1/((1-2x)(1-5*x)).

Original entry on oeis.org

1, 7, 39, 203, 1031, 5187, 25999, 130123, 650871, 3254867, 16275359, 81378843, 406898311, 2034499747, 10172515119, 50862608363, 254313107351, 1271565667827, 6357828601279, 31789143530683, 158945718701991, 794728595607107, 3973642982229839, 19868214919537803, 99341074614466231

Offset: 0

N. J. A. Sloane

With leading zero, binomial transform of A002450. - Paul Barry, Apr 11 2003
The sequence of fractions a(n)/(n+1) is the 3rd binomial transform of the sequence of fractions J(n+1)/(n+1) where J(n) is A001045(n). - Paul Barry, Aug 05 2005
Equals term (1,2) in M^n, M = the 3 X 3 matrix [1, 1, 3; 1, 3, 1; 3, 1, 1]. a(n)/a(n-1) tends to 5, a root to the charpoly x^3 - 5*x^2 - 4*x + 20. - Gary W. Adamson, Mar 12 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, The sequence of higher order Mersenne numbers and associated binomial transforms, arXiv:2307.08073 [math.NT], 2023.
Index entries for linear recurrences with constant coefficients, signature (7,-10).

Cf. A001045, A002450, A005057, A020989.

[(5^(n+1)-2^(n+1))/3: n in [0..30]]; // Vincenzo Librandi, Jun 08 2011

Join[{a=1,b=7},Table[c=7*b-10*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
CoefficientList[Series[1/((1 - 2 x) (1 - 5 x)), {x, 0, 19}], x] (* Michael De Vlieger, Jan 31 2018 *)
LinearRecurrence[{7,-10},{1,7},30] (* Harvey P. Dale, Aug 18 2020 *)

Vec(1/((1-2*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,7,10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

[(5^n - 2^n)/3 for n in range(1,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (5^(n+1) - 2^(n+1))/3 = Sum_{i=0..n} 5^i*2^(n-1) = 5*a(n-1) + 2^n = 2*a(n-1) + 5^n. - Henry Bottomley, Apr 07 2003
Binomial transform of A020989. - Paul Barry, May 18 2003
From Paul Barry, Aug 05 2005: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..n} 5^(n-j)*binomial(j,k).
a(n) = Sum_{k=0..n} 2^k*5^(n-k) = Sum_{k=0..n} 5^k*2^(n-k). (End)
For n > 2, a(n) = 9*a(n-1) - 24*a(n-2) + 20*a(n-3). - Gary W. Adamson, Dec 26 2007
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: exp(2*x)*(5*exp(3*x) - 2)/3.
a(n) = A005057(n+1)/3.
a(n) = 7*a(n-1) - 10*a(n-2). (End)

A155590 a(n) = 7^n + 2^n - 1.

Original entry on oeis.org

1, 8, 52, 350, 2416, 16838, 117712, 823670, 5765056, 40354118, 282476272, 1977328790, 13841291296, 96889018598, 678223089232, 4747561542710, 33232930635136, 232630514118278, 1628413598172592, 11398895185897430, 79792266298660576, 558545864085381158, 3909821048587182352

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23,14).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590.

Table[7^n + 2^n - 1, {n, 0, 25}] (* Paolo Xausa, Jul 30 2024 *)

a(n)=7^n+2^n-1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-7*x)+1/(1-2*x)-1/(1-x).
E.g.f.: exp(7*x)+exp(2*x)-exp(x).
a(n) = 9*a(n-1)-14*a(n-2)-6 with a(0) = 1, a(1) = 8. - Vincenzo Librandi, Jul 21 2010
a(n) = A074602(n)-1. - R. J. Mathar, Mar 10 2022

A155592 8^n+2^n-1^n.

Original entry on oeis.org

1, 9, 67, 519, 4111, 32799, 262207, 2097279, 16777471, 134218239, 1073742847, 8589936639, 68719480831, 549755822079, 4398046527487, 35184372121599, 281474976776191, 2251799813816319, 18014398509744127, 144115188076380159

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Index entries for linear recurrences with constant coefficients, signature (11,-26,16).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590.

Table[8^n+2^n-1^n,{n,0,30}] (* or *) LinearRecurrence[{11,-26,16},{1,9,67},30] (* Harvey P. Dale, Jul 09 2017 *)

a(n)=8^n+2^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-8*x)+1/(1-2*x)-1/(1-x). E.g.f.: e^(8*x)+e^(2*x)-e^x.
a(n)=10*a(n-1)-16*a(n-2)-7 with a(0)=1, a(1)=9 - Vincenzo Librandi, Jul 21 2010
a(n) = A074603(n)-1. - R. J. Mathar, Mar 10 2022

A155593 a(n) = 9^n + 2^n - 1.

Original entry on oeis.org

1, 10, 84, 736, 6576, 59080, 531504, 4783096, 43046976, 387421000, 3486785424, 31381061656, 282429540576, 2541865836520, 22876792471344, 205891132127416, 1853020188917376, 16677181699797640, 150094635297261264

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-29,18).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590, A155592.

Table[9^n+2^n-1,{n,0,30}] (* or *) LinearRecurrence[{12,-29,18},{1,10,84},30] (* Harvey P. Dale, Sep 09 2022 *)

a(n)=9^n+2^n-1^n \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-9*x)+1/(1-2*x)-1/(1-x).
E.g.f.: e^(9*x)+e^(2*x)-e^x.
a(n) = 11*a(n-1)-18*a(n-2)-8 with a(0)=1, a(1)=10. - Vincenzo Librandi, Jul 21 2010
a(n) = A074604(n)-1. - R. J. Mathar, Mar 10 2022

A155594 10^n+2^n-1.

Original entry on oeis.org

1, 11, 103, 1007, 10015, 100031, 1000063, 10000127, 100000255, 1000000511, 10000001023, 100000002047, 1000000004095, 10000000008191, 100000000016383, 1000000000032767, 10000000000065535, 100000000000131071

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Index entries for linear recurrences with constant coefficients, signature (13,-32,20).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590, A155592, A155593.

LinearRecurrence[{13,-32,20},{1,11,103},20] (* Harvey P. Dale, Mar 07 2015 *)

a(n)=10^n+2^n-1^n \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-10*x)+1/(1-2*x)-1/(1-x). E.g.f.: e^(10*x)+e^(2*x)-e^x.
a(n)=12*a(n-1)-20*a(n-2)-9 with a(0)=1, a(1)=11 - Vincenzo Librandi, Jul 21 2010
a(0)=1, a(1)=11, a(2)=103, a(n)=13*a(n-1)-32*a(n-2)+20*a(n-3). - Harvey P. Dale, Mar 07 2015
a(n) = A050621(n+1)-1. - R. J. Mathar, Mar 10 2022

A225466 Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 21, 9, 8, 117, 135, 27, 16, 609, 1431, 702, 81, 32, 3093, 13275, 12015, 3240, 243, 64, 15561, 115479, 171990, 81405, 13851, 729, 128, 77997, 970515, 2238327, 1655640, 479682, 56133, 2187, 256, 390369, 7998111, 27533142, 29893941, 13121514, 2561706

Offset: 0

Peter Luschny, May 08 2013

The definition of the Stirling-Frobenius subset numbers of order m is in A225468.
From Wolfdieter Lang, Apr 09 2017: (Start)
This is the Sheffer triangle (exp(2*x), exp(3*x) - 1), denoted by S2[3,2]. See also A282629 for S2[3,1]. The stirling2 triangle A048993 is in this notation denoted by S2[1,0].
The a-sequence for this Sheffer triangle has e.g.f. 3*x/log(1+x) and is 3*A006232(n)/A006233(n) (Cauchy numbers of the first kind). For a- and z-sequences for Sheffer triangles see the W. Lang link under A006232, also with references).
The z-sequence has e.g.f. (3/(log(1+x)))*(1 - 1/(1+x)^(2/3)) and gives 2*A284862/A284863.
The first column k sequences divided by 3^k are A000079, A016127, A016297, A025999. For the e.g.f.s and o.g.f.s see below.
The row sums give A284864. The alternating row sums give A284865.
This triangle appears in the o.g.f. G(n, x) of the sequence {(2 + 3*m)^n}{m>=0}, as G(n, x) = Sum{k=0..n} T(n, k)*k!*x^k/(1-x)^(k+1), n >= 0. Hence the corresponding e.g.f. is, by the linear inverse Laplace transform, E(n, t) = Sum_{m >=0} (2 + 3*m)^n t^m/m! = exp(t)*Sum_{k=0..n} T(n, k)*t^k.
The corresponding Eulerian number triangle is A225117(n, k) = Sum_{m=0..k} (-1)^(k-m)*binomial(n-m, k-m)*T(n, m)*m!, 0 <= k <= n. (End)

			[n\k][ 0,     1,      2,       3,       4,      5,     6,    7]
[0]    1,
[1]    2,     3,
[2]    4,    21,      9,
[3]    8,   117,    135,      27,
[4]   16,   609,   1431,     702,      81,
[5]   32,  3093,  13275,   12015,    3240,    243,
[6]   64, 15561, 115479,  171990,   81405,  13851,   729,
[7]  128, 77997, 970515, 2238327, 1655640, 479682, 56133, 2187.
...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence (see the Maple program): T(4, 2) = 3*T(3, 1) + (3*2+2)*T(3, 2) = 3*117 + 8*135 = 1431.
Boas-Buck recurrence for column k = 2, and n = 4: T(4,2) = (1/2)*(2*(4 + 3*2)*T(3, 2) + 2*6*(-3)^2*Bernoulli(2)*T(2, 2)) = (1/2)*(20*135 + 12*9*(1/6)*9) = 1431. (End)

Vincenzo Librandi, Rows n = 0..50, flattened
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 9.
Peter Luschny, Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork, Normal ordering problem and the extensions of the Stirling grammar, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv:1308.0169 [math.CO], 2013, p. 12.

Cf. A048993 (m=1), A154537 (m=2), A225467 (m=4), A225468.

Cf. A000079, A000244, A005057, A016127, A016297, A025999, A006232/A006233, A225117, A225472, A225468, A282629, A284862/A284863, A284864, A284865.

SF_SS := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or  k < 0 then return(0) fi;
m*SF_SS(n-1, k-1, m) + (m*(k+1)-1)*SF_SS(n-1, k, m) end:
seq(print(seq(SF_SS(n, k, 3), k=0..n)), n=0..5);

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFSS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/k!; Table[ SFSS[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from Sage *)

T(n, k) = sum(j=0, k, binomial(k, j)*(-1)^(j - k)*(2 + 3*j)^n/k!);
for(n=0, 10, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 10 2017

from sympy import binomial, factorial
def T(n, k): return sum(binomial(k, j)*(-1)**(j - k)*(2 + 3*j)**n//factorial(k) for j in range(k + 1))
for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 10 2017

@CachedFunction
def EulerianNumber(n, k, m) :
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1,k-1,m) + (m*k+1)*EulerianNumber(n-1,k,m)
def SF_SS(n, k, m):
    return add(EulerianNumber(n,j,m)*binomial(j,n-k) for j in (0..n))/ factorial(k)
def A225466(n): return SF_SS(n, k, 3)

T(n, k) = (1/k!)*Sum_{j=0..n} binomial(j, n-k)*A_3(n, j) where A_m(n, j) are the generalized Eulerian numbers A225117.
For a recurrence see the Maple program.
T(n, 0) ~ A000079; T(n, 1) ~ A005057; T(n, n) ~ A000244.
From Wolfdieter Lang, Apr 09 2017: (Start)
T(n, k) = Sum_{j=0..k} binomial(k,j)*(-1)^(j-k)*(2 + 3*j)^n/k!, 0 <= k <= n.
E.g.f. of triangle: exp(2*z)*exp(x*(exp(3*z)-1)) (Sheffer type).
E.g.f. for sequence of column k is exp(2*x)*((exp(3*x) - 1)^k)/k! (Sheffer property).
O.g.f. for sequence of column k is 3^k*x^k/Product_{j=0..k} (1 - (2+3*j)*x).
A nontrivial recurrence for the column m=0 entries T(n, 0) = 2^n from the z-sequence given above: T(n,0) = n*Sum_{k=0..n-1} z(k)*T(n-1,k), n >= 1, T(0, 0) = 1.
The corresponding recurrence for columns k >= 1 from the a-sequence is T(n, k) = (n/k)* Sum_{j=0..n-k} binomial(k-1+j, k-1)*a(j)*T(n-1, k-1+j).
Recurrence for row polynomials R(n, x) (Meixner type): R(n, x) = ((3*x+2) + 3*x*d_x)*R(n-1, x), with differentiation d_x, for n >= 1, with input R(0, x) = 1.
(End)
Boas-Buck recurrence for column sequence k: T(n, k) = (1/(n - k))*((n/2)*(4 + 3*k)*T(n-1, k) + k*Sum_{p=k..n-2} binomial(n, p)*(-3)^(n-p)*Bernoulli(n-p)*T(p, k)), for n > k >= 0, with input T(k, k) = 3^k. See a comment and references in A282629, An example is given below. - Wolfdieter Lang, Aug 11 2017

A155589 a(n) = 6^n + 2^n - 1.

Original entry on oeis.org

1, 7, 39, 223, 1311, 7807, 46719, 280063, 1679871, 10078207, 60467199, 362799103, 2176786431, 13060702207, 78364180479, 470185017343, 2821109972991, 16926659575807, 101559956930559, 609359740534783, 3656158441111551, 21936950642475007, 131621703846461439

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A074501, A074600, A005057, A005056, A099393, A155588.

Table[6^n + 2^n - 1, {n, 0, 25}] (* Paolo Xausa, Jul 30 2024 *)

a(n)=6^n+2^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-6*x)+1/(1-2*x)-1/(1-x).
E.g.f.: exp(6*x)+exp(2*x)-exp(x).
a(n) = 8*a(n-1)-12*a(n-2)-5 with a(0) = 1, a(1) = 7. - Vincenzo Librandi, Jul 21 2010

cp's OEIS Frontend

A024036 a(n) = 4^n - 1.

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A155588 a(n) = 5^n + 2^n - 1^n.

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A155596 a(n) = 5^n - 2^n + 1^n.

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A016127 Expansion of g.f. 1/((1-2*x)*(1-5*x)).

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A155590 a(n) = 7^n + 2^n - 1.

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A155592 8^n+2^n-1^n.

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A155593 a(n) = 9^n + 2^n - 1.

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A016127 Expansion of g.f. 1/((1-2x)(1-5*x)).