A100774 -id:A100774 - cp's OEIS frontend

A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).

2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974

Offset: 0

N. J. A. Sloane, David W. Wilson

Definitions of Pisot and related sequences:
Pisot sequence E(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2) + 1/2) = nearest integer to a(n-1)^2/a(n-2), with 0 < x < y.
Pisot sequence L(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2)).
Pisot sequence P(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2) - 1/2).
Pisot sequence T(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)).
Pisot/Shallit sequence S(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)+1).
A025192 is the main entry for the sequence of numbers 2*3^n.
Number of tilings of a 4 X (4n+4) rectangle into T tetrominoes.
Numbers n such that 3^n = n/2 mod n. Cf. A066601 3^n mod n. - Zak Seidov, Aug 26 2006, Nov 20 2008
For n >= 1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
a(n+1) is the number of compositions of n when there are 2 types of each natural number. - Milan Janjic, Aug 13 2010
2*Sum_{n>=2} 1/A083667(n) = 2*Sum_{n>=2} 2^(-n)*3^(-((n*(n-1))/2)) = Sum_{n>=1} 1/Product_{k=1..n} A008776(k) = Sum_{n>=1} 1/Product_{k=1..n} 2*3^k = 0.17609845431233461692099660022134... . - Alexander R. Povolotsky, Aug 08 2011
Number of monic squarefree polynomials over F_3 of degree n+1. - Charles R Greathouse IV, Feb 07 2012
a(n) is the sum of the elements of the n-th power of the matrix {{1, 2}, {2, 1}}. - Griffin N. Macris, Mar 25 2016
Let D(m) denote the set of divisors of a number m, and consider s1(m) and s2(m) the sums of those divisors that are congruent to 1 and 2 (mod 3) respectively. This sequence lists the numbers m such that s1(m) = 1 and s2(m) = 2. - Michel Lagneau, Feb 09 2017
a(n) is the multiplicative order of k modulo 3^(n+1), where k is any number congruent to 2 or 5 modulo 9. Note that for n > 0, k is a primitive root modulo 3^(n+1) if and only if k == 2, 5 (mod 9). - Jianing Song, Apr 20 2021

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 203).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..200
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 26.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 170
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Craig Knecht, Sphinx tiling of a repetitive shape.
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scu. Norm. Sup. Pisa 2 ser, vol 7. no 3-4 (1938) p 205-248.
Index entries for linear recurrences with constant coefficients, signature (3).

Apart from initial term, same as A025192.
Cf. A080643.
Cf. A000244.

List([0..30], n-> 2*3^n); # G. C. Greubel, Sep 11 2019

a008776 = (* 2) . (3 ^)
a008776_list = iterate (* 3) 2  -- Reinhard Zumkeller, Oct 19 2015

[2*3^n: n in [0..30]]; // G. C. Greubel, Sep 11 2019

# E(x,y) is f(n,x,y,1/2), T(x,y) is f(n,x,y,0), and S(x,y) is f(n,x,y,1).
f:=proc(n,x,y,r) option remember;
if n=0 then x
elif n=1 then y
else floor(f(n-1,x,y,r)^2/f(n-2,x,y,r) + r); fi; end;
[seq(f(n,2,6,1/2),n=0..30)];
# N. J. A. Sloane, Jul 30 2016

Table[EulerPhi[3^n], {n, 0, 100}] (* Artur Jasinski, Nov 19 2008 *)
Table[MatrixPower[{{1,2},{1,2}},n][[1]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
NestList[3#&,2,50] (* Harvey P. Dale, Nov 28 2022 *)

a(n)=3^n<<1 \\ corrected by Michel Marcus, Aug 03 2015

def A008776(n): return 3**n<<1 # Chai Wah Wu, Apr 02 2025

[2*3^n for n in (0..30)] # G. C. Greubel, Sep 11 2019

a(n) = 2*3^n.
a(n) = 3*a(n-1).
G.f.: 2/(1-3*x). - Philippe Deléham, Oct 08 2007
a(n-1) = phi(3^n). - Artur Jasinski, Nov 19 2008
E.g.f.: 2*exp(3*x). - Mohammad K. Azarian, Jan 15 2009
From Paul Curtz, Jan 20 2009: (Start)
a(n) = A048473(n) + 1.
a(n) = A052919(n+1)-1.
a(n) = A115099(n) - 2.
a(n) = A100774(n) + 2. (End)
If p[i]=2, (i >= 1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1)=det A. - Milan Janjic, Apr 29 2010
G.f.: ((1/2)/G(0)-1)/x^2 where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: -G(0)/x where G(k) = 1 - 1/(1-2*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2013
G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k-2)/(1 - x*(2*k+5)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: W(0), where W(k) = 1 + 1/(1 - x*(2*k+3)/(x*(2*k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

Jasinski formula corrected by Charles R Greathouse IV, Feb 18 2011

A052548 a(n) = 2^n + 2.

Original entry on oeis.org

3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The most "compact" sequence that satisfies Bertrand's Postulate. Begin with a(1) = 3 = n, then 2n - 2 = 4 = n_1, 2n_1 - 2 = 6 = n_2, 2n_2 - 2 = 10, etc. = a(n), hence there is guaranteed to be at least one prime between successive members of the sequence. - Andrew S. Plewe, Dec 11 2007

Number of 2-sided prudent polygons of area n, for n>0, see Beaton, p. 5. - Jonathan Vos Post, Nov 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..240
Nicholas R. Beaton, Philippe Flajolet, and Anthony J. Guttmann, The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics, arXiv:1011.6195 [math.CO], Nov 29, 2010.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 485
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=2. [Annotated and scanned copy]
Eric Weisstein's World of Mathematics, Bertrand's Postulate
Index entries for sequences generated by sieves
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from initial term, same as A056469.
Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774, A079004, A058481, A100585, A100586, A058896, A000918, A173786.
Cf. also A000079, A000051, A100314.

a052548 = (+ 2) . a000079
a052548_list = iterate ((subtract 2) . (* 2)) 3
-- Reinhard Zumkeller, Sep 05 2015

[2^n + 2: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011

spec := [S,{S=Union(Sequence(Union(Z,Z)),Sequence(Z),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

2^Range[0,40]+2 (* Harvey P. Dale, Jun 26 2012 *)

a(n)=1<Charles R Greathouse IV, Nov 20 2011

G.f.: (3-5*x)/((1-2*x)*(1-x)) = (3-5*x)/(1 - 3*x + 2*x^2) = 2/(1-x) + 1/(1-2*x).
a(0)=3, a(1)=4, a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A058896(n)/A000918(n), for n>0. - Reinhard Zumkeller, Feb 14 2009
a(n) = A173786(n,1), for n>0. - Reinhard Zumkeller, Feb 28 2010
a(n)*A000918(n) = A028399(2*n), for n>0. - Reinhard Zumkeller, Feb 28 2010
a(0)=3, a(n) = 2*a(n-1) - 2. - Vincenzo Librandi, Aug 06 2010
E.g.f.: (2 + exp(x))*exp(x). - Ilya Gutkovskiy, Aug 16 2016

More terms from James Sellers, Jun 06 2000

A058481 a(n) = 3^n - 2.

Original entry on oeis.org

1, 7, 25, 79, 241, 727, 2185, 6559, 19681, 59047, 177145, 531439, 1594321, 4782967, 14348905, 43046719, 129140161, 387420487, 1162261465, 3486784399, 10460353201, 31381059607, 94143178825, 282429536479, 847288609441

Offset: 1

Vladeta Jovovic, Nov 26 2000

a(n) = number of 2 X n binary matrices with no zero rows or columns.
a(n)^2 + 2*a(n+1) + 1 is a square number, i.e., a(n)^2 + 2*a(n+1) + 1 = (a(n)+3)^2: for n=2, a(2)^2 + 2*a(3) + 1 = 7^2 + 2*25 + 1 = 100 = (7+3)^2; for n=3, a(3)^2 + 2*a(4) + 1 = 25^2 + 2*79 + 1 = 784 = (25+3)^2. - Bruno Berselli, Apr 23 2010
Sum of n-th row of triangle of powers of 3: 1; 3 1 3; 9 3 1 3 9; 27 9 3 1 3 9 27; ... . - Philippe Deléham, Feb 24 2014
a(n) = least k such that k*3^n + 1 is a square. Thus, the square is given by (3^n-1)^2. - Derek Orr, Mar 23 2014
Binomial transform of A058481: (1, 6, 12, 24, 48, 96, ...) and second binomial transform of (1, 5, 1, 5, 1, 5, ...). - Gary W. Adamson, Aug 24 2016
Number of ordered pairs of nonempty sets whose union is [n]. a(2) = 7: ({1,2},{1,2}), ({1,2},{1}), ({1,2},{2}), ({1},{1,2}), ({1},{2}), ({2},{1,2}), ({2},{1}). If "nonempty" is omitted we get A000244. - Manfred Boergens, Mar 29 2023

			G.f. = x + 7*x^2 + 25*x^3 + 79*x^4 + 241*x^5 + 727*x^6 + 2185*x^7 + 6559*x^8 + ...
a(1) = 1;
a(2) = 3 + 1 + 3 = 7;
a(3) = 9 + 3 + 1 + 3 + 9 = 25;
a(4) = 27 + 9 + 3 + 1 + 3 + 9 + 27 = 79; etc. - _Philippe Deléham_, Feb 24 2014

G. C. Greubel, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Union of Triple-Sierpinski Triangles
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A055602, A024206 (unlabeled case), A055609, A058482, A000244.

Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774, A079004.

A058481:=n->3^n-2; seq(A058481(n), n=1..30); # Wesley Ivan Hurt, Mar 24 2014

a=1;lst={a};Do[a=a*3+4;AppendTo[lst,a],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
3^Range[30]-2  (* Harvey P. Dale, Mar 28 2011 *)
LinearRecurrence[{4, -3}, {1, 7}, 25] (* G. C. Greubel, Aug 25 2016 *)

a(n)=3^n-2 \\ Charles R Greathouse IV, Feb 06 2017

{a(n) = if( n<1, 0, 3^n - 2)}; /* Michael Somos, Feb 17 2017 */

Number of m X n binary matrices with no zero rows or columns is Sum_{j=0..m} (-1)^j*C(m, j)*(2^(m-j)-1)^n.
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-3*x)-2/(1-x)+1.
E.g.f.: e^(3*x)-2*(e^x)+1. (End)
a(n) = 3*a(n-1) + 4 (with a(1)=1). - Vincenzo Librandi, Aug 07 2010
a(n) = 4*a(n-1) - 3*a(n-2). - G. C. Greubel, Aug 25 2016

More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

Original entry on oeis.org

3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151

Offset: 0

Philippe Deléham, Feb 16 2014

a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020
First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Radius.
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* Start from Eric W. Weisstein, Mar 13 2018 *)
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = A000244(n+1) + A003462(n).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
a(n) = A199109(n) - 1.
a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018
From Klaus Purath, Apr 13 2020: (Start)
a(n) = A057198(n+1) + A024023(n).
a(n) = A029858(n+2) - A024023(n).
a(n) = A052919(n+1) + A029858(n+1).
a(n) = (A000244(n+1) + A171498(n))/2.
a(n) = 7*A003462(n) + 3.
a(n) = A116952(n) + 2. (End)
a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020
E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A164123 Partial sums of A162436.

Original entry on oeis.org

1, 4, 7, 16, 25, 52, 79, 160, 241, 484, 727, 1456, 2185, 4372, 6559, 13120, 19681, 39364, 59047, 118096, 177145, 354292, 531439, 1062880, 1594321, 3188644, 4782967, 9565936, 14348905, 28697812, 43046719, 86093440, 129140161, 258280324, 387420487, 774840976, 1162261465

Offset: 1

Klaus Brockhaus, Aug 10 2009

Interleaving of A058481 and A100774 without initial term 0.
Apparently a(n) = A062318(n+2) - 1.
The terms beginning with a(2) are the row numbers in Pascal's Triangle where every 3rd element in those rows is divisible by 3 and none of the other elements in those rows are divisible by 3. - Thomas M. Green, Apr 03 2013

			For n = 3, a(3) = 7. The binomial coefficients of the 7th row of Pascal's Triangle are 1 7 21 35 35 21 7 1 and every 3rd element is a multiple of 3. - _Thomas M. Green_, Apr 03 2013

Thomas M. Green, Prime Patterns in Pascal's Triangle, paper in review process, 2013.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A162436, A058481 (3^n-2), A100774 (2*(3^n - 1)), A062318, A038754, A038754.

T:=[ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..33] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

Accumulate[Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]]] (* Harvey P. Dale, Feb 17 2012 *)

a(n) = (2+n%2)*3^(n\2)-2 \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A038754(n+1) - 2.
a(n) = 3*a(n-2) + 4 for n > 2; a(1) = 1, a(2) = 4.
a(n) = (5 - (-1)^n)*3^(1/4*(2*n - 1 + (-1)^n))/2 - 2.
G.f.: x*(1 + 3*x)/((1 - x)*(1 - 3*x^2)).
E.g.f.: 2*(cosh(sqrt(3)*x) - cosh(x)) + sqrt(3)*sinh(sqrt(3)*x) - 2*sinh(x). - Stefano Spezia, Dec 31 2022

Incorrect formula removed by Stefano Spezia, Dec 31 2022

A079004 Least x>=3 such that F(x)==1 (mod 3^n) where F(x) denotes the x-th Fibonacci number (A000045).

Original entry on oeis.org

7, 10, 10, 34, 106, 322, 970, 2914, 8746, 26242, 78730, 236194, 708586, 2125762, 6377290, 19131874, 57395626, 172186882, 516560650, 1549681954, 4649045866, 13947137602, 41841412810, 125524238434, 376572715306, 1129718145922

Offset: 1

Benoit Cloitre, Feb 01 2003

R. L. Graham, D. E. Knuth and O. Patashnick, "Concrete Mathematics", second edition, Addison Wesley, ex. 6.59.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000045, A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774.

7, 10, seq(4*3^(n-2)-2,n=3..50); # Robert Israel, Jan 15 2015

a=2;lst={7,10};Do[a=a*3+4;AppendTo[lst,a],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
LinearRecurrence[{4,-3},{7,10,10,34},40] (* Harvey P. Dale, Aug 16 2024 *)

a(n)=if(n<0,0,x=3; while((fibonacci(x)-1)%(3^n)>0,x++); x)

a(1)=7, a(2)=10, a(3)=10; for n>3, a(n) = 3*a(n-1) + 4.
a(n) = 4*3^(n-2)-2 for n >= 3.
G.f.: 8*x^2+(23/3)*x+14/9+2/(x-1)-4/(9*(3*x-1)). - Robert Israel, Jan 15 2015

Formula corrected by Robert Israel, Jan 15 2015

A132753 a(n) = 2^(n+1) - n + 1.

Original entry on oeis.org

3, 4, 7, 14, 29, 60, 123, 250, 505, 1016, 2039, 4086, 8181, 16372, 32755, 65522, 131057, 262128, 524271, 1048558, 2097133, 4194284, 8388587, 16777194, 33554409, 67108840, 134217703, 268435430, 536870885, 1073741796, 2147483619

Offset: 0

Gary W. Adamson, Aug 28 2007

Apart from a(0): Row sums of triangle A132752 (old name).

Apart from a(0): Binomial transform of [1, 3, 0, 4, 0, 4, 0, 4, ...].

			a(3) = 14 = sum of row 3 terms of triangle A132752: (3 + 5 + 5 + 1).
a(3) = 14 = (1, 3, 3, 1) dot (1, 3, 0, 4) = (1 + 9 + 0 + 4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A132752.

Cf. A003462, A007051, A024023, A029858, A034472, A052548, A058481, A067771, A079004, A100774, A115099, A134931. - Vladimir Joseph Stephan Orlovsky, Dec 25 2008

[2^(n+1) -n+1: n in [0..40]]; // G. C. Greubel, Feb 16 2021

A132753:= n-> 2^(n+1) -n+1; seq(A132753(n), n=0..40) # G. C. Greubel, Feb 16 2021

Table[2^(n+1) -n+1, {n, 0, 30}] (* Bruno Berselli, Aug 31 2013 *)

PARI
```
a(n)=2^(n+1)-n+1
```

Vec( (3-8*x+6*x^2)/((1-x)^2*(1-2*x)) + O(x^40)) \\ Colin Barker, Mar 14 2014

[2^(n+1) -n+1 for n in (0..40)] # G. C. Greubel, Feb 16 2021

From Colin Barker, Mar 14 2014: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (3 - 8*x + 6*x^2)/((1-x)^2 * (1-2*x)). (End)
E.g.f.: (1-x)*exp(x) + 2*exp(2*x). - G. C. Greubel, Feb 16 2021

More terms Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Changed first member, and better name from Ralf Stephan, Aug 31 2013

A146882 a(n) = 5*(4^(n+1) - 1)/3.

Original entry on oeis.org

5, 25, 105, 425, 1705, 6825, 27305, 109225, 436905, 1747625, 6990505, 27962025, 111848105, 447392425, 1789569705, 7158278825, 28633115305, 114532461225, 458129844905, 1832519379625, 7330077518505, 29320310074025

Offset: 0

Roger L. Bagula, Nov 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000051, A000225, A002450, A100774.

[5*(4^(n+1)-1)/3: n in [0..40]]; // Vincenzo Librandi, Jul 15 2011

Mathematica
```
Table[5*(4^(n+1) -1)/3, {n,0,40}]
```

a(n)=20<<(2*n)\/3-2 \\ Charles R Greathouse IV, Jul 15 2011

[(5/3)*(2^(2*n+2)-1) for n in range(41)] # G. C. Greubel, Oct 12 2022

a(n) = 5*A002450(n+1).
G.f.: 5 / ( (1-x)*(1-4*x) ). - R. J. Mathar, Jun 26 2011
From G. C. Greubel, Oct 12 2022: (Start)
a(n) = (5/3)*A000051(n+1)*A000225(n+1).
E.g.f.: (5/3)*(4*exp(4*x) - exp(x)). (End)

A146883 a(n) = 6 * Sum_{m=0..n} 5^m.

Original entry on oeis.org

6, 36, 186, 936, 4686, 23436, 117186, 585936, 2929686, 14648436, 73242186, 366210936, 1831054686, 9155273436, 45776367186, 228881835936, 1144409179686, 5722045898436, 28610229492186, 143051147460936, 715255737304686

Offset: 0

Roger L. Bagula, Nov 02 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A003463, A100774.

[n le 2 select 6^n else 6*Self(n-1) -5*Self(n-2): n in [1..31]]; // G. C. Greubel, Oct 12 2022

a[n_]:= 6*Sum[5^m, {m,0,n}]; Table[a[n], {n,0,30}]
(* Second programs: *)
Array[3(5^(#+1) -1)/2 &, 21, 0] (* or *)
CoefficientList[Series[6/((1-x)(1-5x)), {x, 0, 20}], x] (* Michael De Vlieger, Nov 02 2017 *)
LinearRecurrence[{6,-5},{6,36},30] (* Harvey P. Dale, Jan 07 2024 *)

a(n) = sum(m=0, n, 6*5^m); \\ Michel Marcus, Nov 03 2017

[3*(5^(n+1) -1)/2 for n in range(41)] # G. C. Greubel, Oct 12 2022

a(n) = Sum_{m=0..n} 6*5^m.
a(n) = 3*(5^(n+1) - 1)/2. - Emeric Deutsch, Nov 02 2017
G.f.: 6/((1-z)*(1-5*z)). - Robert Israel, Nov 02 2017
a(n) = 6 * A003463(n+1). - Alois P. Heinz, Nov 03 2017
E.g.f.: (3/2)*(5*exp(5*x) - exp(x)). - G. C. Greubel, Oct 12 2022

A146884 a(n) = 7*Sum_{k=0..n} 6^k.

Original entry on oeis.org

7, 49, 301, 1813, 10885, 65317, 391909, 2351461, 14108773, 84652645, 507915877, 3047495269, 18284971621, 109709829733, 658258978405, 3949553870437, 23697323222629, 142183939335781, 853103636014693, 5118621816088165

Offset: 0

Roger L. Bagula, Nov 02 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A003464, A005843, A068156, A100774, A132583, A146882, A146883, A146885.

[n le 2 select 7^n else 7*Self(n-1) -6*Self(n-2): n in [1..31]]; // G. C. Greubel, Oct 12 2022

a[n_]:= Sum[7*6^m, {m,0,n}]; Table[a[n], {n,0,30}]
Accumulate[7*6^Range[0,20]] (* Harvey P. Dale, Dec 18 2021 *)

[(7/5)*(6^(n+1)-1) for n in range(41)] # G. C. Greubel, Oct 12 2022

From G. C. Greubel, Oct 12 2022: (Start)
a(n) = (7/5)*(6^(n+1) - 1).
a(n) = 7*A003464(n+1).
a(n) = 7*a(n-1) - 6*a(n-2).
G.f.: 7/((1-x)*(1-6*x)).
E.g.f.: (7/5)*(6*exp(6*x) - exp(x)). (End)

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