A048471 -id:A048471 - cp's OEIS frontend

A036543 a(n) = T(3,n), array T given by A048471.

1, 9, 33, 105, 321, 969, 2913, 8745, 26241, 78729, 236193, 708585, 2125761, 6377289, 19131873, 57395625, 172186881, 516560649, 1549681953, 4649045865, 13947137601, 41841412809, 125524238433, 376572715305, 1129718145921

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+2) for n=1, 2, 3, ...

Cf. A146541 (inv. bin. transf.)

[4*3^n-3: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

4*3^Range[0,25]-3 (* or *) LinearRecurrence[{4,-3},{1,9},25] (* Harvey P. Dale, Aug 16 2011 *)

vector(30, n, n--; 4*3^n-3) \\ G. C. Greubel, Nov 23 2018

[4*3^n-3 for n in range(30)] # G. C. Greubel, Nov 23 2018

Binomial transform of A084242. Second binomial transform of periodic sequence A010688. - Paul Barry, May 23 2003
From Paul Barry, May 23 2003: (Start)
a(n) = 4*3^n - 3;
G.f.: (1+5*x)/((1-x)*(1-3*x));
E.g.f.: 4*exp(3*x) - 3*exp(x). (End)
a(n) = 4*a(n-1) - 3*a(n-2); a(0)=1, a(1)=9. - Harvey P. Dale, Aug 16 2011
a(n) = 3*a(n-1) + 6. - Vincenzo Librandi, Nov 11 2011
a(n) = A171498(n) - 2. - Philippe Deléham, Apr 13 2013

A036545 a(n) = T(4,n), array T given by A048471.

Original entry on oeis.org

1, 17, 65, 209, 641, 1937, 5825, 17489, 52481, 157457, 472385, 1417169, 4251521, 12754577, 38263745, 114791249, 344373761, 1033121297, 3099363905, 9298091729, 27894275201, 83682825617, 251048476865, 753145430609, 2259436291841, 6778308875537, 20334926626625

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+3) for n=1, 2, 3, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A048471.

[8*3^n-7: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

LinearRecurrence[{4,-3},{1,17},30] (* Harvey P. Dale, Mar 05 2022 *)

a(n) = 8*3^n - 7. - Ralf Stephan, Feb 17 2004
From Vincenzo Librandi, Nov 11 2011: (Start)
a(n) = 3*a(n-1) + 14.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (1+13*x)/((1-x)*(1-3*x)). (End)
E.g.f.: exp(x)*(8*exp(2*x) - 7). - Elmo R. Oliveira, Aug 29 2024

A036546 a(n) = T(5,n), array T given by A048471.

Original entry on oeis.org

1, 33, 129, 417, 1281, 3873, 11649, 34977, 104961, 314913, 944769, 2834337, 8503041, 25509153, 76527489, 229582497, 688747521, 2066242593, 6198727809, 18596183457, 55788550401, 167365651233, 502096953729, 1506290861217, 4518872583681, 13556617751073, 40669853253249

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+4) for n=1, 2, 3, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A048471.

[16*3^n-15: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

LinearRecurrence[{4,-3},{1,33},30] (* Harvey P. Dale, Mar 16 2012 *)

a(n) = 16*3^n - 15. - Ralf Stephan, Feb 17 2004
From Vincenzo Librandi, Nov 11 2011: (Start)
a(n) = 3*a(n-1) + 30.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (1+29*x)/((1-x)*(1-3*x)). (End)
E.g.f.: exp(x)*(16*exp(2*x) - 15). - Elmo R. Oliveira, Aug 29 2024

A036547 a(n) = T(6,n), array T given by A048471.

Original entry on oeis.org

1, 65, 257, 833, 2561, 7745, 23297, 69953, 209921, 629825, 1889537, 5668673, 17006081, 51018305, 153054977, 459164993, 1377495041, 4132485185, 12397455617, 37192366913, 111577100801, 334731302465, 1004193907457, 3012581722433, 9037745167361, 27113235502145

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+5) for n=1, 2, 3, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A048471.

[32*3^n-31: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

a(n) = 32*3^n - 31. - Ralf Stephan, Feb 17 2004
From Vincenzo Librandi, Nov 11 2011: (Start)
a(n) = 3*a(n-1) + 62.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (1+61*x)/((1-x)*(1-3*x)). (End)
E.g.f.: exp(x)*(32*exp(2*x) - 31). - Elmo R. Oliveira, Aug 29 2024

A036548 a(n) = T(7,n), array T given by A048471.

Original entry on oeis.org

1, 129, 513, 1665, 5121, 15489, 46593, 139905, 419841, 1259649, 3779073, 11337345, 34012161, 102036609, 306109953, 918329985, 2754990081, 8264970369, 24794911233, 74384733825, 223154201601, 669462604929, 2008387814913, 6025163444865, 18075490334721, 54226471004289

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+6) for n=1, 2, 3, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A048471.

[64*3^n-63: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

a(n) = 64*3^n - 63. - Ralf Stephan, Feb 17 2004
From Vincenzo Librandi, Nov 11 2011: (Start)
a(n) = 3*a(n-1) + 126.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (1+125*x)/((1-x)*(1-3*x)). (End)
E.g.f.: exp(x)*(64*exp(2*x) - 63). - Elmo R. Oliveira, Aug 29 2024

A036549 a(n) = T(8,n), array T given by A048471.

Original entry on oeis.org

1, 257, 1025, 3329, 10241, 30977, 93185, 279809, 839681, 2519297, 7558145, 22674689, 68024321, 204073217, 612219905, 1836659969, 5509980161, 16529940737, 49589822465, 148769467649, 446308403201, 1338925209857, 4016775629825, 12050326889729, 36150980669441

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+7) for n=1, 2, 3, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A048471.

[128*3^n-127: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

From Vincenzo Librandi, Nov 11 2011: (Start)
a(n) = 128*3^n - 127.
a(n) = 3*a(n-1) + 254.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (1+253*x)/((1-x)*(1-3*x)). (End)
E.g.f.: exp(x)*(128*exp(2*x) - 127). - Elmo R. Oliveira, Aug 29 2024

A036550 a(n) = T(0,n) + T(1,n-1) + ... + T(n,0), array T given by A048471.

Original entry on oeis.org

1, 3, 9, 29, 95, 307, 973, 3033, 9339, 28511, 86537, 261637, 788983, 2375115, 7141701, 21457841, 64439027, 193448119, 580606465, 1742343645, 5228079471, 15686335523, 47063200829, 141197991049, 423610750315, 1270865805327

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Partial sums of A083323.

[(1/2) * (3^(n+1) - 2^(n+2) + 2*n + 3): n in [0..30]]; // Vincenzo Librandi, Nov 12 2011

a(n) = (1/2) * (3^(n+1) - 2^(n+2) + 2n + 3). - Ralf Stephan, Feb 17 2004

G.f: (1/2)*(3/(1-3*x) - 4/(1-2*x) + 2*x/(1-x)^2 + 3/(1-x)). - Vincenzo Librandi, Nov 12 2011

Corrected by T. D. Noe, Nov 07 2006

A048473 a(0)=1, a(n) = 3a(n-1) + 2; a(n) = 23^n - 1.

Original entry on oeis.org

1, 5, 17, 53, 161, 485, 1457, 4373, 13121, 39365, 118097, 354293, 1062881, 3188645, 9565937, 28697813, 86093441, 258280325, 774840977, 2324522933, 6973568801, 20920706405, 62762119217, 188286357653, 564859072961, 1694577218885, 5083731656657, 15251194969973, 45753584909921

Offset: 0

Clark Kimberling

The number of triangles (of all sizes, including holes) in Sierpiński's triangle after n inscriptions. - Lee Reeves, May 10 2004
The sequence is not only related to Sierpiński's triangle, but also to "Floret's cube" and the quaternion factor space Q X Q / {(1,1), (-1,-1)}. It can be written as a_n = ves((A+1)x)^n) as described at the Math Forum Discussions link. - Creighton Dement, Jul 28 2004
Relation to C(n) = Collatz function iteration using only odd steps: If we look for record subsequences where C(n) > n, this subsequence starts at 2^n - 1 and stops at the local maximum of 2*3^n - 1. Examples: [3,5], [7,11,17], [15,23,35,53], ..., [127,191,287,431,647,971,1457]. - Lambert Klasen, Mar 11 2005
Group the natural numbers so that the (2n-1)-th group sum is a multiple of the (2n)-th group containing one term. (1,2),(3),(4,5,6,7,8,9,10,11),(12),(13,14,15,16,17,18,19,...,38),(39),(40,41,...,118,119),(120), (121,122,123,...) ... a(n) = {the sum of the terms of (2n-1)-th group}/{the term of (2n)th group}. The first term of the odd numbered group is given by A003462. The only term of even numbered group is given by A029858. - Amarnath Murthy, Aug 01 2005
a(n)+1 = A008776(n); it appears that this gives the number of terms in the (n+1)-th "gap" of numbers missing in A171884. - M. F. Hasler, May 09 2013
Sum of n-th row of triangle of powers of 3: 1; 1 3 1; 1 3 9 3 1; 1 3 9 27 9 3 1; ... - Philippe Deléham, Feb 23 2014
For n >= 3, also the number of dominating sets in the n-helm graph. - Eric W. Weisstein, May 28 2017
The number of elements of length <= n in the free group on two generators. - Anton Mellit, Aug 10 2017
In general, a first order inhomogeneous recurrence of the form s(0) = a, s(n) = m*s(n-1) + k, n>0, will have a closed form of a*m^n + ((m^n-1)/(m-1))*k. - Gary Detlefs, Jun 07 2024

			a(0) = 1;
a(1) = 1 + 3 + 1 = 5;
a(2) = 1 + 3 + 9 + 3 + 1 = 17;
a(3) = 1 + 3 + 9 + 27 + 9 + 3 + 1 = 53; etc. - _Philippe Deléham_, Feb 23 2014

Theoni Pappas, Math Stuff, Wide World Publ/Tetra, San Carlos CA, page 15, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Creighton Dement, A paper on floretions and quaternions, some questions, The Math Forum.
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, Helm Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

a(n)=T(2,n), array T given by A048471.
Cf. A003462, A029858. A column of A119725.
Cf. A008776, A010684, A046055, A112468, A112739, A134347, A171884.

[2*3^n - 1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

g:= ((1+x)/(1-3*x)/(1-x)): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=0..30); # Zerinvary Lajos, Jan 11 2009; typo fixed by Marko Mihaily, Mar 07 2009

NestList[3 # + 2 &, 1, 30] (* Harvey P. Dale, Mar 06 2012 *)
LinearRecurrence[{4, -3}, {1, 5}, 30] (* Harvey P. Dale, Mar 06 2012 *)
Table[2 3^n - 1, {n, 20}] (* Eric W. Weisstein, May 28 2017 *)
2 3^Range[20] - 1 (* Eric W. Weisstein, May 28 2017 *)

first(m)=vector(m,n,n--;2*3^n - 1) \\ Anders Hellström, Dec 11 2015

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+1) for n=1, 2, 3, ...
a(0)=1, a(n) = a(n-1) + 3^n + 3^(n-1). - Lee Reeves, May 10 2004
a(n) = (3^n + 3^(n+1) - 2)/2. - Creighton Dement, Jul 31 2004
(1, 5, 17, 53, 161, ...) = Ternary (1, 12, 122, 1222, 12222, ...). - Gary W. Adamson, May 02 2005
Row sums of triangle A134347. Also, binomial transform of A046055: (1, 4, 8, 16, 32, 64, ...); and double binomial transform of A010684: (1, 3, 1, 3, 1, 3, ...). - Gary W. Adamson, Oct 21 2007
G.f.: (1+x)/((1-3*x)*(1-x)). - Zerinvary Lajos, Jan 11 2009; corrected by R. J. Mathar, Jan 21 2009
a(0)=1, a(1)=5, a(n) = 4*a(n-1) - 3*a(n-2). - Harvey P. Dale, Mar 06 2012
a(n) = Sum_{k=0..n} A112468(n,k)*4^k. - Philippe Deléham, Feb 23 2014
E.g.f.: exp(x)*(2*exp(2*x) - 1). - Elmo R. Oliveira, Mar 08 2025

Better description from Amarnath Murthy, May 27 2001

A036551 a(n) = 2^(n-1)*(3^n-1) + 1.

Original entry on oeis.org

1, 3, 17, 105, 641, 3873, 23297, 139905, 839681, 5038593, 30232577, 181397505, 1088389121, 6530342913, 39182073857, 235092475905, 1410554920961, 8463329656833, 50779978203137, 304679869743105, 1828079219507201, 10968475319140353, 65810851919036417

Offset: 0

Clark Kimberling

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

Cf. A048471.

Vec((1-6*x+10*x^2)/((1-x)*(1-2*x)*(1-6*x)) + O(x^30)) \\ Colin Barker, Aug 24 2016

a(n) = T(n, n), array T given by A048471.
From Colin Barker, Aug 24 2016: (Start)
a(n) = 9*a(n-1)-20*a(n-2)+12*a(n-3) for n>2.
G.f.: (1-6*x+10*x^2) / ((1-x)*(1-2*x)*(1-6*x)).
(End)

Simpler definition from Ralf Stephan, Feb 17 2004

Corrected by T. D. Noe, Nov 07 2006

cp's OEIS Frontend

A036543 a(n) = T(3,n), array T given by A048471.

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A036545 a(n) = T(4,n), array T given by A048471.

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A036546 a(n) = T(5,n), array T given by A048471.

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A036547 a(n) = T(6,n), array T given by A048471.

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A036548 a(n) = T(7,n), array T given by A048471.

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Magma

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A036549 a(n) = T(8,n), array T given by A048471.

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A036550 a(n) = T(0,n) + T(1,n-1) + ... + T(n,0), array T given by A048471.

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A048473 a(0)=1, a(n) = 3*a(n-1) + 2; a(n) = 2*3^n - 1.

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A048473 a(0)=1, a(n) = 3a(n-1) + 2; a(n) = 23^n - 1.