A178248 -id:A178248 - cp's OEIS frontend

A000051 a(n) = 2^n + 1.

2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593

Offset: 0

N. J. A. Sloane

Same as Pisot sequence L(2,3).
Length of the continued fraction for Sum_{k=0..n} 1/3^(2^k). - Benoit Cloitre, Nov 12 2003
See also A004119 for a(n) = 2a(n-1)-1 with first term = 1. - Philippe Deléham, Feb 20 2004
From the second term on (n>=1), in base 2, these numbers present the pattern 1000...0001 (with n-1 zeros), which is the "opposite" of the binary 2^n-2: (0)111...1110 (cf. A000918). - Alexandre Wajnberg, May 31 2005
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)* charpoly(A,3). - Milan Janjic, Jan 27 2010
First differences of A006127. - Reinhard Zumkeller, Apr 14 2011
The odd prime numbers in this sequence form A019434, the Fermat primes. - David W. Wilson, Nov 16 2011
Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012
Is the mentioned Pisano period lengths (see above) the same as A007733? - Omar E. Pol, Aug 10 2012
Only positive integers that are not 1 mod (2k+1) for any k>1. - Jon Perry, Oct 16 2012
For n >= 1, a(n) is the total length of the segments of the Hilbert curve after n iterations. - Kival Ngaokrajang, Mar 30 2014
Frénicle de Bessy (1657) proved that a(3) = 9 is the only square in this sequence. - Charles R Greathouse IV, May 13 2014
a(n) is the number of distinct possible sums made with at most two elements in {1,...,a(n-1)} for n > 0. - Derek Orr, Dec 13 2014
For n > 0, given any set of a(n) lattice points in R^n, there exist 2 distinct members in this set whose midpoint is also a lattice point. - Melvin Peralta, Jan 28 2017
Also the number of independent vertex sets, irredundant sets, and vertex covers in the (n+1)-star graph. - Eric W. Weisstein, Aug 04 and Sep 21 2017
Also the number of maximum matchings in the 2(n-1)-crossed prism graph. - Eric W. Weisstein, Dec 31 2017
Conjecture: For any integer n >= 0, a(n) is the permanent of the (n+1) X (n+1) matrix with M(j, k) = -floor((j - k - 1)/(n + 1)). This conjecture is inspired by the conjecture of Zhi-Wei Sun in A036968. - Peter Luschny, Sep 07 2021

Paul Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 46, 60, 244.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 141.

Ivan Panchenko, Table of n, a(n) for n = 0..100
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Bartomeu Fiol, Jairo Martínez-Montoya, and Alan Rios Fukelman, The planar limit of N=2 superconformal field theories, arXiv:2003.02879 [hep-th], 2020.
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 114
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 362
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
Kival Ngaokrajang, Illustration of Hilbert curve for n = 1..5
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 1.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Crossed Prism Graph.
Eric Weisstein's World of Mathematics, Cunningham Number.
Eric Weisstein's World of Mathematics, Fermat-Lucas Number.
Eric Weisstein's World of Mathematics, Hilbert curve.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Irredundant Set.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence.
Eric Weisstein's World of Mathematics, Star Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from the initial 1, identical to A094373.
See A008776 for definitions of Pisot sequences.
Column 2 of array A103438.
Cf. A007583 (a((n-1)/2)/3 for odd n).
Cf. A000079, A000225, A000918, A004119, A005126, A006217, A006521, A007583.
Cf. A007689, A007733, A019434, A024036, A027641, A027642, A034472, A034474.
Cf. A034491, A034524, A036968, A052539, A052944, A062394, A062395, A062396.
Cf. A062397, A063376, A063481, A074600 - A074624, A127904, A173786, A176691.
Cf. A178248, A194455, A228081, A323482.

a000051 = (+ 1) . a000079
a000051_list = iterate ((subtract 1) . (* 2)) 2
-- Reinhard Zumkeller, May 03 2012

[2^n+1: n in [0..40]]; // G. C. Greubel, Jan 18 2025

A000051:=-(-2+3*z)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation
a := n -> add(binomial(n,k)*bernoulli(n-k,1)*2^(k+1)/(k+1),k=0..n); # Peter Luschny, Apr 20 2009

Table[2^n + 1, {n,0,40}]
2^Range[0,40] + 1 (* Eric W. Weisstein, Jul 17 2017 *)
LinearRecurrence[{3, -2}, {2, 3}, 40] (* Eric W. Weisstein, Sep 21 2017 *)

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

first(n) = Vec((2 - 3*x)/((1 - x)*(1 - 2*x)) + O(x^n)) \\ Iain Fox, Dec 31 2017

def A000051(n): return (1<Chai Wah Wu, Dec 21 2022

a(n) = 2*a(n-1) - 1 = 3*a(n-1) - 2*a(n-2).
G.f.: (2-3*x)/((1-x)*(1-2*x)).
First differences of A052944. - Emeric Deutsch, Mar 04 2004
a(0) = 1, then a(n) = (Sum_{i=0..n-1} a(i)) - (n-2). - Gerald McGarvey, Jul 10 2004
Inverse binomial transform of A007689. Also, V sequence in Lucas sequence L(3, 2). - Ross La Haye, Feb 07 2005
a(n) = A127904(n+1) for n>0. - Reinhard Zumkeller, Feb 05 2007
Equals binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008
a(n) = A000079(n)+1. - Omar E. Pol, May 18 2008
E.g.f.: exp(x) + exp(2*x). - Mohammad K. Azarian, Jan 02 2009
a(n) = A024036(n)/A000225(n). - Reinhard Zumkeller, Feb 14 2009
From Peter Luschny, Apr 20 2009: (Start)
A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).
a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) (End)
a(n) is the a(n-1)-th odd number for n >= 1. - Jaroslav Krizek, Apr 25 2009
From Reinhard Zumkeller, Feb 28 2010: (Start)
a(n)*A000225(n) = A000225(2*n).
a(n) = A173786(n,0). (End)
If p[i]=Fibonacci(i-4) and if A is the 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, May 08 2010
a(n+2) = a(n) + a(n+1) + A000225(n). - Ivan N. Ianakiev, Jun 24 2012
a(A006521(n)) mod A006521(n) = 0. - Reinhard Zumkeller, Jul 17 2014
a(n) = 3*A007583((n-1)/2) for n odd. - Eric W. Weisstein, Jul 17 2017
Sum_{n>=0} 1/a(n) = A323482. - Amiram Eldar, Nov 11 2020

A034472 a(n) = 3^n + 1.

Original entry on oeis.org

2, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050, 177148, 531442, 1594324, 4782970, 14348908, 43046722, 129140164, 387420490, 1162261468, 3486784402, 10460353204, 31381059610, 94143178828, 282429536482, 847288609444, 2541865828330, 7625597484988

Offset: 0

N. J. A. Sloane

Companion numbers to A003462.
a(n) = A024101(n)/A024023(n). - Reinhard Zumkeller, Feb 14 2009
Mahler exhibits this sequence with n>=2 as a proof that there exists an infinite number of x coprime to 3, such that x belongs to A005836 and x^2 belong to A125293. - Michel Marcus, Nov 12 2012
a(n-1) is the number of n-digit base 3 numbers that have an even number of digits 0. - Yifan Xie, Jul 13 2024

			a(3)=28 because 4*a(2)-3*a(1)=4*10-3*4=28 (28 is also 3^3 + 1).
G.f. = 2 + 4*x + 10*x^2 + 28*x^3 + 82*x^4 + 244*x^5 + 730*x^5 + ...

Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, pages 148 and 220, Problem 191.
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, pp. 35-36, 53.

T. D. Noe, Table of n, a(n) for n=0..200
T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066, eq 5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 454
Kurt Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.
Burkard Polster, Special numbers in 3-coloring of Pascal's triangle, Mathologer video (2019).
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
D. Suprijanto and I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2211 - 2217.
Eric Weisstein's World of Mathematics, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A000204, A007051, A000051, A052539, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081, A279396.

[3^n+1: n in [0..30]]; // Vincenzo Librandi, Jan 11 2017

ZL:= [S, {S=Union(Sequence(Z), Sequence(Union(Z, Z, Z)))}, unlabeled]: seq(combstruct[count](ZL, size=n), n=0..25); # Zerinvary Lajos, Jun 19 2008
g:=1/(1-3*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)+1, n=0..31); # Zerinvary Lajos, Jan 09 2009

Mathematica
```
Table[3^n + 1, {n, 0, 24}]
```
PARI
```
a(n) = 3^n + 1
```

Vec(2*(1-2*x)/((1-x)*(1-3*x)) + O(x^50)) \\ Altug Alkan, Nov 15 2015

[lucas_number2(n,4,3) for n in range(27)] # Zerinvary Lajos, Jul 08 2008

[sigma(3,n) for n in range(27)] # Zerinvary Lajos, Jun 04 2009

[3^n+1 for n in range(30)] # Bruno Berselli, Jan 11 2017

a(n) = 3*a(n-1) - 2 = 4*a(n-1) - 3*a(n-2). (Lucas sequence, with A003462, associated to the pair (4, 3).)
G.f.: 2*(1-2*x)/((1-x)*(1-3*x)). Inverse binomial transforms yields 2,2,4,8,16,... i.e., A000079 with the first entry changed to 2. Binomial transform yields A063376 without A063376(-1). - R. J. Mathar, Sep 05 2008
E.g.f.: exp(x) + exp(3*x). - Mohammad K. Azarian, Jan 02 2009
a(n) = A279396(n+3,3). - Wolfdieter Lang, Jan 10 2017
a(n) = 2*A007051(n). - R. J. Mathar, Apr 07 2022

Additional comments from Rick L. Shepherd, Feb 13 2002

A052539 a(n) = 4^n + 1.

Original entry on oeis.org

2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417

Offset: 0

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

The sequence is a Lucas sequence V(P,Q) with P = 5 and Q = 4, so if n is a prime number, then V_n(5,4) - 5 is divisible by n. The smallest pseudoprime q which divides V_q(5,4) - 5 is 15.
Also the edge cover number of the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 20 2017
First bisection of A000051, A049332, A052531 and A014551. - Klaus Purath, Sep 23 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..175
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 470.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences 8(10) (2019).
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A164346 (first differences).
Other powers: A000051, A034472, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081.
Cf. A019434, A274903.

List([0..30], n-> 4^n+1); # G. C. Greubel, May 09 2019

[4^n+1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011

spec := [S,{S=Union(Sequence(Union(Z,Z,Z,Z)),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
A052539:=n->4^n + 1; seq(A052539(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014

Table[4^n + 1, {n, 0, 30}]
(* From Eric W. Weisstein, Sep 20 2017 *)
4^Range[0, 30] + 1
LinearRecurrence[{5, -4}, {2, 5}, 30]
CoefficientList[Series[(2-5x)/(1-5x+4x^2), {x, 0, 30}], x] (* End *)

a(n)=4^n+1 \\ Charles R Greathouse IV, Nov 20 2011

[4^n+1 for n in (0..30)] # G. C. Greubel, May 09 2019

a(n) = 4^n + 1.
a(n) = 4*a(n-1) - 3 = 5*a(n-1) - 4*a(n-2).
G.f.: (2 - 5*x)/((1 - 4*x)*(1 - x)).
E.g.f.: exp(x) + exp(4*x). - Mohammad K. Azarian, Jan 02 2009
From Klaus Purath, Sep 23 2020: (Start)
a(n) = 3*4^(n-1) + a(n-1).
a(n) = (a(n-1)^2 + 9*4^(n-2))/a(n-2).
a(n) = A178675(n) - 3. (End)

A001021 Powers of 12.

Original entry on oeis.org

1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, 1283918464548864, 15407021574586368, 184884258895036416, 2218611106740436992

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 12), L(1, 12), P(1, 12), T(1, 12). Essentially same as Pisot sequences E(12, 144), L(12, 144), P(12, 144), T(12, 144). See A008776 for definitions of Pisot sequences.
Central terms of the triangle in A100851. - Reinhard Zumkeller, Mar 04 2006
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 12-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Starting with 12, this sequence appears in the film "Vollmond" (1998, dir. Fredi Murer), when the children write it on the sidewalk at night. - Alonso del Arte, Dec 21 2011

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 276.
Tanya Khovanova, Recursive Sequences.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A024140 and A178248, 12^n +/- 1; A016125.

Cf. A000005, A000244, A000302, A000351, A000384, A001222, A008585, A008776, A100851, A159991.

a001021 = (12 ^)
a001021_list = iterate (* 12) 1  -- Reinhard Zumkeller, Mar 31 2012

A001021:=-1/(-1+12*z); # Simon Plouffe in his 1992 dissertation

Table[12^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=12^n \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 1/(1-12*x).
E.g.f.: exp(12x).
a(n) = 12*a(n-1). - Zerinvary Lajos, Apr 27 2009
a(n) = A159991(n)/A000351(n). - Reinhard Zumkeller, May 02 2009
From Reinhard Zumkeller, Mar 31 2012: (Start)
a(n) = A000302(n) * A000244(n). - Reinhard Zumkeller, Mar 31 2012
A001222(a(n)) = A008585(n); A000005(a(n)) = A000384(a(n)). (End)
a(n) = det(|ps(i+2, j)|, 1 <= i, j <= n), where ps(n, k) are Legendre-Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013

A034474 a(n) = 5^n + 1.

Original entry on oeis.org

2, 6, 26, 126, 626, 3126, 15626, 78126, 390626, 1953126, 9765626, 48828126, 244140626, 1220703126, 6103515626, 30517578126, 152587890626, 762939453126, 3814697265626, 19073486328126, 95367431640626, 476837158203126

Offset: 0

N. J. A. Sloane

a(n) is the deficiency of 3*5^n (see A033879). - Patrick J. McNab, May 28 2017

			G.f. = 2 + 6*x + 26*x^2 + 126*x^3 + 626*x^4 + 3126*x^5 + 15626*x^6 + ...

T. D. Noe, Table of n, a(n) for n = 0..200
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000051, A000351, A007689, A024049, A033879, A034472, A034478, A034491, A034524, A052539, A062394, A062395, A062396, A062397, A063376, A063481, A074600-A074624, A178248, A228081, A279396.

[5^n+1: n in [0..30]]; // Vincenzo Librandi, Jan 11 2017

Table[5^n + 1, {n, 0, 25}]
LinearRecurrence[{6,-5},{2,6},30] (* Harvey P. Dale, Jul 29 2015 *)

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

[lucas_number2(n,6,5) for n in range(25)] # Zerinvary Lajos, Jul 08 2008

[sigma(5,n) for n in range(25)] # Zerinvary Lajos, Jun 04 2009

[5^n+1 for n in range(30)] # Bruno Berselli, Jan 11 2017

a(n) = 5*a(n-1) - 4 with a(0) = 2.
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1.
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-5*x) = (2-6*x)/((1-x)*(1-5*x)).
E.g.f.: exp(x) + exp(5*x). (End)
a(n) = A279396(n+5,5). - Wolfdieter Lang, Jan 10 2017
From Elmo R. Oliveira, Dec 06 2023: (Start)
a(n) = A000351(n) + 1.
a(n) = 2*A034478(n). (End)

A016125 Expansion of 1/((1-x)(1-12x)).

Original entry on oeis.org

1, 13, 157, 1885, 22621, 271453, 3257437, 39089245, 469070941, 5628851293, 67546215517, 810554586205, 9726655034461, 116719860413533, 1400638324962397, 16807659899548765, 201691918794585181

Offset: 0

N. J. A. Sloane

Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=12, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det(A). - Milan Janjic, Feb 21 2010
Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=13, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-2)=(-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010
Numbers that are repunits in duodecimal representation. - Reinhard Zumkeller, Dec 12 2012
a(n) is the total number of holes in a certain box fractal (start with 12 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015

			For n=5, a(5) = 1*6 + 11*15 + 121*20 + 1331*15 + 14641*6 + 161051*1 = 271453. - _Bruno Berselli_, Nov 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Duodecimal
Wikipedia, Duodecimal
Wikipedia, Repunit
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A001021, A024140, A178248.

Cf. A001020, A135278.

a016125 n = a016125_list !! n
a016125_list = iterate ((+ 1) . (* 12)) 1
-- Reinhard Zumkeller, Dec 12 2012

[(12^(n+1)-1)/11: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011

a:=n->sum(12^(n-j),j=1..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=13},Table[c=13*b-12*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[1/((1-x)(1-12x)),{x,0,20}],x] (* or *) LinearRecurrence[{13,-12},{1,13},20] (* Harvey P. Dale, Aug 20 2022 *)

A016125(n):=(12^(n+1) - 1)/11$
makelist(A016125(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

Vec(1/(1-13*x+12*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011

[lucas_number1(n,13,12) for n in range(1,18)] # Zerinvary Lajos, Apr 29 2009

[gaussian_binomial(n,1,12) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

[(12^(n+1)-1)/11 for n in (0..20)] # Bruno Berselli, Nov 11 2015

a(n) = (12^(n+1) - 1)/11.
a(n) = 12*a(n-1)+1 for n>0, a(0)=1. - Vincenzo Librandi, Nov 19 2010
a(n) =  Sum_{i=0...n} 11^i*binomial(n+1,n-i). - Bruno Berselli, Nov 11 2015
E.g.f.: exp(x)*(12*exp(11*x) - 1)/11. - Stefano Spezia, Mar 11 2023

A062394 a(n) = 6^n + 1.

Original entry on oeis.org

2, 7, 37, 217, 1297, 7777, 46657, 279937, 1679617, 10077697, 60466177, 362797057, 2176782337, 13060694017, 78364164097, 470184984577, 2821109907457, 16926659444737, 101559956668417, 609359740010497, 3656158440062977

Offset: 0

Henry Bottomley, Jun 22 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Sequences of the form m^n + 1: A000012 (m=0), A007395 (m=1), A000051 (m=2), A034472 (m=3), A052539 (m=4), A034474 (m=5), this sequence (m=6), A034491 (m=7), A062395 (m=8), A062396 (m=9), A062397 (m=10), A034524 (m=11), A178248 (m=12), A141012 (m=13), A228081 (m=64).

Cf. A000400.

[6^n + 1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011

6^Range[0,30] +1
LinearRecurrence[{7,-6},{2,7},30] (* Harvey P. Dale, Aug 11 2015 *)

vector(20, n, n--; 6^n + 1) \\ Michel Marcus, Jun 11 2015

[6^n+1 for n in range(31)] # G. C. Greubel, Mar 11 2023

a(n) = 6*a(n-1) - 5.
a(n) = A000400(n) + 1.
a(n) = 7*a(n-1) - 6*a(n-2).
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-6*x).
E.g.f.: exp(x) + exp(6*x). (End)

A034491 a(n) = 7^n + 1.

Original entry on oeis.org

2, 8, 50, 344, 2402, 16808, 117650, 823544, 5764802, 40353608, 282475250, 1977326744, 13841287202, 96889010408, 678223072850, 4747561509944, 33232930569602, 232630513987208, 1628413597910450, 11398895185373144

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000051, A007689, A034472, A034474, A034494, A034524, A052539.
Cf. A062394, A062395, A062396, A062397, A063376, A063481.
Cf. A074600 - A074624, A178248, A228081.

[7^n +1: n in [0..30]]; // G. C. Greubel, Mar 11 2023

7^Range[0,30] +1
LinearRecurrence[{8,-7},{2,8},20] (* Harvey P. Dale, Aug 18 2018 *)

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

[sigma(7,n) for n in range(0,20)] # Zerinvary Lajos, Jun 04 2009

a(n) = 7*a(n-1) - 6.
a(n) = 8*a(n-1) - 7*a(n-2).
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-7*x).
E.g.f.: exp(x) + exp(7*x). (End)
a(n) = 2*A034494(n). - G. C. Greubel, Mar 11 2023

A062395 a(n) = 8^n + 1.

Original entry on oeis.org

2, 9, 65, 513, 4097, 32769, 262145, 2097153, 16777217, 134217729, 1073741825, 8589934593, 68719476737, 549755813889, 4398046511105, 35184372088833, 281474976710657, 2251799813685249, 18014398509481985, 144115188075855873

Offset: 0

Henry Bottomley, Jun 22 2001

Any number of the form b^k+1 is composite for b>2 and k odd since b+1 algebraically divides b^k+1. - Robert G. Wilson v, Aug 25 2002

D. M. Burton, Elementary Number Theory, Allyn and Bacon, Boston, MA, 1976, pp. 51.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Vincenzo Librandi, Table of n, a(n) for n = 0..140
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081 for numbers one more than powers.

[8^n + 1: n in [0..40] ]; // Vincenzo Librandi, Apr 30 2011

Table[8^n + 1, {n, 0, 20}]
LinearRecurrence[{9,-8},{2,9},20] (* Harvey P. Dale, Jan 24 2019 *)

PARI
```
for(n=0,22,print(8^n+1))
```

a(n) = 8a(n-1)-7 = A001018(n)+1 = 9a(n-1) - 8a(n-2).
G.f.: -(-2+9*x)/(-1+x)/(-1+8*x). - R. J. Mathar, Nov 16 2007
E.g.f.: e^x+e^(8*x). - Mohammad K. Azarian, Jan 02 2009

A062396 a(n) = 9^n + 1.

Original entry on oeis.org

2, 10, 82, 730, 6562, 59050, 531442, 4782970, 43046722, 387420490, 3486784402, 31381059610, 282429536482, 2541865828330, 22876792454962, 205891132094650, 1853020188851842, 16677181699666570, 150094635296999122

Offset: 0

Henry Bottomley, Jun 22 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062397, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081 for numbers one more than powers.

[9^n + 1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

Table[9^n + 1, {n, 0, 20}]
LinearRecurrence[{10,-9},{2,10},20] (* Harvey P. Dale, May 30 2013 *)

a(n)=9^n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 9*a(n-1) - 8 = A001019(n) + 1 = 10*a(n-1) - 9*a(n-2).
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-9*x).
E.g.f.: e^x + e^(9*x). (End)

cp's OEIS Frontend

A000051 a(n) = 2^n + 1.

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A034472 a(n) = 3^n + 1.

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A052539 a(n) = 4^n + 1.

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A001021 Powers of 12.

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A034474 a(n) = 5^n + 1.

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A016125 Expansion of 1/((1-x)(1-12x)).