A125857 -id:A125857 - cp's OEIS frontend

A002452 a(n) = (9^n - 1)/8.

0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011, 1519708182382116100, 13677373641439044901, 123096362772951404110

Offset: 0

N. J. A. Sloane

From David W. Wilson: Numbers triangular, differences square.
To be precise, the differences are the squares of the powers of three with positive indices. Hence a(n+1) - a(n) = (A000244(n+1))^2 = A001019(n+1). [Added by Ant King, Jan 05 2011]
Partial sums of A001019. This is m-th triangular number, where m is partial sums of A000244. a(n) = A000217(A003462(n)). - Lekraj Beedassy, May 25 2004
With offset 0, binomial transform of A003951. - Philippe Deléham, Jul 22 2005
Numbers in base 9: 1, 11, 111, 1111, 11111, 111111, 1111111, etc. - Zerinvary Lajos, Apr 26 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=9, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 2, a(n-1) = (-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010
From Hieronymus Fischer, Jan 30 2013: (Start)
Least index k such that A052382(k) >= 10^(n-1), for n > 0.
Also index k such that A052382(k) = (10^n-1)/9, n > 0.
A052382(a(n)) is the least zerofree number with n digits, for n > 0.
For n > 1: A052382(a(n)-1) is the greatest zerofree number with n-1 digits. (End)
For n > 0, 4*a(n) is the total number of holes in a certain triangle fractal (start with 9 triangles, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Feb 21 2015
For n > 0, a(n) is the sum of the numerators and denominators of the reduced fractions 0 < (b/3^(n-1)) < 1 plus 1. Example for n=3 gives fractions 1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, and 8/9 plus 1 has sum of numerators and denominators +1 = a(3) = 91. - J. M. Bergot, Jul 11 2015
Except for 0 and 1, all terms are Brazilian repunits numbers in base 9, so belong to A125134. All these terms are composite because a(n) is the ((3^n - 1)/2)-th triangular number. - Bernard Schott, Apr 23 2017
These are also the second steps after the junctions of the Collatz trajectories of 2^(2k-1)-1 and 2^2k-1. - David Rabahy, Nov 01 2017

			a(4) = (9^4 - 1)/8 = 820 = 1111_9 = (1/2) * 40 * 41 is the ((3^4 - 1)/2)-th = 40th triangular number. - _Bernard Schott_, Apr 23 2017

A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 36.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Kival Ngaokrajang, Illustration of initial terms
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
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. 4.
A. G. Shannon, Letter to N. J. A. Sloane, Dec 06 1974.
M. Ward, Note on divisibility sequences, Bull. Amer. Math. Soc., 42 (1936), 843-845.
Eric Weisstein's World of Mathematics, Repunit.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Right-hand column 1 in triangle A008958.
Cf. A217094, A011540, A052382, A125857.
Cf. A125134, A000217.

[(9^n - 1)/8 : n in [0..25]]; // Vincenzo Librandi, Jun 01 2011

A002452 := 1/(9*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation

(9^# & /@ Range[0, 18] // Accumulate) (* Ant King, Jan 06 2011 *)
LinearRecurrence[{10,-9},{0,1},30] (* Harvey P. Dale, Sep 23 2018 *)

A002452(n):=floor((9^n-1)/8)$
makelist(A002452(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=9^n>>3 \\ Charles R Greathouse IV, Jul 25 2011

From Philippe Deléham, Mar 13 2004: (Start)
a(n) = 9*a(n-1) + 1; a(1) = 1.
G.f.: x / ((1-x)*(1-9*x)). (End)
a(n) = 10*a(n-1) - 9*a(n-2). - Ant King, Jan 05 2011
a(n) = floor(A000217(3^n)/4) - A033113(n-1). - Arkadiusz Wesolowski, Feb 14 2012
Sum_{n>0} a(n)*(-1)^(n+1)*x^(2*n+1)/(2*n+1)! = (1/6)*sin(x)^3. - Vladimir Kruchinin, Sep 30 2012
a(n) = A011540(A217094(n-1)) - A217094(n-1) + 2, n > 0. - Hieronymus Fischer, Jan 30 2013
a(n) = 10^(n-1) + 2 - A217094(n-1). - Hieronymus Fischer, Jan 30 2013
a(n) = det(|v(i+2,j+1)|, 1 <= i,j <= n-1), where v(n,k) are central factorial numbers of the first kind with odd indices (A008956) and n > 0. - Mircea Merca, Apr 06 2013
a(n) = Sum_{k=0..n-1} 9^k. - Doug Bell, May 26 2017
E.g.f.: exp(5*x)*sinh(4*x)/4. - Stefano Spezia, Mar 11 2023

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

Offset changed from 1 to 0 and added 0 by Vincenzo Librandi, Jun 01 2011

A066443 Number of distinct paths of length 2n+1 along edges of a unit cube between two fixed adjacent vertices.

Original entry on oeis.org

1, 7, 61, 547, 4921, 44287, 398581, 3587227, 32285041, 290565367, 2615088301, 23535794707, 211822152361, 1906399371247, 17157594341221, 154418349070987, 1389765141638881, 12507886274749927, 112570976472749341

Offset: 0

John W. Layman, Aug 12 2002

All members of sequence are also hex, or central hexagonal, numbers (A003215). (If n is a hex number, 9n - 2 is always a hex number; see recurrence.) - Matthew Vandermast, Mar 30 2003
The sequence 1,1,7,61,547,... with g.f. (1-9x+6x^2)/((1-x)(1-9x)) and a(n) = A054879(n)/3 + 2*0^n/3 gives the denominators in the probability that a random walk on the cube returns to its starting corner on the 2n-th step. - Paul Barry, Mar 11 2004
Equals row sums of even row terms of triangle A158303. - Gary W. Adamson, Mar 15 2009
It appears that a(n) is the n-th record value in A120437, which gives the  differences of A037314 (positive integers n such that the sum of the base 3 digits of n equals the sum of the base 9 digits of n). - John W. Layman, Dec 14 2010
Numbers in base 9 are 1, 6+1, 66+1, 666+1, 6666+1, 66666+1, etc.; that is, n 6's + 1. - Yuchun Ji, Aug 15 2019
All prime factors of a(n) are 1 mod 6. In addition, if n is not 1 mod 3 (first index being n=0), then 3 is a cubic residue modulo all prime factors of a(n). This provides a simple proof that there are infinitely many primes 1 mod 6 that have 3 as a cubic residue. - William Hu, Jul 26 2024

			From _Michael B. Porter_, Aug 22 2016: (Start)
Give coordinates (a,b,c) to the vertices of the cube, where a, b, and c are either 0 or 1. For n = 1, the a(1) = 7 paths of length 2n + 1 = 3 from (0,0,0) to (0,0,1) are:
(0,0,0) -> (0,0,1) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (0,0,1) -> (0,1,1) -> (0,0,1)
(0,0,0) -> (0,0,1) -> (1,0,1) -> (0,0,1)
(0,0,0) -> (0,1,0) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (0,1,0) -> (0,1,1) -> (0,0,1)
(0,0,0) -> (1,0,0) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (1,0,0) -> (1,0,1) -> (0,0,1) (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
G. Benkart and D. Moon, A Schur-Weyl Duality Approach to Walking on Cubes, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and Ann. Combin. 20 (3) (2016) 397-417.
E. Estrada and J. A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176 [math.CO], 2013. - _N. J. A. Sloane_, Feb 28 2013
E. Estrada and J. A. de la Pena, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54:369-391, 1947.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A158303, A037314, A120437, A083234 (binomial transform), A083233 (inverse binomial transform), A054879 (recurrent walks), A125857 (walks ending on face diagonal), A054880 (walks ending on space diagonal).

[(3^(2*n+1)+1)/4: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

seq((3^(2*n+1) + 1)/4, n=0..18); # Zerinvary Lajos, Jun 16 2007

NestList[9 # - 2 &, 1, 18] (* or *)
Table[(3^(2 n + 1) + 1)/4, {n, 0, 18}] (* or *)
CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 9 x)), {x, 0, 18}], x] (* Michael De Vlieger, Aug 22 2016 *)

a(n)=3^(2*n+1)\/4 \\ Charles R Greathouse IV, Jul 02 2013

Vec((1-3*x)/((1-x)*(1-9*x)) + O(x^50)) \\ Altug Alkan, Nov 13 2015

a(n) = (3^(2*n+1)+1)/4. - Vladeta Jovovic, Dec 22 2002
a(n) = 9*a(n-1) - 2. - Matthew Vandermast, Mar 30 2003
From Paul Barry, Apr 21 2003: (Start)
G.f.: (1-3*x)/((1-x)*(1-9*x)).
E.g.f.: (3*exp(9*x) + exp(x))/4. (End)
a(n) = (-1)^n times the (i, i)-th element of M^n (for any i), where M = ((1, 1, 1, -2), (1, 1, -2, 1), (1, -2, 1, 1), (-2, 1, 1, 1)). - Simone Severini, Nov 25 2004
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*4^(n-k). - Paul Barry, Jan 22 2005
a(n) = A054880(n) + 1.
a(n) = A057660(3^n). - Henry Bottomley, Nov 08 2015
a(n) = Sum_{k=0..2n} (-3)^k == 1 + Sum_{k=1..n} 2*3^(2k-1). - Bob Selcoe, Aug 21 2016
a(n) = 3^(2*n+1) * a(-1-n) for all n in Z. - Michael Somos, Jul 02 2017
a(n) = 6*A002452(n) + 1. - Yuchun Ji, Aug 15 2019

Corrected by Vladeta Jovovic, Dec 22 2002

A191681 a(n) = (9^n - 1)/2.

Original entry on oeis.org

0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044

Offset: 0

Adi Dani, Jun 11 2011

Number of compositions of odd numbers into n parts < 9.
These are also the junctions of the Collatz trajectories of 2^(2k-1)-1 and 2^2k-1. - David Rabahy, Nov 01 2017
a(n) gives the number of turns in the n-th iteration of the Peano curve given by plotting (A163528, A163529) or by (Siromoney 1982). - Jason V. Morgan, Oct 08 2021

			a(2)=40: there are 40 compositions of odd numbers into 2 parts < 9:
1:  (0,1),(1,0);
3:  (0,3),(3,0),(1,2),(2,1);
5:  (0,5),(5,0),(1,4),(4,1),(2,3),(3,2);
7:  (0,7),(7,0),(1,6),(6,1),(2,5),(5,2),(3,4),(4,3);
9:  (1,8),(8,1),(2,7),(7,2),(3,6),(6,3),(4,5),(5,4);
11: (3,8),(8,3),(4,7),(7,4),(5,6),(6,5);
13: (5,8),(8,5),(6,7),(7,6);
15: (7,8),(8,7).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Adi Dani, Restricted compositions of natural numbers
Rani Siromoney and K. G. Subramanian, Space-filling curves and infinite graphs, International Workshop on Graph Grammars and Their Application to Computer Science, 1982.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A002452, A096053, A125857, A138894, A198964.

[(9^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 16 2011

Table[(9^n - 1)/2, {n, 0, 19}]
LinearRecurrence[{10,-9},{0,4},30] (* Harvey P. Dale, Jun 19 2011 *)

a(n)=9^n\2 \\ Charles R Greathouse IV, Oct 16 2015

a(0)=0, a(1)=4, a(n) = 10*a(n-1) - 9*a(n-2). - Harvey P. Dale, Jun 19 2011
G.f.: 4*x / ((x-1)*(9*x-1)). - Colin Barker, May 16 2013
a(n) = 2 * A125857(n+1) = 4 * A002452(n). - Bernard Schott, Oct 29 2021

Example corrected by L. Edson Jeffery, Feb 13 2015

A273514 a(n) is the number of arithmetic progressions m < n < p (three numbers in arithmetic progression) such that m and p contain no 2's in their ternary representation.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 8, 2, 2, 8, 8, 8, 8, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2

Offset: 0

Max Barrentine, May 23 2016

This is a recursive sequence that gives the number of times n is rejected from A005836, if n is the middle member of an arithmetic triple whose first and last terms are contained in A005836.
Also, a(n) is the number of unordered pairs of members of A005836 whose average (arithmetic mean) is n.
It appears that when A273513(n) and A262097(n) are coprime, a(n) = 2.
Local maxima occur at a(A125857(n)).

			a(2) = 2 because there are two arithmetic triples a < 2 < b such that a and b are members of A005836: 0, 2, 4 and 1, 2, 3.

Max Barrentine, Table of n, a(n) for n = 0..19683

Cf. A005836, A125857, A262097, A273513.

precCantor(n)=my(v=digits(n,3)); for(i=1,#v, if(v[i]==2, for(j=i,#v,v[j]=1); break)); fromdigits(v,2)
a(n)=if(n==0, return(0)); sum(i=0,precCantor(n-1), my(m=fromdigits(digits(i,2),3)); vecmax(digits(2*n-m,3))<2) \\ Charles R Greathouse IV, Jun 17 2016

a(0) = 0, a(n) = a(3n) = a(3n+1); if a(n) = 0, a(9n + 2) = 2, otherwise a(9n + 2) = 4a(n); a(9n + 5) = a(9n + 6) = a(9n + 7) = a(9n + 8) = a(3n + 2).

A125725 Numbers whose base-7 representation is 222....2.

Original entry on oeis.org

0, 2, 16, 114, 800, 5602, 39216, 274514, 1921600, 13451202, 94158416, 659108914, 4613762400, 32296336802, 226074357616, 1582520503314, 11077643523200, 77543504662402, 542804532636816, 3799631728457714, 26597422099204000

Offset: 1

Zerinvary Lajos, Feb 02 2007

			base 7.......decimal
0..................0
2..................2
22................16
222..............114
2222.............800
22222...........5602
222222.........39216
2222222.......274514
22222222.....1921600
222222222...13451202
etc...........etc.

Davis Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A007310, A023000, A047522, A165759.

Cf. also A002276, A005610, A020988, A024023, A125831, A125835, A125857 for related or similarly constructed sequences.

List([1..25], n-> (7^(n-1) -1)/3); # G. C. Greubel, May 23 2019

[0] cat [n:n in [1..15000000]| Set(Intseq(n,7)) subset [2]]; // Marius A. Burtea, May 06 2019

[(7^(n-1)-1)/3: n in [1..25]]; // Marius A. Burtea, May 06 2019

Maple
```
seq(2*(7^n-1)/6, n=0..25);
```

FromDigits[#,7]&/@Table[PadLeft[{2},n,2],{n,0,25}]  (* Harvey P. Dale, Apr 13 2011 *)
(7^(Range[25]-1) - 1)/3 (* G. C. Greubel, May 23 2019 *)

vector(25, n, (7^(n-1)-1)/3) \\ Davis Smith, Apr 04 2019

[(7^(n-1) -1)/3 for n in (1..25)] # G. C. Greubel, May 23 2019

a(n) = (7^(n-1) - 1)/3 = 2*A023000(n-1).
a(n) = 7*a(n-1) + 2, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 2*x^2 / ( (1-x)*(1-7*x) ). - R. J. Mathar, Sep 30 2013
From Davis Smith, Apr 04 2019: (Start)
A007310(a(n) + 1) = 7^(n - 1).
A047522(a(n + 1)) = -1*A165759(n). (End)
E.g.f.: (exp(7*x) - 7*exp(x) + 6)/21. - Stefano Spezia, Jan 12 2025

Offset corrected by N. J. A. Sloane, Oct 02 2010

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A002452 a(n) = (9^n - 1)/8.

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A066443 Number of distinct paths of length 2n+1 along edges of a unit cube between two fixed adjacent vertices.

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A191681 a(n) = (9^n - 1)/2.

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A273514 a(n) is the number of arithmetic progressions m < n < p (three numbers in arithmetic progression) such that m and p contain no 2's in their ternary representation.

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A125725 Numbers whose base-7 representation is 222....2.

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