A102901 -id:A102901 - cp's OEIS frontend

A015441 Generalized Fibonacci numbers.

0, 1, 1, 7, 13, 55, 133, 463, 1261, 4039, 11605, 35839, 105469, 320503, 953317, 2876335, 8596237, 25854247, 77431669, 232557151, 697147165, 2092490071, 6275373061, 18830313487, 56482551853, 169464432775, 508359743893, 1525146340543, 4575304803901, 13726182847159

Offset: 0

Olivier Gérard

a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x,y) = xF(n-1)(x,y) + yF(n-2)(x,y), F(0)(x,y)=0, F(1)(x,y)=1, when y=6x^2. - Mario Catalani (mario.catalani(AT)unito.it), Dec 06 2002
For n>=1: number of length-(n-1) words with letters {0,1,2,3,4,5,6,7} where no two consecutive letters are nonzero, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Starting with offset 1 and convolved with (1, 3, 3, 3, ...) = A003462: (1, 4, 13, 40, ...). - Gary W. Adamson, May 28 2009
a(n) is identical to its inverse binomial transform signed. Differences: A102901. - Paul Curtz, Feb 23 2010
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>=2, 7*a(n-2) equals the number of 7-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
Pisano period lengths: 1, 1, 1, 2, 20, 1, 6, 2, 3, 20, 5, 2, 12, 6, 20, 4, 16, 3, 18, 20, ... - R. J. Mathar, Aug 10 2012
A015441 and A015518 are the only integer sequences (from the family of homogeneous linear recurrence relation of order 2 with positive integer coefficients with initial values a(0)=0 and a(1)=1) whose ratio a(n+1)/a(n) converges to 3 as n approaches infinity. - Felix P. Muga II, Mar 14 2014
This is an autosequence of the first kind: the array of successive differences shows a main diagonal of zeros and the inverse binomial transform is identical to the sequence (with alternating signs). - Pointed out by Paul Curtz, Dec 05 2016
First two upper diagonals: A000400(n).
This is a variation on the Starhex honeycomb configuration A332243, see illustration in links. It is an alternating pattern of the 2nd iteration of the centered hexagonal numbers A003215 and centered 12-gonal 'Star' numbers A003154. - John Elias, Oct 06 2021

			G.f. = x + x^2 + 7*x^3 + 13*x^4 + 55*x^5 + 133*x^6 + 463*x^7 + 1261*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), p. 317-318
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
John Elias, Illustration: A Starhex honeycomb variation
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
H. Li and T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 48.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 2, (2015), 35-42.
Index entries for linear recurrences with constant coefficients, signature (1,6).

Cf. A000079, A003462, A016153, A109466, A122117. Cf. A001656, A087451. Cf. A000400.

I:=[0,1]; [n le 2 select I[n] else Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

A015441:=n->(1/5)*((3^n)-((-2)^n)); seq(A015441(n), n=0..30); # Wesley Ivan Hurt, Mar 14 2014

a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]], {n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1,6},{0,1},30] (* Harvey P. Dale, Apr 26 2011 *)
CoefficientList[Series[x/((1 + 2 x) (1 - 3 x)), {x, 0, 29}], x] (* Michael De Vlieger, Dec 05 2016 *)

PARI
```
{a(n) = (3^n - (-2)^n) / 5};
```

[lucas_number1(n,1,-6) for n in range(0, 27)] # Zerinvary Lajos, Apr 22 2009

G.f.: x/((1+2*x)*(1-3*x)).
a(n) = a(n-1) + 6*a(n-2).
a(n) = (1/5)*((3^n)-((-2)^n)). - henryk.wicke(AT)stud.uni-hannover.de
E.g.f.: (exp(3*x) - exp(-2*x))/5. - Paul Barry, Apr 20 2003
a(n+1) = Sum_{k=0..ceiling(n/2)} 6^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
a(n) = (A000244(n) - A001045(n+1)(-1)^n - A001045(n)(-1)^n)/5. - Paul Barry, Apr 27 2004
The binomial transform of [1,1,7,13,55,133,463,...] is A122117. - Philippe Deléham, Oct 19 2006
a(n+1) = Sum_{k=0..n} A109466(n,k)*(-6)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = 3a(n-1) + (-1)^(n+1)*A000079(n-1). - Paul Curtz, Feb 23 2010
G.f.: Q(0) -1, where Q(k) = 1 + 6*x^2 + (k+2)*x - x*(k+1 + 6*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013
a(n) = (Sum_{1<=k<=n, k odd} binomial(n,k)*5^(k-1))/2^(n-1). - Vladimir Shevelev, Feb 05 2014
a(-n) = -(-1)^n * a(n) / 6^n for all n in Z. - Michael Somos, Mar 18 2014
From Peter Bala, Apr 01 2015: (Start)
Sum_{n >= 0} a(n+1)*x^n = exp( Sum_{n >= 1} A087451(n)*x^n/n ).
For k = 0, 1, 2, ... and for n >= 1, (5^k)*a(n) | a((5^k)*n).
The expansion of exp( Sum_{n >= 1} a(5*n)/(5*a(n))*x^n/n ) has integral coefficients. Cf. A001656. (End)
From  Peter Bala, Jun 27 2025: (Start)
Sum_{n >= 1} (-6)^n/(a(n)*a(n+1)) = -2, since (-6)^n/(a(n)*a(n+1)) = (-2)^n/a(n) - (-2)^(n+1)/a(n+1) for n >= 1.
The following are examples of telescoping infinite products:
Product_{n >= 0} (1 + 6^n/a(2*n+2)) = 6, since (1 + 6^(2*n-1)/a(4*n))*(1 + 6^(2*n)/a(4*n+2)) = (6 - 4^(n+1)/b(n)) / (6 - 4^n/b(n-1)), where b(n) = (2*4^n + 3*9^n)/5 = A096951(n). Similarly,
Product_{n >= 1} (1 - 6^n/a(2*n+2)) = 3/13.
Product_{n >= 0} (1 + (-6)^n/a(2*n+2)) = 6/5.
Product_{n >= 1} (1 - (-6)^n/a(2*n+2)) = 15/13.
exp( Sum_{n >= 1} a(2*n)/a(n)*x^n/n ) = Sum_{n >= 0} a(n+1)*x^n. (End)

A102900 a(n) = 3a(n-1) + 4a(n-2), a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 7, 25, 103, 409, 1639, 6553, 26215, 104857, 419431, 1677721, 6710887, 26843545, 107374183, 429496729, 1717986919, 6871947673, 27487790695, 109951162777, 439804651111, 1759218604441, 7036874417767, 28147497671065

Offset: 0

Paul Barry, Jan 17 2005

Binomial transform of A102901.

Hankel transform is = 1,6,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A001045, A004171, A046717, A086901, A102901, A247666 (which appears to be the run length transform of this sequence).

a102900 n = a102900_list !! n
a102900_list = 1 : 1 : zipWith (+)
               (map (* 4) a102900_list) (map (* 3) $ tail a102900_list)
-- Reinhard Zumkeller, Feb 13 2015

[n le 2 select 1 else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 28 2015

a[n_]:=(MatrixPower[{{2,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{3, 4}, {1, 1}, 30] (* Vincenzo Librandi, Dec 28 2015 *)

a(n)=([0,1; 4,3]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Mar 28 2016

A102900=BinaryRecurrenceSequence(3,4,1,1)
[A102900(n) for n in range(51)] # G. C. Greubel, Dec 09 2022

G.f.: (1-2*x)/(1-3*x-4*x^2).
a(n) = (2*4^n + 3*(-1)^n)/5.
a(n) = ceiling(4^n/5) + floor(4^n/5) = (ceiling(4^n/5))^2 - (floor(4^n/5))^2.
a(n) + a(n+1) = 2^(2*n+1) = A004171(n).
a(n) = Sum_{k=0..n} binomial(2*n-k, 2*k)*2^k. - Paul Barry, Jan 20 2005
a(n) = upper left term in the 2 X 2 matrix [1,3; 2,2]^n. - Gary W. Adamson, Mar 14 2008
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(8*4^k-3*(-1)^k)/(x*(8*4^k-3*(-1)^k) + (2*4^k+3*(-1)^k)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
a(n) = 2^(2*n-1) - a(n-1), a(1)=1. - Ben Paul Thurston, Dec 27 2015; corrected by Klaus Purath, Aug 02 2020
From Klaus Purath, Aug 02 2020: (Start)
a(n) = 4*a(n-1) + 3*(-1)^n.
a(n) = 6*4^(n-2) + a(n-2), n>=2. (End)

A347237 Möbius transform of A347236.

Original entry on oeis.org

1, 0, 1, 6, 1, 0, 3, 6, 17, 0, 1, 6, 3, 0, 1, 42, 1, 0, 3, 6, 3, 0, 5, 6, 37, 0, 49, 18, 1, 0, 5, 78, 1, 0, 3, 102, 3, 0, 3, 6, 1, 0, 3, 6, 17, 0, 5, 42, 89, 0, 1, 18, 5, 0, 1, 18, 3, 0, 1, 6, 5, 0, 51, 330, 3, 0, 3, 6, 5, 0, 1, 102, 5, 0, 37, 18, 3, 0, 3, 42, 353, 0, 5, 18, 1, 0, 1, 6, 7, 0, 9, 30, 5, 0, 3, 78, 3, 0, 17

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of A003972 (prime shifted phi) with A061019.
Dirichlet convolution of A003961 with A158523.
Multiplicative because A003972 and A061019 are, and also because A347236 is.
From Antti Karttunen, Aug 25 2021: (Start)
All terms are nonnegative because sequence is multiplicative and a(p^k) >= 0 for all primes p and k >= 0.
Proof: For any prime p, sequence a(p^k), k>=0, is obtained as an ordinary convolution of sequences (-p)^k and the first differences of q^k, where q = A151800(p). (E.g., for powers of 2, the sequences convolved are A122803 and A025192, giving A102901.) This convolution is an alternating sum, with the terms 1*(q-1)*q^(k-1), -(p)*(q-1)*q^(k-2), (p^2)*(q-1)*q^(k-3), -(p^3)*(q-1)*q^(k-4), ..., (p^(k-1))*(q-1), -(p^k), for odd k, with sum of each consecutive pair being nonnegative because q >= p+1, while with an even exponent k, the leftover term p^k at the end is also positive, thus the whole sum is nonnegative also in that case.
(End)

Cf. A000040, A001223, A003961, A003972, A008683, A008836, A016825 (positions of zeros), A061019, A102901, A120612, A158523, A347236.
Cf. also A347137.
Cf. A151800.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));
\\ Or alternatively as:
A158523(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((-1)^e)*(p+1)*(p^(e-1))); };
A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

a(n) = Sum_{d|n} A008683(n/d) * A347236(d).
a(n) = Sum_{d|n} A003972(n/d) * A061019(d).
a(n) = Sum_{d|n} A003961(n/d) * A158523(d).
For all n >= 1, a(A000040(n)) = A001223(n) - 1.
For all n >= 0, a(2^n) = A102901(n).
For all n >= 0, a(3^n) = A120612(n).
Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A140796 a(n)=a(n-1)+6a(n-2), n>2.

Original entry on oeis.org

1, 5, 14, 44, 128, 392, 1160, 3512, 10472, 31544, 94376, 283640, 849896, 2551736, 7651112, 22961528, 68868200, 206637368, 619846568, 1859670776, 5578750184, 16736774840, 50209275944, 150629924984, 451885580648, 1355665130552

Offset: 0

Paul Curtz, Jul 15 2008

nonn

The binomial transform is A037481.

The recurrence of the definition is also satisfied by A087451, A102901 and A140725.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 6).

Cf. A140725

Join[{1},LinearRecurrence[{1,6},{5,14},30]] (* Harvey P. Dale, Nov 20 2011 *)

a(n+1)-3a(n) = (-1)^n*A000079(n-1), n>0.
d(n+1)-3d(n) = (-1)^(n+1)*A000079(n-1), n>0, where d(n) is the sequence of pair sums d(n)= a(n)+a(n+1)=6, 19, 58, 172,...
O.g.f.: (1+x)(3x+1)/((2x+1)(1-3x)). - R. J. Mathar, Jul 29 2008
a(n) = (-1)^(n+1)*2^n/10+8*3^n/5, n>0. - R. J. Mathar, Jul 29 2008
a(n) = A140725(n)+A140725(n+1). - Philippe Deléham, Nov 17 2013

Edited and extended by R. J. Mathar, Jul 29 2008

A232015 Expansion of (1-2x)/((1+2x)(1-3x)).

Original entry on oeis.org

1, -1, 5, -1, 29, 23, 197, 335, 1517, 3527, 12629, 33791, 109565, 312311, 969701, 2843567, 8661773, 25723175, 77693813, 232032863, 698195741, 2090392919, 6279567365, 18821924879, 56499329069, 169430878343, 508426852757, 1525012122815, 4575573239357

Offset: 0

Philippe Deléham, Nov 17 2013

Index entries for linear recurrences with constant coefficients, signature (1,6).

Cf. A078008, A108561.

Table[(3^n + 4 (-2)^n)/5, {n, 0, 30}] (* Bruno Berselli, Nov 18 2013 *)
CoefficientList[Series[(1-2x)/((1+2x)(1-3x)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,6},{1,-1},30] (* Harvey P. Dale, Apr 20 2017 *)

Vec((1-2*x)/((1+2*x)*(1-3*x))+O(x^20)) \\ Edward Jiang, Sep 06 2014

G.f.: (1 - 2*x) / (1 - x - 6*x^2).
a(n) = a(n-1) + 6*a(n-2) for n>1, a(0)=1, a(1)=-1.
a(n) = sum_{k=0..n} A108561(n,k)*2^k.
a(n) = A102901(n) - A015441(n).
From Bruno Berselli, Nov 18 2013: (Start)
a(n) = (3^n + 4*(-2)^n)/5.
a(n+1) + a(n) = 4*A015441(n).
a(n+1) - a(n) = -2*(-1)^n*A165405(n).
Sum(a(i), i=0..n) = A091001(n+1). (End)

A208459 Triangle T_x = T(n,k) given by (0, 1/x, 1-1/x, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (x, 1/x-1, -1/x, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938, for x = 0.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, -3, 0, 1, 0, 3, -1, 0, 5, 0, 1, 0, 4, -2, 3, 2, -8, 0, 1, 0, 5, -3, 7, -2, -5, 13, 0, 1, 0, 6, -4, 12, -8, 2, 12, -21, 0, 1, 0, 7, -5, 18, -16, 15, 3, -25, 34

Offset: 0

Philippe Deléham, Feb 27 2012

Triangle T_x : T_1 = A103631, T_2 = A208343, T_3 = A208345.

			Triangle begins :
1
0, 0
0, 1, 1
0, 1, 0, -1
0, 1, 0, 1, 2
0, 1, 0, 2, 0, -3
0, 1, 0, 3, -1, 0, 5
0, 1, 0, 4, -2, 3, 2, -8
0, 1, 0, 5, -3, 7, -2, -5, 13
0, 1, 0, 6, -4, 12, -8, 2, 12, -21
0, 1, 0, 7, -5, 18, -16, 15, 3, -25, 34

Cf. A103631, A208343, A208345, A000045 (Fibonacci)

T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) with T(0,0) = 1 T(1,0) = 0, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-x+y*x)/(1-x+y*x- y^2*x^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = 12*A015548(n-1), 6*A085939(n-1), A106434(n), A000007(n), A000007(n), A077957(n), (-1)^n*A102901(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sm_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A034834(n-1), A077957(n), A052533(n), (-1)^n*A086344(n) for x = -1, 0, 1, 2, 3 respectively.

cp's OEIS Frontend

A015441 Generalized Fibonacci numbers.

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A102900 a(n) = 3*a(n-1) + 4*a(n-2), a(0)=a(1)=1.

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A347237 Möbius transform of A347236.

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A140796 a(n)=a(n-1)+6a(n-2), n>2.

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A232015 Expansion of (1-2*x)/((1+2*x)*(1-3*x)).

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A208459 Triangle T_x = T(n,k) given by (0, 1/x, 1-1/x, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (x, 1/x-1, -1/x, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938, for x = 0.

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A102900 a(n) = 3a(n-1) + 4a(n-2), a(0)=a(1)=1.

A232015 Expansion of (1-2x)/((1+2x)(1-3x)).