A041041 -id:A041041 - cp's OEIS frontend

A154597 a(n) = 15*a(n-1) + a(n-2) with a(0) = 0, a(1) = 1.

0, 1, 15, 226, 3405, 51301, 772920, 11645101, 175449435, 2643386626, 39826248825, 600037119001, 9040383033840, 136205782626601, 2052127122432855, 30918112619119426, 465823816409224245, 7018275358757483101, 105739954197771470760, 1593117588325329544501

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009

Limit_{n -> infinity} a(n)/a(n-1) = (15 + sqrt(229))/2. - Klaus Brockhaus, Oct 07 2009
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 15's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n - 1 on alphabet {0,1,...,15} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Apr 30 2023: (Start)
Also called the 15-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 15 kinds of squares available. (End)

G. C. Greubel, Table of n, a(n) for n = 0..840 (a(0) = 0 added by Jianing Song)
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive sequences. [From _Johannes W. Meijer_, Jun 12 2010]
Index entries for linear recurrences with constant coefficients, signature (15,1).

Row n=15 of A073133, A172236 and A352361 and column k=15 of A157103.
First bisection is A098247.
Cf. A166125 (decimal expansion of sqrt(229)), A166126 (decimal expansion of (15 + sqrt(229))/2).
Cf. also A041427, A090301, A098245.
Sequences with g.f. 1/(1-k*x-x^2) or x/(1-k*x-x^2): A000045 (k=1), A000129 (k=2), A006190 (k=3), A001076 (k=4), A052918 (k=5), A005668 (k=6), A054413 (k=7), A041025 (k=8), A099371 (k=9), A041041 (k=10), A049666 (k=11), A041061 (k=12), A140455 (k=13), A041085 (k=14), this sequence (k=15), A041113 (k=16), A178765 (k=17), A041145 (k=18), A243399 (k=19), A041181 (k=20).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-229); S:=[ ((15+r)^n-(15-r)^n)/(2^n*r): n in [1..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009

[n le 2 select n-1 else 15*Self(n-1) +Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 20 2024

LinearRecurrence[{15,1}, {0,1}, 31] (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
CoefficientList[Series[x/(1-15*x-x^2), {x,0,50}], x] (* G. C. Greubel, Apr 16 2017 *)

my(x='x+O('x^50)); concat([0], Vec(x/(1-15*x-x^2))) \\ G. C. Greubel, Apr 16 2017

def A154597(n): return (-i)^(n-1)*chebyshev_U(n-1, 15*i/2)
[A154597(n) for n in range(31)] # G. C. Greubel, Sep 20 2024

G.f.: x/(1 - 15*x - x^2). - Klaus Brockhaus, Jan 12 2009, corrected Oct 07 2009
a(n) = ((15 + sqrt(229))^n - (15 - sqrt(229))^n)/(2^n*sqrt(229)).
From Johannes W. Meijer, Jun 12 2010: (Start)
Limit_{k -> infinity} a(n+k)/a(k) = (A090301(n) + a(n)*sqrt(229))/2.
Limit_{n -> infinity} A090301(n)/a(n) = sqrt(229).
a(2n+1) = 15*A098245(n-1).
a(3n+1) = A041427(5n), a(3n+2) = A041427(5n+3), a(3n+3) = 2*A041427(5n+4). (End)
E.g.f.: (2/sqrt(229))*exp(15*x/2)*sinh(sqrt(229)*x/2). - G. C. Greubel, Sep 20 2024

Extended beyond a(7) by Klaus Brockhaus and Philippe Deléham, Jan 12 2009
Name from Philippe Deléham, Jan 12 2009
Edited by Klaus Brockhaus, Oct 07 2009
Missing a(0) added by Jianing Song, Jan 29 2019

A040020 Continued fraction for sqrt(26).

Original entry on oeis.org

5, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane

			5.09901951359278483002822... = 5 + 1/(10 + 1/(10 + 1/(10 + 1/(10 + ...)))). - _Harry J. Smith_, Jun 03 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010481 (decimal expansion), A041040/A041041 (convergents), A248253 (Egyptian fraction).

Cf. A040000, A010692.

numtheory[cfrac](sqrt(26), 100, 'quotients'); # obsolete code updated by Alois P. Heinz, Feb 24 2018

ContinuedFraction[Sqrt[26],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{5},120,{10}] (* Harvey P. Dale, Apr 22 2021 *)

contfrac(sqrt(26)) \\ Harry J. Smith, Jun 03 2009 [Edited by M. F. Hasler, Feb 24 2018]

A040020(n)=if(n,10,5) \\ M. F. Hasler, Feb 24 2018

From Elmo R. Oliveira, Feb 06 2024: (Start)
a(n) = 10 for n >= 1.
G.f.: 5*(1+x)/(1-x).
E.g.f.: 10*exp(x) - 5.
a(n) = 5*A040000(n). (End)

A041040 Numerators of continued fraction convergents to sqrt(26).

Original entry on oeis.org

5, 51, 515, 5201, 52525, 530451, 5357035, 54100801, 546365045, 5517751251, 55723877555, 562756526801, 5683289145565, 57395647982451, 579639768970075, 5853793337683201, 59117573145802085, 597029524795704051

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,1).

Cf. A010481, A041041.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[26],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
CoefficientList[Series[(5 + x)/(1 - 10 x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

From Philippe Deléham, Nov 20 2008: (Start)
a(n) = 10*a(n-1) + a(n-2), a(0)=5, a(1)=51.
G.f.: (5+x)/(1-10*x-x^2). (End)

A165293 Inverse of A038303, and generalization of A130595.

Original entry on oeis.org

1, 10, -1, 100, -20, 1, 1000, -300, 30, -1, 10000, -4000, 600, -40, 1, 100000, -50000, 10000, -1000, 50, -1, 1000000, -600000, 150000, -20000, 1500, -60, 1, 10000000, -7000000, 2100000, -350000, 35000, -2100, 70

Offset: 1

Mark Dols, Sep 13 2009

Rows sum up to A001019 (powers of 9), diagonals to A004189, a generalization of A010892 (the inverse Fibonacci). Ratio of diagonal sums converges to a decimal sequence: A000108 (Catalan numbers), which is the squared difference of sqrt(2) and sqrt(3), or 5-sqrt(24). Ratio between first binomial transform (A054320 and A138288)of A004189, converges to sqrt(2/3). 1/(2*sqrt(24)) gives A000984 (central binomial coefficients) as a decimal sequence.

Triangle T(n,k), read by rows, given by [10,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009

			Triangle begins:
      1;
     10,    -1;
    100,   -20,   1;
   1000,  -300,  30,  -1;
  10000, -4000, 600, -40, 1;

Cf. A007318, A130595, A038303, A004189, A010892, A001079, A054320, A138288, A041041, A000108.

Sum_{k=0..n} T(n,k)*x^k = (10-x)^n. - Philippe Deléham, Dec 15 2009

G.f.: x*y/(1-10*x+x*y). - R. J. Mathar, Aug 11 2015

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A174768 y-values in the solution to x^2 - 26*y^2 = 1.

Original entry on oeis.org

0, 10, 1020, 104030, 10610040, 1082120050, 110365635060, 11256212656070, 1148023325284080, 117087122966320090, 11941738519239365100, 1217940241839448920110, 124217962929104550486120, 12669014278526824700664130, 1292115238446807014917255140, 131783085307295788696859360150

Offset: 1

Vincenzo Librandi, Apr 14 2010

The corresponding values of x of this Pell equation are in A099397.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (102,-1).

Cf. A041041, A099397.

I:=[0, 10]; [n le 2 select I[n] else 102*Self(n-1)-Self(n-2): n in [1..20]];

LinearRecurrence[{102,-1}, {0,10}, 30]
a[ n_] := Fibonacci[2*n-2, 10]; (* Michael Somos, Oct 25 2022 *)

a(n) = 102*a(n-1)-a(n-2) with a(1)=0, a(2)=10.
G.f.: 10*x^2/(1-102*x+x^2).
a(n+1) = A041041(2*n-1). - Michael Somos, Oct 25 2022

A317055 Triangle read by rows: T(0,0) = 1; T(n,k) = 10*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 10, 100, 1, 1000, 20, 10000, 300, 1, 100000, 4000, 30, 1000000, 50000, 600, 1, 10000000, 600000, 10000, 40, 100000000, 7000000, 150000, 1000, 1, 1000000000, 80000000, 2100000, 20000, 50, 10000000000, 900000000, 28000000, 350000, 1500, 1, 100000000000, 10000000000, 360000000, 5600000, 35000, 60

Offset: 0

Zagros Lalo, Jul 21 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013617 ((1+10*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038303 ((10+x)^n).
The coefficients in the expansion of 1/(1-10*x-x^2) are given by the sequence generated by the row sums.
The row sums are Denominators of continued fraction convergents to sqrt(26), see A041041.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 10.09901951359278483002... (a metallic mean) when n approaches infinity (see A176537: (5+sqrt(26))).

			Triangle begins:
  1;
  10;
  100, 1;
  1000, 20;
  10000, 300, 1;
  100000, 4000, 30;
  1000000, 50000, 600, 1;
  10000000, 600000, 10000, 40;
  100000000, 7000000, 150000, 1000, 1;
  1000000000, 80000000, 2100000, 20000, 50;
  10000000000, 900000000, 28000000, 350000, 1500, 1;
  100000000000, 10000000000, 360000000, 5600000, 35000, 60;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 102

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 10x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (10 + x)^n

Row sums give A041041.
Cf. A013617, A038303, A176537.
Cf. A011557 (column 0), A053541 (column 1), A081140 (column 2).

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 10 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

A109107 a(n) = (1/sqrt(26))((5+sqrt(26))^(n+1)-(5-sqrt(26))^(n+1)).

Original entry on oeis.org

2, 20, 202, 2040, 20602, 208060, 2101202, 21220080, 214302002, 2164240100, 21856703002, 220731270120, 2229169404202, 22512425312140, 227353422525602, 2296046650568160, 23187819928207202, 234174245932640180

Offset: 0

Emeric Deutsch, Jun 19 2005

nonn

a(n) = 2*A041041(n). Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 284, K{Q(n)}).

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10, 1).

Cf. A041041.

a:=n->(1/sqrt(26))*((5+sqrt(26))^(n+1)-(5-sqrt(26))^(n+1)): seq(expand(a(n)),n=0..20);

G.f.: 2/(1-10z-z^2).

A109108 a(n) = 10a(n-1) + a(n-2), a(0)=1, a(1)=9.

Original entry on oeis.org

1, 9, 91, 919, 9281, 93729, 946571, 9559439, 96540961, 974969049, 9846231451, 99437283559, 1004219067041, 10141627953969, 102420498606731, 1034346614021279, 10445886638819521, 105493213002216489

Offset: 0

Emeric Deutsch, Jun 19 2005

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 284, K{Q_1(n)}).

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,1).

First differences of A041041.

a:=n->(1/2/sqrt(26))*((sqrt(26)+4)*(5+sqrt(26))^n+(sqrt(26)-4)*(5-sqrt(26))^n): seq(expand(a(n)),n=0..20);

LinearRecurrence[{10,1},{1,9},20] (* Harvey P. Dale, Jan 04 2024 *)

a(n) = (1/2/sqrt(26))((sqrt(26)+4)(5+sqrt(26))^n+(sqrt(26)-4)(5-sqrt(26))^n).

G.f.: (1-z)/(1-10z-z^2).

A305534 Index of the smallest prime in the n-Fibonacci sequence, or the Lucas U(n,-1) sequence.

Original entry on oeis.org

3, 2, 2, 3, 2, 3, 2, 5, 29, 3, 2, 5, 2, 3, 23, 3, 2, 7, 2, 3, 29, 19, 2, 3, 83, 3, 53, 19, 2, 5, 2, 5, 5, 5479, 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5

Offset: 1

Eric Chen, Jun 04 2018

Smallest k such that the k-th Fibonacci polynomial evaluated at x=n is prime. (The first few Fibonacci polynomials are 1, x, x^2 + 1, x^3 + 2*x, x^4 + 3*x^2 + 1, x^5 + 4*x^3 + 3*x, ...)
All terms are primes, since if a divides b, then the a-th term of the n-Fibonacci sequence also divides the b-th term of the n-Fibonacci sequence.
Corresponding primes are 2, 2, 3, 17, 5, 37, 7, 4289, 726120289954448054047428229, 101, 11, 21169, 13, 197, 82088569942721142820383601, 257, 17, 34539049, 19, 401, ...
a(n) = 2 if and only if n is prime.
a(n) = 3 if and only if n^2 + 1 is prime (A005574), except n=2 (since 2 is the only prime p such that p^2 + 1 is also prime).
a(34) > 1024, does a(n) exist for all n >= 1? (However, 17 is the only prime in the first 1024 terms of the 4-Fibonacci sequence, and it seems that 17 is the only prime in the 4-Fibonacci sequence.)
a(35)..a(48) = 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5, a(50)..a(54) = 11, 11, 23, 2, 3, a(56) = 3, a(58)..a(75) = 5, 2, 47, 2, 5, 311, 13, 233, 3, 2, 5, 11, 5, 2, 7, 2, 3, 5. Unknown terms a(34), a(49), a(55), a(57), exceed 1024, if they exist.
a(49) > 20000, if it exists. - Giovanni Resta, Jun 06 2018

Cf. A001605, A096650, A209493, which are the indices of the primes in the n-Fibonacci sequence for n = 1, 2, 3.
Cf. A005478, A086383, A201001, which are the primes in the n-Fibonacci sequence for n = 1, 2, 3.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061 (the n-Fibonacci sequence for n = 1 to 12).
Cf. A302990 (for n-step Fibonacci sequence instead of n-Fibonacci sequence).

b(n,k)=([n,1;1,0]^k)[1,2]
a(n)=for(k=1,2^12,if(ispseudoprime(b(n,k)),return(k)))

a(34)-a(48) from Giovanni Resta, Jun 06 2018

cp's OEIS Frontend

A154597 a(n) = 15*a(n-1) + a(n-2) with a(0) = 0, a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A040020 Continued fraction for sqrt(26).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A041040 Numerators of continued fraction convergents to sqrt(26).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A165293 Inverse of A038303, and generalization of A130595.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A174768 y-values in the solution to x^2 - 26*y^2 = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A317055 Triangle read by rows: T(0,0) = 1; T(n,k) = 10*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.