A100545 -id:A100545 - cp's OEIS frontend

A054486 Expansion of (1+2x)/(1-3x+x^2).

1, 5, 14, 37, 97, 254, 665, 1741, 4558, 11933, 31241, 81790, 214129, 560597, 1467662, 3842389, 10059505, 26336126, 68948873, 180510493, 472582606, 1237237325, 3239129369, 8480150782, 22201322977, 58123818149, 152170131470, 398386576261, 1042989597313

Offset: 0

Barry E. Williams, May 06 2000

Binomial transform of A000285. - R. J. Mathar, Oct 26 2011

			G.f. = 1 + 5*x + 14*x^2 + 37*x^3 + 97*x^4 + 254*x^5 + 665*x^6 + 1741*x^7 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A000045, A000285, A002878, A100545, A104449, A228208.

F:=Fibonacci;; List([0..30], n-> F(2*n+2) +2*F(2*n) ); # G. C. Greubel, Nov 08 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+2*x)/(1-3*x+x^2)) ); // Marius A. Burtea, Nov 05 2019

a:=[1,5]; [n le 2 select a[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // Marius A. Burtea, Nov 05 2019

with(combinat); f:=fibonacci; seq(f(2*n+2)+2*f(2*n), n=0..30); # G. C. Greubel, Nov 08 2019

CoefficientList[Series[(2*z+1)/(z^2-3*z+1), {z, 0, 30}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)
a[ n_]:= 3 Fibonacci[2n] + Fibonacci[2n+1]; (* Michael Somos, Mar 17 2015 *)
LinearRecurrence[{3,-1},{1,5},40] (* Harvey P. Dale, Apr 24 2019 *)

Vec((1+2*x)/(1-3*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 15 2011

{a(n) = 3*fibonacci(2*n) + fibonacci(2*n+1)}; /* Michael Somos, Mar 17 2015 */

f=fibonacci; [f(2*n+2) +2*f(2*n) for n in (0..30)] # G. C. Greubel, Nov 08 2019

a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=5.
a(n) = (5*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - (((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)))/sqrt(5).
a(n) + 7*A001519(n) = A005248(n). - Creighton Dement, Oct 30 2004
a(n) = Lucas(2*n+1) + Fibonacci(2*n) = A002878(n) + A001906(n) = A025169(n-1) + A001906(n+1).
a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-6)^k. - Philippe Deléham, Mar 05 2014
0 = -11 + a(n)^2 - 3*a(n)*a(n+1) + a(n+1)^2 for all n in Z. - Michael Somos, Mar 17 2015
a(n) = -2*F(n)^2 + 6*F(n)*F(n+1) + F(n+1)^2 for all n in Z where F = Fibonacci. - Michael Somos, Mar 17 2015
a(n) = 3*F(2*n) + F(2*n+1) for all n in Z where F = Fibonacci. - Michael Somos, Mar 17 2015
a(n) = -A100545(-2-n) for all n in Z. - Michael Somos, Mar 17 2015
a(n) = A000285(2*n) = A228208(2*n+1) = A104449(2*n+1) for all n in Z. - Michael Somos, Mar 17 2015
From Klaus Purath, Nov 05 2019: (Start)
a(n) = (a(n-m) + a(n+m))/Lucas(2*m), m <= n.
a(n) = sum of 2*m+1 consecutive terms starting with a(n-m) divided by Lucas(2*m+1), m <= n.
a(n) = alternating sum of 2*m+1 consecutive terms starting with a(n-m) divided by Fibonacci(2*m+1), m <= n.
a(n) + a(n+1) = sum of 2*m+2 consecutive terms starting with a(n-m) divided by Fibonacci(2*m+2), m <= n.
a(n) + a(n+1) = (a(n-m) + a(n+m+1))/Fibonacci(2*m+1), m <= n.
The following formulas are extended to negative indexes:
a(n) = 3*Fibonacci(2*n+1) - Fibonacci(2*n-3).
a(n) = (Fibonacci(2*n+5) - 3* Fibonacci(2*n-1))/2.
a(n) = (4*Lucas(2*n+2) - Lucas(2*n-4))/5.
a(n) = Fibonacci(2*n+5) - 4*Fibonacci(2*n+1).
a(n) = (5*Fibonacci(2*n+5) - Fibonacci(2*n-7))/12. (End)
E.g.f.: exp(-(1/2)*(-3+sqrt(5))*x)*(-7 + sqrt(5) + (7 + sqrt(5))*exp(sqrt(5)*x))/(2*sqrt(5)). - Stefano Spezia, Nov 19 2019
a(n) = 3*n + 1 + Sum_{k=1..n} k*a(n-k). - Yu Xiao, Jun 20 2020

"a(1)=5", not "a(0)=5" from Dan Nielsen (nielsed(AT)uah.edu), Sep 10 2009

A244472 2nd-largest term in n-th row of Stern's diatomic triangle A002487.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804, 6155, 9959, 16114, 26073, 42187, 68260, 110447, 178707, 289154, 467861, 757015, 1224876, 1981891, 3206767, 5188658, 8395425, 13584083, 21979508, 35563591, 57543099

Offset: 1

N. J. A. Sloane, Jul 01 2014

Colin Barker, Table of n, a(n) for n = 1..1000
Jennifer Lansing, Largest Values for the Stern Sequence, J. Integer Seqs., 17 (2014), #14.7.5.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A002487, A013655, A100545 (bisection).

Cf. A244473, A244474, A244475, A244476.

I:=[1, 2, 4, 7, 12]; [n le 5 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Wesley Ivan Hurt, Jul 10 2015

A244472 := proc(n)
    if n < 4 then
        op(n,[1,2,4]) ;
    else
        combinat[fibonacci](n+2)-combinat[fibonacci](n-3) ;
    end if;
end proc:
seq(A244472(n),n=1..50) ; # R. J. Mathar, Jul 05 2014

CoefficientList[Series[-(x^4 + x^3 + x^2 + x + 1)/(x^2 + x - 1), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jul 10 2015 *)
Join[{1, 2, 4}, LinearRecurrence[{1, 1}, {7, 12}, 50]] (* Vincenzo Librandi, Jul 11 2015 *)

Vec(-x*(x^4+x^3+x^2+x+1)/(x^2+x-1) + O(x^100)) \\ Colin Barker, Jul 10 2015

a(n) = A013655(n-1), n>3.
a(n) = a(n-1)+a(n-2), n>5. - Colin Barker, Jul 10 2015
G.f.: -x*(x^4+x^3+x^2+x+1) / (x^2+x-1). - Colin Barker, Jul 10 2015

A271357 a(n) = kFibonacci(2n+1) + (k+1)Fibonacci(2n), where k=3.

Original entry on oeis.org

3, 10, 27, 71, 186, 487, 1275, 3338, 8739, 22879, 59898, 156815, 410547, 1074826, 2813931, 7366967, 19286970, 50493943, 132194859, 346090634, 906077043, 2372140495, 6210344442, 16258892831, 42566334051, 111440109322, 291753993915, 763821872423, 1999711623354

Offset: 0

Colin Barker, Apr 05 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000045.

Cf. A001906 (k=0), A002878 (k=1), A100545 (k=2, without the initial 2), this sequence (k=3), A271358 (k=4), A271359 (k=5).

k:=3; [k*Fibonacci(2*n+1)+(k+1)*Fibonacci(2*n): n in [0..30]]; // Bruno Berselli, Apr 06 2016

Table[3Fibonacci[2n+1]+4Fibonacci[2n],{n,0,30}] (* or *) LinearRecurrence[ {3,-1},{3,10},30] (* Harvey P. Dale, Apr 05 2019 *)

a(n) = 3*fibonacci(2*n+1) + 4*fibonacci(2*n)

PARI
```
Vec((3+x)/(1-3*x+x^2) + O(x^50))
```

G.f.: (3+x) / (1-3*x+x^2).
a(n) = 3*a(n-1)-a(n-2) for n>1.
a(n) = (2^(-2-n)*((9-sqrt(5))*(3+sqrt(5))^(n+1) - (9+sqrt(5))*(3-sqrt(5))^(n+1))) / sqrt(5).
a(n) = 4*Fibonacci(2*n+2) - Fibonacci(2*n+1).

Changed offset and adapted definition, programs and formulas by Bruno Berselli, Apr 06 2016

A271358 a(n) = kFibonacci(2n+1) + (k+1)Fibonacci(2n), where k=4.

Original entry on oeis.org

4, 13, 35, 92, 241, 631, 1652, 4325, 11323, 29644, 77609, 203183, 531940, 1392637, 3645971, 9545276, 24989857, 65424295, 171283028, 448424789, 1173991339, 3073549228, 8046656345, 21066419807, 55152603076, 144391389421, 378021565187, 989673306140

Offset: 0

Colin Barker, Apr 05 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. On the cyclically fully commutative elements of Coxeter groups, J. Algebr. Comb. 36, No. 1, 123-148 (2012) Table 1 CFC Type D.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000045.

Cf. A001906 (k=0), A002878 (k=1), A100545 (k=2, without the initial 2), A271357 (k=3), this sequence (k=4), A271359 (k=5).

k:=4; [k*Fibonacci(2*n+1)+(k+1)*Fibonacci(2*n): n in [0..30]]; // Bruno Berselli, Apr 06 2016

a(n) = 4*fibonacci(2*n+1) + 5*fibonacci(2*n)

PARI
```
Vec((4+x)/(1-3*x+x^2) + O(x^50))
```

G.f.: (4+x) / (1-3*x+x^2).
a(n) = 3*a(n-1)-a(n-2) for n>1.
a(n) = (2^(-2-n)*((11-sqrt(5))*(3+sqrt(5))^(n+1) - (11+sqrt(5))*(3-sqrt(5))^(n+1))) / sqrt(5).
a(n) = 5*Fibonacci(2*n+2) - Fibonacci(2*n+1).
a(n) = 4*A001906(n+1) + A001906(n-1).

Changed offset and adapted definition, programs and formulas by Bruno Berselli, Apr 06 2016

A271359 a(n) = kFibonacci(2n+1) + (k+1)Fibonacci(2n), where k=5.

Original entry on oeis.org

5, 16, 43, 113, 296, 775, 2029, 5312, 13907, 36409, 95320, 249551, 653333, 1710448, 4478011, 11723585, 30692744, 80354647, 210371197, 550758944, 1441905635, 3774957961, 9882968248, 25873946783, 67738872101, 177342669520, 464289136459, 1215524739857

Offset: 0

Colin Barker, Apr 05 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000045.

Cf. A001906 (k=0), A002878 (k=1), A100545 (k=2, without the initial 2), A271357 (k=3), A271358 (k=4), this sequence (k=5).

k:=5; [k*Fibonacci(2*n+1)+(k+1)*Fibonacci(2*n): n in [0..30]]; // Bruno Berselli, Apr 06 2016

a(n) = 5*fibonacci(2*n+1) + 6*fibonacci(2*n)

PARI
```
Vec((5+x)/(1-3*x+x^2) + O(x^50))
```

G.f.: (5+x) / (1-3*x+x^2).
a(n) = 3*a(n-1)-a(n-2) for n>1.
a(n) = (2^(-2-n)*((13-sqrt(5))*(3+sqrt(5))^(n+1) - (13+sqrt(5))*(3-sqrt(5))^(n+1))) / sqrt(5).
a(n) = 6*Fibonacci(2*n+2) - Fibonacci(2*n+1) = 5*A001906(n+1) +A001906(n).

Changed offset and adapted definition, programs and formulas by Bruno Berselli, Apr 06 2016

A102714 Expansion of (x+2) / ((x+1)(x^2-3x+1)).

Original entry on oeis.org

2, 5, 14, 36, 95, 248, 650, 1701, 4454, 11660, 30527, 79920, 209234, 547781, 1434110, 3754548, 9829535, 25734056, 67372634, 176383845, 461778902, 1208952860, 3165079679, 8286286176, 21693778850, 56795050373, 148691372270, 389279066436, 1019145827039

Offset: 0

Creighton Dement, Feb 06 2005

A floretion-generated sequence relating Fibonacci numbers.

Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 2dia[I]forseq[ + .5'i + .5'ii' + .5'ij' + .5'ik' ], 2dia[J]forseq = 2dia[K]forseq = A001654, mixforseq = A001519, tesforseq = A099016, vesforseq = A000004. Identity used: dia[I] + dia[J] + dia[K] + mix + tes = ves

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A001519, A099016, A001654, A100545, A055849, A000004.

CoefficientList[Series[(x+2)/((x+1)(x^2-3x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-1},{2,5,14},30] (* Harvey P. Dale, Apr 22 2012 *)

a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

Vec((x+2)/((x+1)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), a(0) = 2, a(1) = 5, a(2) = 14.
a(n) + a(n+1) = A100545(n).
a(n) + 2*a(n+1) + a(n+2) = A055849(n+2).
a(n) + 2*A001654(n) - A099016(n+2) + 2*A001519(n) = 0.
a(n) = (2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5. - Colin Barker, Oct 01 2016
a(n) = (-1)^n +9*A001906(n+1) -A001906(n) . - R. J. Mathar, Sep 11 2019

Corrected by T. D. Noe, Nov 02 2006, Nov 07 2006

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A054486 Expansion of (1+2*x)/(1-3*x+x^2).

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A244472 2nd-largest term in n-th row of Stern's diatomic triangle A002487.

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A271357 a(n) = k*Fibonacci(2*n+1) + (k+1)*Fibonacci(2*n), where k=3.

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A271358 a(n) = k*Fibonacci(2*n+1) + (k+1)*Fibonacci(2*n), where k=4.

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A271359 a(n) = k*Fibonacci(2*n+1) + (k+1)*Fibonacci(2*n), where k=5.

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A102714 Expansion of (x+2) / ((x+1)*(x^2-3*x+1)).

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A054486 Expansion of (1+2x)/(1-3x+x^2).

A271357 a(n) = kFibonacci(2n+1) + (k+1)Fibonacci(2n), where k=3.

A271358 a(n) = kFibonacci(2n+1) + (k+1)Fibonacci(2n), where k=4.

A271359 a(n) = kFibonacci(2n+1) + (k+1)Fibonacci(2n), where k=5.

A102714 Expansion of (x+2) / ((x+1)(x^2-3x+1)).