A103435 -id:A103435 - cp's OEIS frontend

A189749 a(1)=5, a(2)=5, a(n)=5a(n-1) + 5a(n-2).

5, 5, 50, 275, 1625, 9500, 55625, 325625, 1906250, 11159375, 65328125, 382437500, 2238828125, 13106328125, 76725781250, 449160546875, 2629431640625, 15392960937500, 90111962890625, 527524619140625, 3088182910156250, 18078537646484375, 105833602783203125

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748.

Mathematica
```
LinearRecurrence[{5,5},{5,5},40]
```

a[1]:5$ a[2]:5$ a[n]:=5*a[n-1]+5*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: 5*x*(1-4*x)/(1-5*x-5*x^2). - Bruno Berselli, May 24 2011

a(n) = 5*A188168(n). - R. J. Mathar, Feb 13 2020

A189737 a(1)=3, a(2)=3, a(n)=3a(n-1) + 3a(n-2).

Original entry on oeis.org

3, 3, 18, 63, 243, 918, 3483, 13203, 50058, 189783, 719523, 2727918, 10342323, 39210723, 148659138, 563609583, 2136806163, 8101247238, 30714160203, 116446222323, 441481147578, 1673782109703, 6345789771843, 24058715644638, 91213516249443, 345816695682243

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{3,3},{3,3},40]
```

a[1]:3$ a[2]:3$ a[n]:=3*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 26); /* Bruno Berselli, May 24 2011 */

G.f.: 3*x*(1-2*x)/(1-3*x-3*x^2). - Bruno Berselli, May 24 2011

A189746 a(1)=5, a(2)=2, a(n) = 5a(n-1) + 2a(n-2).

Original entry on oeis.org

5, 2, 20, 104, 560, 3008, 16160, 86816, 466400, 2505632, 13460960, 72316064, 388502240, 2087143328, 11212721120, 60237892256, 323614903520, 1738550302112, 9339981317600, 50177007192224, 269564998596320, 1448179007366048, 7780025034022880, 41796483184846496

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,2).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{5,2},{5,2},40]
```

a[1]:5$ a[2]:2$ a[n]:=5*a[n-1]+2*a[n-2]$ makelist(a[n], n, 1, 24); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-23*x)/(1-5*x-2*x^2). - Bruno Berselli, May 24 2011

A189747 a(1)=5, a(2)=3, a(n)=5a(n-1) + 3a(n-2).

Original entry on oeis.org

5, 3, 30, 159, 885, 4902, 27165, 150531, 834150, 4622343, 25614165, 141937854, 786531765, 4358472387, 24151957230, 133835203311, 741631888245, 4109665051158, 22773220920525, 126195099756099, 699295161542070, 3875061106978647, 21473191019519445

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,3).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189748, A189749.

Mathematica
```
LinearRecurrence[{5,3},{5,3},40]
```

a[1]:5$ a[2]:3$ a[n]:=5*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-22*x)/(1-5*x-3*x^2). - Bruno Berselli, May 24 2011

A189748 a(n) = 5a(n-1) + 4a(n-2) with a(1)=5, a(2)=4.

Original entry on oeis.org

5, 4, 40, 216, 1240, 7064, 40280, 229656, 1309400, 7465624, 42565720, 242691096, 1383718360, 7889356184, 44981654360, 256465696536, 1462255100120, 8337138286744, 47534711834200, 271022112317976, 1545249408926680, 8810335493905304, 50232675105233240

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,4).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189749.

Mathematica
```
LinearRecurrence[{5,4},{5,4},40]
```

a[1]:5$ a[2]:4$ a[n]:=5*a[n-1]+4*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: x*(5-21*x)/(1-5*x-4*x^2). - Bruno Berselli, May 24 2011

A014334 Exponential convolution of Fibonacci numbers with themselves.

Original entry on oeis.org

0, 0, 2, 6, 22, 70, 230, 742, 2406, 7782, 25190, 81510, 263782, 853606, 2762342, 8939110, 28927590, 93611622, 302933606, 980313702, 3172361830, 10265978470, 33221404262, 107506722406, 347899061862, 1125825013350, 3643246274150, 11789792601702

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. A. Church and Marjorie Bicknell, Exponential generating functions for Fibonacci identities, Fibonacci Quarterly 11, no. 3 (1973), 275-281.
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics, Vol. 26, No. 3 (2020), pp. 96-106.
Helmut Prodinger, Convolution identities for Tribonacci numbers via the diagonal of a bivariate generating function, arXiv:1910.08323 [math.NT], 2019.
Charles R. Wall, Problem B-573, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 24, No. 2 (1986), p. 181; Solution to Problem B-573 by Bob Prielipp, ibid., Vol. 25, No. 2 (1987), p. 184.
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Cf. A000032, A000045, A014335, A103435.

[(2^n*Lucas(n) -2)/5: n in [0..40]]; // Vincenzo Librandi, Jul 15 2018

LinearRecurrence[{3,2,-4}, {0,0,2}, 30] (* Harvey P. Dale, Oct 24 2015 *)
Table[(2^n LucasL[n] -2)/5, {n,0,100}] (* Vladimir Reshetnikov, May 18 2016 *)

a(n)=if(n<1,0,sum(k=0,n-1,fibonacci(k)*2^k))

[(2^n*lucas_number2(n,1,-1) -2)/5 for n in range(41)] # G. C. Greubel, Jan 06 2023

From Benoit Cloitre, May 29 2003: (Start)
a(n) = 3*a(n-1) + 2*a(n-2) - 4*a(n-3), a(0)=0, a(1)=0, a(2)=2.
a(n) = Sum_{k=0..n-1} 2^k*Fibonacci(k) for n > 0.
a(n) = (-2 + ((1+sqrt(5))^n + (1-sqrt(5))^n))/5. (End)
a(n) = Sum_{k=0..n} Fibonacci(k)*Fibonacci(n-k)*binomial(n, k). - Benoit Cloitre, May 11 2005
From R. J. Mathar, Sep 29 2010: (Start)
a(n) = 2*A014335(n).
G.f.: 2*x^2/((1-x)*(1-2*x-4*x^2)).
a(n) = Sum_{k=1..n-1} A103435(k). (End)
a(n) = (2^n*A000032(n) - 2)/5. - Vladimir Reshetnikov, May 18 2016
E.g.f.: 2*(cosh(sqrt(5)*x)-1)*exp(x)/5. - Ilya Gutkovskiy, May 18 2016
a(n) = ((Sum_{k=0..n} Lucas(k)*Lucas(n-k)*binomial(n, k)) - 4)/5 (Wall, 1986). - Amiram Eldar, Jan 27 2022

A006483 a(n) = Fibonacci(n)*2^n + 1.

Original entry on oeis.org

1, 3, 5, 17, 49, 161, 513, 1665, 5377, 17409, 56321, 182273, 589825, 1908737, 6176769, 19988481, 64684033, 209321985, 677380097, 2192048129, 7093616641, 22955425793, 74285318145, 240392339457, 777925951489, 2517421260801, 8146546327553, 26362777698305

Offset: 0

Dennis S. Kluk (mathemagician(AT)ameritech.net)

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979.
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
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Cf. A000045, A063727, A087206.

Equals A103435 + 1.

[Fibonacci(n)*2^n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 09 2013

A006483:=-(-1+6*z**2)/(z-1)/(4*z**2+2*z-1); # Simon Plouffe in his 1992 dissertation

lst={};Do[AppendTo[lst, Fibonacci[n]*2^n+1], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
CoefficientList[Series[(-(- 1 + 6 x^2)) / ((1 - x) (1 - 2 x - 4 x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
LinearRecurrence[{3,2,-4},{1,3,5},40] (* Harvey P. Dale, Aug 01 2021 *)

G.f.: -(-1+6*x^2)/((1-x)*(1-2*x-4*x^2)).

G.f. in Formula field corrected by Vincenzo Librandi, Jun 09 2013

A269991 Decimal expansion of Sum_{n >= 1} 2^(1-n)/Fibonacci(n).

Original entry on oeis.org

1, 6, 8, 4, 8, 1, 3, 1, 4, 4, 4, 8, 9, 5, 7, 6, 0, 9, 6, 3, 1, 6, 5, 5, 4, 3, 3, 7, 3, 8, 0, 0, 7, 8, 2, 3, 0, 2, 3, 7, 0, 6, 3, 8, 8, 2, 4, 5, 7, 0, 8, 6, 8, 2, 0, 9, 4, 3, 1, 7, 6, 1, 8, 8, 5, 9, 5, 0, 5, 6, 0, 0, 2, 8, 0, 4, 9, 4, 5, 4, 9, 8, 9, 1, 0, 8

Offset: 1

Clark Kimberling, Mar 12 2016

			1.684813144489576096316554337380078230...

Cf. A000045, A269992.

Cf. A063727, A085449, A103435, A209084.

x = N[Sum[2^(1 - n)/Fibonacci[n], {n, 1, 500}], 100]
RealDigits[x][[1]]

suminf(n=1, 2^(1-n)/fibonacci(n)) \\ Michel Marcus, Feb 01 2021

Equals Sum_{n>=0} 1/A063727(n) = Sum_{n>=1} 1/A085449(n) = 2 * Sum_{n>=1} 1/A103435(n) = 4 * Sum_{n>=1} 1/A209084(n). - Amiram Eldar, Feb 01 2021

A086344 a(n) = -2a(n-1) + 4a(n-2), a(0) = 1, a(1) = 0.

Original entry on oeis.org

1, 0, 4, -8, 32, -96, 320, -1024, 3328, -10752, 34816, -112640, 364544, -1179648, 3817472, -12353536, 39976960, -129368064, 418643968, -1354760192, 4384096256, -14187233280, 45910851584, -148570636288, 480784678912, -1555851902976, 5034842521600, -16293092655104

Offset: 0

Paul Barry, Jul 17 2003

Inverse binomial transform of (1,1,5,5,25,25,.....).

The absolute values are the constant terms of the reduction by x^2->x+1 of the polynomial p(n,x) given for d=sqrt(x+1) by p(n,x)=((x+d)^n-(x-d)^n)/(2d), for n>=1. The coefficient of x under this reduction is given by A103435. See A192232 for a discussion of reduction. - Clark Kimberling, Jun 29 2011

Index entries for linear recurrences with constant coefficients, signature (-2,4).

seq((-2)^n * combinat:-fibonacci(n-1), n = 0 .. 100); # Robert Israel, Oct 02 2014

LinearRecurrence[{-2,4},{1,0},40] (* Harvey P. Dale, Oct 10 2018 *)

G.f.: (1+2*x)/((1+(1+sqrt(5))*x)(1+(1-sqrt(5))*x)) = ( -1-2*x ) / ( -1-2*x+4*x^2 ).
E.g.f.: exp(-x)*(cosh(sqrt(5)*x)+sinh(sqrt(5)*x)/sqrt(5)).
a(n)=(sqrt(5)-1)^n*(sqrt(5)/10+1/2)+(-sqrt(5)-1)^n*(1/2-sqrt(5)/10).
(-1)^n*a(n) = A063727(n) - 2*A063727(n-1). - R. J. Mathar, Jul 19 2012
(-1)^n*a(n) = sum(k=0..n, binomial(n,k)*(F(n+1)-F(n))), F(n) Fibonacci number A000045. - Peter Luschny, Oct 01 2014
a(n) = (-2)^n *A000045(n-1). - Robert Israel, Oct 02 2014

A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.

Original entry on oeis.org

0, 4, 8, 32, 96, 320, 1024, 3328, 10752, 34816, 112640, 364544, 1179648, 3817472, 12353536, 39976960, 129368064, 418643968, 1354760192, 4384096256, 14187233280, 45910851584, 148570636288, 480784678912, 1555851902976, 5034842521600, 16293092655104

Offset: 0

Seiichi Kirikami, Mar 06 2012

a(n)/A063727(n) are convergents for A134972.

Abs(Sum_{i=0..n} C(n,n-i)*a(i)-(sqrt(5)-1)* A033887(n))->0. - Seiichi Kirikami, Jan 20 2016

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A063727, A033887, A085449, A103435, A134972, A269991.

Cf. A086344 (this sequence with signs).

I:=[0,4]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 16 2016

RecurrenceTable[{a[n]==2*a[n-1]+4*a[n-2], a[0]==0, a[1]==4}, a, {n, 30}]
LinearRecurrence[{2, 4}, {0, 4}, 40] (* Vincenzo Librandi, Jan 16 2016 *)

concat(0, Vec(4*x/(1-2*x-4*x^2) + O(x^40))) \\ Michel Marcus, Jan 16 2016

a(n) = (2/sqrt(5))*((1+sqrt(5))^n-(1-sqrt(5))^n).
G.f.: 4*x/(1-2*x-4*x^2). - Bruno Berselli, Mar 08 2012
a(n) = 4*A085449(n) = 2*A103435(n). - Bruno Berselli, Mar 08 2012
Sum_{n>=1} 1/a(n) = (1/4) * A269991. - Amiram Eldar, Feb 01 2021

cp's OEIS Frontend

A189749 a(1)=5, a(2)=5, a(n)=5*a(n-1) + 5*a(n-2).

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A189737 a(1)=3, a(2)=3, a(n)=3*a(n-1) + 3*a(n-2).

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A189746 a(1)=5, a(2)=2, a(n) = 5*a(n-1) + 2*a(n-2).

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A189747 a(1)=5, a(2)=3, a(n)=5*a(n-1) + 3*a(n-2).

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A189748 a(n) = 5*a(n-1) + 4*a(n-2) with a(1)=5, a(2)=4.

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Maxima

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A014334 Exponential convolution of Fibonacci numbers with themselves.

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A006483 a(n) = Fibonacci(n)*2^n + 1.

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A269991 Decimal expansion of Sum_{n >= 1} 2^(1-n)/Fibonacci(n).

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A189749 a(1)=5, a(2)=5, a(n)=5a(n-1) + 5a(n-2).

A189737 a(1)=3, a(2)=3, a(n)=3a(n-1) + 3a(n-2).

A189746 a(1)=5, a(2)=2, a(n) = 5a(n-1) + 2a(n-2).

A189747 a(1)=5, a(2)=3, a(n)=5a(n-1) + 3a(n-2).

A189748 a(n) = 5a(n-1) + 4a(n-2) with a(1)=5, a(2)=4.

A086344 a(n) = -2a(n-1) + 4a(n-2), a(0) = 1, a(1) = 0.

A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.