A014551 -id:A014551 - cp's OEIS frontend

A087288 a(n)=2a(n-1)+a(n-2)-2a(n-3).

4, 4, 10, 16, 34, 64, 130, 256, 514, 1024, 2050, 4096, 8194, 16384, 32770, 65536, 131074, 262144, 524290, 1048576, 2097154, 4194304, 8388610, 16777216, 33554434, 67108864, 134217730, 268435456, 536870914, 1073741824, 2147483650

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 31 2003

a(n)=(A000051(n)-A014551(n))/2.

Indranil Ghosh, Table of n, a(n) for n = 0..3314
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

CoefficientList[Series[(4 - 4x - 2x^2)/(1 - 2x - x^2 + 2x^3), {x, 0, 30}], x]

a(n)=2<Charles R Greathouse IV, Aug 20 2013

G.f.: (4-4x-2x^2)/(1-2x-x^2+2x^3). a(n)=2^(n+1)+1+(-1)^n.

A105578 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.

Original entry on oeis.org

1, 1, 0, -1, 0, 3, 4, -1, -8, -5, 12, 23, 0, -45, -44, 47, 136, 43, -228, -313, 144, 771, 484, -1057, -2024, 91, 4140, 3959, -4320, -12237, -3596, 20879, 28072, -13685, -69828, -42457, 97200, 182115, -12284, -376513, -351944, 401083, 1104972, 302807, -1907136, -2512749, 1301524, 6327023, 3723976

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: ibaseiseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-3,2).

Cf. A002249, A014551, A078020, A105577, A105579.

Equals (A107920(n) + 1)/2.

-Join[{-1,-1,a=0,b=1},Table[c=1*b-2*a-1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
LinearRecurrence[{2,-3,2},{1,1,0},50] (* Harvey P. Dale, Mar 28 2019 *)

Vec(-(x^2-x+1)/((x-1)*(2*x^2-x+1)) + O(x^100)) \\ Colin Barker, Feb 08 2015

a(n) - a(n+1) = A001607(n); a(n+2) - 2a(n+1) + a(n) = - A078020(n).

G.f.: -(x^2-x+1) / ((x-1)*(2*x^2-x+1)). - Colin Barker, Feb 08 2015

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 17, 63, 257, 1023, 4097, 16383, 65537, 262143, 1048577, 4194303, 16777217, 67108863, 268435457, 1073741823, 4294967297, 17179869183, 68719476737, 274877906943, 1099511627777, 4398046511103, 17592186044417, 70368744177663, 281474976710657

Offset: 0

Bruno Berselli, Jan 09 2013

This is the Lucas sequence V(3,-4).

Inverse binomial transform of this sequence is A087451.

Bruno Berselli, Table of n, a(n) for n = 0..200
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. for the same recurrence with initial values (i,i+1): A015521 (Lucas sequence U(3,-4); i=0), A122117 (i=1), A189738 (i=3).
Cf. for similar closed form: A014551 (2^n+(-1)^n), A102345 (3^n+(-1)^n), A087404 (5^n+(-1)^n).
Cf. A052539, A024036.

[n le 1 select n+2 else 3*Self(n)+4*Self(n-1): n in [0..25]];

RecurrenceTable[{a[n] == 3 a[n - 1] + 4 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]

a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+4*a[n-2]$ makelist(a[n], n, 0, 25);

Vec((2-3*x)/((1+x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Jun 26 2015

G.f.: (2-3*x)/((1+x)*(1-4*x)).
a(n) = 4^n+(-1)^n.
a(n) = A086341(A047524(n)) for n>0, a(0)=2.
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 25*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = (2/4^n) * Sum_{k = 0..n} binomial(4*n+1, 4*k). - Peter Bala, Feb 06 2019

A083332 a(n) = 10a(n-2) - 16a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.

Original entry on oeis.org

1, 5, 14, 34, 124, 260, 1016, 2056, 8176, 16400, 65504, 131104, 524224, 1048640, 4194176, 8388736, 33554176, 67109120, 268434944, 536871424, 2147482624, 4294968320, 17179867136, 34359740416, 137438949376, 274877911040

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

a(n)/A083333(n) converges to 3.

Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -16).

Cf. A147590, A081342 (bisections). [R. J. Mathar, Jul 13 2009]

Cf. A199710. [Bruno Berselli, Nov 11 2011]

CoefficientList[Series[(1+5x+4x^2-16x^3)/(1-10x^2+16x^4), {x, 0, 30}], x]

(a[0] : 1, a[1] : 5, a[2] : 14, a[3] : 34, a[n] := 10*a[n - 2] - 16*a[n - 4], makelist(a[n], n, 0, 50));/* Franck Maminirina Ramaharo, Nov 12 2018 */

G.f.: (1 + 5*x + 4*x^2 - 16*x^3)/(1 - 10*x^2 + 16*x^4).
a(n) = A016116(n)*A014551(n+1). - R. J. Mathar, Jul 08 2009
From Franck Maminirina Ramaharo, Nov 12 2018: (Start)
a(n) = sqrt(2)^(3*n - 1)*(1 + sqrt(2) + (-1)^n*(sqrt(2) - 1)) + sqrt(2)^(n - 3)*(1 - sqrt(2) - (-1)^n*(sqrt(2) + 1)).
E.g.f.: (sinh(sqrt(2)*x) + 2*sinh(2*sqrt(2)*x))/sqrt(2) - cosh(sqrt(2)*x) + 2*cosh(2*sqrt(2)*x). (End)

A099430 a(n) = 2^n+(-1)^n-1.

Original entry on oeis.org

1, 0, 4, 6, 16, 30, 64, 126, 256, 510, 1024, 2046, 4096, 8190, 16384, 32766, 65536, 131070, 262144, 524286, 1048576, 2097150, 4194304, 8388606, 16777216, 33554430, 67108864, 134217726, 268435456, 536870910, 1073741824, 2147483646

Offset: 0

Paul Barry, Oct 15 2004

Jacobsthal-Lucas numbers less 1.

For n >= 1, a(n) is also the number of periodic points of period n of the Period Doubling and the Thue-Morse Chain. [Natascha Neumaerker (naneumae(AT)math.uni-bielefeld.de), Apr 06 2009]

N. Neumarker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, JIS 12 (2009) 09.4.5, Example 9.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A014551.

LinearRecurrence[{2,1,-2},{1,0,4},40] (* Harvey P. Dale, Nov 07 2017 *)

a(n)=2^n+(-1)^n-1 \\ Charles R Greathouse IV, May 09 2016

G.f.: (1-2x+3x^2)/((1-x)(1-x-2x^2)) = (1-2x+3x^2)/((1-x^2)(1-2x)).
a(n) = A014551(n)-1.
zeta(z) = (1-z)/((1+z)(1-2z)). [Natascha Neumaerker (naneumae(AT)math.uni-bielefeld.de), Apr 06 2009]
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3). - Wesley Ivan Hurt, Jun 09 2023

A105577 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 1, a(1) = 5, a(2) = 6.

Original entry on oeis.org

1, 5, 6, -1, -10, -5, 18, 31, -2, -61, -54, 71, 182, 43, -318, -401, 238, 1043, 570, -1513, -2650, 379, 5682, 4927, -6434, -16285, -3414, 29159, 35990, -22325, -94302, -49649, 138958, 238259, -39654, -516169, -436858, 595483, 1469202, 278239, -2660162, -3216637, 2103690, 8536967

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: 2lesseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Index entries for linear recurrences with constant coefficients, signature (2,-3,2).

Cf. A002249, A014551, A105578.

Equals (1/4) [A107920(n+4) + 2*A107920(n-1) + 3].

LinearRecurrence[{2,-3,2},{1,5,6},50] (* Harvey P. Dale, Apr 13 2019 *)

G.f.: (1+3*x-x^2)/((1-x)*(1-x+2*x^2)). - Colin Barker, Mar 26 2012

E.g.f.: exp(x/2)*(21*exp(x/2) - 7*cos(sqrt(7)*x/2) + 15*sqrt(7)*sin(sqrt(7)*x/2))/14. - Stefano Spezia, May 22 2025

A259713 a(n) = 32^n - 2(-1)^n.

Original entry on oeis.org

1, 8, 10, 26, 46, 98, 190, 386, 766, 1538, 3070, 6146, 12286, 24578, 49150, 98306, 196606, 393218, 786430, 1572866, 3145726, 6291458, 12582910, 25165826, 50331646, 100663298, 201326590, 402653186, 805306366, 1610612738, 3221225470, 6442450946, 12884901886

Offset: 0

Paul Curtz, Jul 03 2015

Inverse binomial transform of 3^n, with 3 (second term) excluded.

a(n) mod 9 gives A010689.

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

Cf. A000244, A005010, A007283, A010689, A096045, A140788, A182460, A201630.

[3*2^n-2*(-1)^n: n in [0..40]]; // Vincenzo Librandi, Jul 04 2015

Table[3 2^n - 2 (-1)^n, {n, 0, 50}] (* Vincenzo Librandi, Jul 04 2015 *)
LinearRecurrence[{1,2},{1,8},40] (* Harvey P. Dale, Aug 19 2020 *)

Vec(-(7*x+1)/((x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jul 03 2015

a(n) = a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(1)=8.
a(n) = 2*a(n-1) - 6*(-1)^n for n>0, a(0)=1.
a(4n+2) = 10*A182460(n); a(2n) = A096045(n), a(2n+1) = A140788(n).
a(n) = 3*A014551(n+1) - A201630(n).
a(n+2) - a(n) = a(n) + a(n+1) = A005010(n).
G.f.: -(7*x+1) / ((x+1)*(2*x-1)). - Colin Barker, Jul 03 2015

Typo in data fixed by Colin Barker, Jul 03 2015

A271494 Expansion of (1+16x)/((1+4x)(1-8x)).

Original entry on oeis.org

1, 20, 112, 1088, 7936, 66560, 520192, 4210688, 33488896, 268697600, 2146435072, 17184063488, 137422176256, 1099578736640, 8795824586752, 70369817919488, 562945658454016, 4503616807239680, 36028728299487232, 288230651029618688, 2305841909702066176, 18446748471756062720

Offset: 0

N. J. A. Sloane, Apr 15 2016

Sixth moments of the Rudin-Shapiro polynomials.

Shalosh B. Ekhad, Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials, Preprint, 2016.
Doron Zeilberger, Personal Communication to N. J. A. Sloane, Apr 15 2016.

Colin Barker, Table of n, a(n) for n = 0..1000
Christophe Doche, Even moments of generalized Rudin-Shapiro polynomials, Mathematics of computation 74.252 (2005): 1923-1935.
Christophe Doche and Laurent Habsieger, Moments of the Rudin-Shapiro polynomials, Journal of Fourier Analysis and Applications 10.5 (2004): 497-505.
Index entries for linear recurrences with constant coefficients, signature (4,32).

Cf. A246036, A271495, A271496.

CoefficientList[Series[(1+16x)/((1+4x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[{4,32},{1,20},30] (* Harvey P. Dale, May 13 2017 *)

Vec((1+16*x)/((1+4*x)*(1-8*x)) + O(x^50)) \\ Colin Barker, Apr 17 2016

From Colin Barker, Apr 17 2016: (Start)
a(n) = 2^(1+3*n)-(-4)^n.
a(n) = 4*a(n-1) + 32*a(n-2) for n>1.
(End)
a(n) = 4^n*A014551(n+1). - R. J. Mathar, Mar 08 2021

A049332 Number of conjugacy classes in Clifford group CL(n).

Original entry on oeis.org

2, 4, 5, 10, 17, 34, 65, 130, 257, 514, 1025, 2050, 4097, 8194, 16385, 32770, 65537, 131074, 262145, 524290, 1048577, 2097154, 4194305, 8388610, 16777217, 33554434, 67108865, 134217730, 268435457, 536870914, 1073741825, 2147483650

Offset: 0

N. J. A. Sloane

Floretion Algebra Multiplication Program, FAMP Code: 4tesforseq[ (- .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e)*( + .5'i + .5i' + .5'ii' + .5'jk' + .5'kj' + .5e ) ], 1vesforseq = (1,1,1,1,1,1,1). (Dement)

B. Simon, Representations of Finite and Compact Groups, Amer. Math. Soc., 1996, p. 69.

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

Cf. A101622, A014551 (first differences)

[(3/2-(-1)^n/2+2^n): n in [0..40]]; // Vincenzo Librandi, Apr 27 2012

A049332 := proc(n) if n mod 2 = 0 then 2^n+1 else 2^n+2; fi; end;

CoefficientList[Series[(2-5*x^2)/((1-x)*(1+x)*(1-2*x)),{x,0,40}],x] (* Vincenzo Librandi, Apr 27 2012 *)
Table[2^n + Mod[n, 2] + 1, {n, 0, 31}] (* Jean-François Alcover, Feb 11 2014 *)
LinearRecurrence[{2,1,-2},{2,4,5},40] (* Harvey P. Dale, Nov 29 2014 *)

a(n) = 3/2 - (-1)^n/2 + 2^n \\ Charles R Greathouse IV, Feb 10 2017

a(n+2) - A101622(n+1) = 4. - Creighton Dement, Mar 07 2005
From Colin Barker, Apr 18 2012: (Start)
a(n) = (3/2 - (-1)^n/2 + 2^n).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
G.f.: (2-5*x^2)/((1-x)*(1+x)*(1-2*x)). (End)
E.g.f.: cosh(x) + cosh(2*x) + 2*sinh(x) + sinh(2*x). - Stefano Spezia, May 27 2022
a(n) = 2*A000975(n+1) -5*A000975(n-1). - R. J. Mathar, Oct 12 2022

A099429 A Jacobsthal-Lucas convolution.

Original entry on oeis.org

0, 0, 2, 3, 12, 25, 66, 147, 344, 765, 1710, 3751, 8196, 17745, 38234, 81915, 174768, 371365, 786438, 1660239, 3495260, 7340025, 15379122, 32156323, 67108872, 139810125, 290805086, 603979767, 1252698804, 2594876065, 5368709130, 11095332171, 22906492256

Offset: 0

Paul Barry, Oct 15 2004

Inverse binomial transform is (-1)^n * a(n). - Michael Somos, Jun 02 2014

If we concatenate the lexicographically ordered bit strings of length n, then a(n) is the number of times 11 appears as a substring, if overlapping substrings are not considered as being separate. - John M. Campbell, Jan 18 2019

			G.f. = 2*x^2 + 3*x^3 + 12*x^4 + 25*x^5 + 66*x^6 + 147*x^7 + 344*x^8 + ...
If we concatenate the lexicographically ordered bit strings of length 4, we obtain the expression 0000000100100011010001010110011110001001101010111100110111101111, and we see that the substring 11 appears a total of a(4) = 12 times, with overlapping substrings not being considered as being separate. - _John M. Campbell_, Jan 18 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Cf. A001045, A014551, A193449.

m:=40; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( x^2*(2-x)/(1-x-2*x^2)^2 )); // G. C. Greubel, Feb 25 2019

CoefficientList[Series[x^2*(2-x)/(1-x-2x^2)^2, {x, 0, 32}], x] (* Michael De Vlieger, Jan 18 2019 *)

{a(n) = if( n>=0, polcoeff( x^2*(2-x)/((1+x)*(1-2*x))^2 + x*O(x^n), n), polcoeff( x*(1-2*x)/((1+x)*(2-x))^2 + x*O(x^-n), -n) )}; /* Michael Somos, Jun 02 2014 */

(x^2*(2-x)/(1-x-2*x^2)^2).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

G.f.: x^2*(2-x)/(1-x-2*x^2)^2. [Typo corrected by Colin Barker, Jun 16 2012]
a(n) = Sum_{k=0..n} J(n-k)*(2^(k-1) -(-1)^k +0^k/2).
a(n) = Sum_{k=0..n+1} J(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = A036289(n)/6 +(-1)^n*n/3. - R. J. Mathar, Sep 21 2012
a(-n) = (-2)^(-n-1) * A193449(n) for all n in Z. - Michael Somos, Jun 02 2014

cp's OEIS Frontend

A087288 a(n)=2a(n-1)+a(n-2)-2a(n-3).

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A105578 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.

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A201455 a(n) = 3*a(n-1) + 4*a(n-2) for n>1, a(0)=2, a(1)=3.

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A083332 a(n) = 10*a(n-2) - 16*a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.

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A099430 a(n) = 2^n+(-1)^n-1.

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A105577 a(n) = 2*a(n-1) - 3*a(n-2) + 2*a(n-3) with a(0) = 1, a(1) = 5, a(2) = 6.

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A259713 a(n) = 3*2^n - 2*(-1)^n.

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A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.

A083332 a(n) = 10a(n-2) - 16a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.

A105577 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 1, a(1) = 5, a(2) = 6.

A259713 a(n) = 32^n - 2(-1)^n.

A271494 Expansion of (1+16x)/((1+4x)(1-8x)).