A048696 -id:A048696 - cp's OEIS frontend

A048772 Partial sums of A048696.

1, 10, 29, 76, 189, 462, 1121, 2712, 6553, 15826, 38213, 92260, 222741, 537750, 1298249, 3134256, 7566769, 18267802, 44102381, 106472572, 257047533, 620567646, 1498182833

Offset: 0

Barry E. Williams

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

Cf. A001333, A000129, A048694-A048696, A105082.

a048772 n = a048772_list !! n
a048772_list = scanl1 (+) a048696_list
-- Reinhard Zumkeller, Dec 15 2013

Accumulate[LinearRecurrence[{2,1},{1,9},30]] (* or *) LinearRecurrence[ {3,-1,-1},{1,10,29},30] (* Harvey P. Dale, Apr 20 2012 *)

a(n)=([0,1,0; 0,0,1; -1,-1,3]^n*[1;10;29])[1,1] \\ Charles R Greathouse IV, Feb 10 2017

a(n)=2*a(n-1)+a(n-2)+8; a(0)=1, a(1)=10.
a(n)=[ {(9+5*sqrt(2))(1+sqrt(2))^n - (9-5*sqrt(2))(1-sqrt(2))^n}/2*sqrt(2) ]-4.
a(0)=1, a(1)=10, a(2)=29, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, Apr 20 2012
G.f. ( 1+7*x ) / ( (x-1)*(x^2+2*x-1) ). a(n)=A048739(n)+7*A048739(n-1). - R. J. Mathar, Nov 08 2012

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

Original entry on oeis.org

1, 10, 38, 132, 452, 1544, 5272, 18000, 61456, 209824, 716384, 2445888, 8350784, 28511360, 97343872, 332352768, 1134723328, 3874187776, 13227304448, 45160842240, 154188760064, 526433355776, 1797355902976, 6136556900352, 20951515795456, 71532949381120

Offset: 0

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

Binomial transform of A048696. Second binomial transform of A164587. Inverse binomial transform of A164299.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

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

Sequences in the class a(n, m): this sequence (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
Cf. A007070, A016921, A048696, A056236, A112032, A164587, A241204.
Cf. A016116(n+1).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+r)^n+(1-4*r)*(2-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+6*x)/(1-4*x+2*x^2) )); // G. C. Greubel, Dec 14 2018

a:=n->((1+4*sqrt(2))*(2+sqrt(2))^n+(1-4*sqrt(2))*(2-sqrt(2))^n)/2: seq(floor(a(n)),n=0..25); # Muniru A Asiru, Dec 15 2018

LinearRecurrence[{4,-2}, {1,10}, 50] (* or *) CoefficientList[Series[(1 + 6*x)/(1 - 4*x + 2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 12 2017 *)

my(x='x+O('x^50)); Vec((1+6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 12 2017

[( (1+6*x)/(1-4*x+2*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 14 2018; Mar 12 2021

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0)=1, a(1)=10.
G.f.: (1+6*x)/(1-4*x+2*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = A056236(n) + 8*A007070(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A164587 a(n) = 2*a(n - 2) for n > 2; a(1) = 1, a(2) = 8.

Original entry on oeis.org

1, 8, 2, 16, 4, 32, 8, 64, 16, 128, 32, 256, 64, 512, 128, 1024, 256, 2048, 512, 4096, 1024, 8192, 2048, 16384, 4096, 32768, 8192, 65536, 16384, 131072, 32768, 262144, 65536, 524288, 131072, 1048576, 262144, 2097152, 524288, 4194304, 1048576

Offset: 1

Klaus Brockhaus, Aug 17 2009

nonn

Interleaving of A000079 and A000079 without initial terms 1, 2, 4.

Binomial transform is A048696. Second binomial transform is A164298.

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

Equals A112032 without initial term 4.

Cf. A000079 (powers of 2), A048696, A164298.

[ n le 2 select 7*n-6 else 2*Self(n-2): n in [1..41] ];

CoefficientList[Series[(1 - x)/(1 - 10*x + 17*x^2), {x,0,50}], x] (* G. C. Greubel, Aug 12 2017 *)

x='x+O('x^50); Vec(x*(1+8*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 12 2017

a(n) = (5 + 3*(-1)^n)*2^((2*n -5 +(-1)^n)/4).
G.f.: x*(1+8*x)/(1-2*x^2).
E.g.f.: 4*cosh(sqrt(2)*x) + (1/sqrt(2))*sinh(sqrt(2)*x) - 4. - G. C. Greubel, Aug 12 2017

A105082 Expansion of (5+4x)/(1-2x-x^2).

Original entry on oeis.org

5, 14, 33, 80, 193, 466, 1125, 2716, 6557, 15830, 38217, 92264, 222745, 537754, 1298253, 3134260, 7566773, 18267806, 44102385, 106472576, 257047537, 620567650, 1498182837, 3616933324, 8732049485, 21081032294, 50894114073

Offset: 0

Creighton Dement, Apr 06 2005

A floretion-generated, Pellian related sequence.
Floretion Algebra Multiplication Program, FAMP Code: lesloop(infty)-tesforseq[ + .25'i + .25i' - .25'ii' - .25'jj' - .25'kk' + .25'jk' + .25'kj' - .25e ], Fortype: 1A.
For n > 0: A048696(n) = a(n) - a(n-1). - Reinhard Zumkeller, Dec 15 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A001333, A048654, A048696.

Cf. A048772.

a105082 n = a105082_list !! n
a105082_list = scanl (+) 5 $ tail a048696_list
-- Reinhard Zumkeller, Dec 15 2013

a(n+2) = 2*a(n+1) + a(n); FAMP result: a(n) = 2*A001333(n) + 3*A048654(n); SuperSeeker results: a(n+1) - a(n) = A048696(n+1); a(n) + a(n+1) = A048696(n+2)

a(n) = ((9+5*sqrt(2))*(1+sqrt(2))^n - (9-5*sqrt(2))*(1-sqrt(2))^n)/(2*sqrt(2)) - Lambert Herrgesell (zero815(AT)googlemail.com), Jan 26 2007

A048773 Partial sums of A048697.

Original entry on oeis.org

1, 11, 32, 84, 209, 511, 1240, 3000, 7249, 17507, 42272, 102060, 246401, 594871, 1436152, 3467184, 8370529, 20208251, 48787040, 117782340, 284351729, 686485807, 1657323352, 4001132520, 9659588401, 23320309331, 56300207072, 135920723484, 328141654049

Offset: 0

Barry E. Williams

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A001333, A000129, A048696, A048697.

Accumulate[LinearRecurrence[{2,1},{1,10},35]] (* Harvey P. Dale, Jul 26 2011 *)
LinearRecurrence[{3, -1, -1},{1, 11, 32},29] (* Ray Chandler, Aug 03 2015 *)

a(n) = 2*a(n-1)+a(n-2)+9; a(0)=1, a(1)=11.
a(n) = (((10+(11/2)*sqrt(2))*(1+sqrt(2))^n - (10-(11/2)*sqrt(2))*(1-sqrt(2))^n)/ 2*sqrt(2))-9/2.
From R. J. Mathar, Nov 08 2012: (Start)
G.f.: ( 1+8*x ) / ( (x-1)*(x^2+2*x-1) ).
a(n) = A048739(n)+8*A048739(n-1). (End)
a(n) = 3*a(n-1)-a(n-2)-a(n-3). - Wesley Ivan Hurt, May 21 2021

More terms from Harvey P. Dale, Jul 26 2011

cp's OEIS Frontend

A048772 Partial sums of A048696.

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A164298 a(n) = ((1+4*sqrt(2))*(2+sqrt(2))^n + (1-4*sqrt(2))*(2-sqrt(2))^n)/2.

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A164587 a(n) = 2*a(n - 2) for n > 2; a(1) = 1, a(2) = 8.

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A105082 Expansion of (5+4x)/(1-2x-x^2).

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A048773 Partial sums of A048697.

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A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.