A048694 -id:A048694 - cp's OEIS frontend

A048770 Partial sums of A048694.

1, 8, 23, 60, 149, 364, 883, 2136, 5161, 12464, 30095, 72660, 175421, 423508, 1022443, 2468400, 5959249, 14386904, 34733063, 83853036, 202439141, 488731324, 1179901795, 2848534920, 6876971641, 16602478208, 40081928063

Offset: 0

Barry E. Williams

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

Cf. A001333, A000129, A048654, A048655, A048694.

Accumulate[LinearRecurrence[{2,1},{1,7},40]] (* Harvey P. Dale, Jul 22 2011 *)
LinearRecurrence[{3, -1, -1},{1, 8, 23},27] (* Ray Chandler, Aug 03 2015 *)

a(n) = ((7+4*sqrt(2))*(1+sqrt(2))^n-(7-4*sqrt(2))*(1-sqrt(2))^n)/(2*sqrt(2))-3.
a(n) = 2*a(n-1)+a(n-2)+6 with n>1, a(0)=1, a(1)=8.
a(n) = 3*a(n-1)-a(n-2)-a(n-3). G.f.: (1+5*x)/((1-x)*(1-2*x-x^2)). - Colin Barker, Jun 23 2012
a(n) = 3*A000129(n)+4*A000129(n+1)-3. - R. J. Mathar, Sep 27 2012

More terms from James Sellers, Jun 20 2000

A163864 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

Original entry on oeis.org

1, 6, 2, 12, 4, 24, 8, 48, 16, 96, 32, 192, 64, 384, 128, 768, 256, 1536, 512, 3072, 1024, 6144, 2048, 12288, 4096, 24576, 8192, 49152, 16384, 98304, 32768, 196608, 65536, 393216, 131072, 786432, 262144, 1572864, 524288, 3145728, 1048576, 6291456

Offset: 1

Klaus Brockhaus, Aug 05 2009

nonn

Interleaving of A000079 and A007283 without initial 3.

Binomial transform is A048694, second binomial transform is A163613, third binomial transform is A163614, fourth binomial transform is A163615, fifth binomial transform is A163616, sixth binomial transform is A081183 without initial 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A048694, A163613, A163614, A163615, A163616, A081183.

[ n le 2 select 5*n-4 else 2*Self(n-2): n in [1..42] ];

LinearRecurrence[{0, 2}, {1, 6, 2, 12}, 50] (* G. C. Greubel, Aug 06 2017 *)

x='x+O('x^50); Vec(x*(1+6*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 06 2017

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

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

A163613 a(n) = ((1 + 3sqrt(2))(2 + sqrt(2))^n + (1 - 3sqrt(2))(2 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 8, 30, 104, 356, 1216, 4152, 14176, 48400, 165248, 564192, 1926272, 6576704, 22454272, 76663680, 261746176, 893657344, 3051137024, 10417233408, 35566659584, 121432171520, 414595366912, 1415517124608, 4832877764608

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

nonn

Binomial transform of A048694. Second binomial transform of A163864. Inverse binomial transform of A163614.

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

Cf. A048694, A163864 (1, 6, 2, 12, 4, 24, ...), A163614.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+r)^n+(1-3*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{4, -2}, {1, 8}, 50] (* G. C. Greubel, Jul 30 2017 *)

x='x+O('x^50); Vec((1+4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 30 2017

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
G.f.: (1+4*x)/(1-4*x+2*x^2).
E.g.f.: exp(2*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A048772 Partial sums of A048696.

Original entry on oeis.org

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

A048771 Partial sums of A048695.

Original entry on oeis.org

1, 9, 26, 68, 169, 413, 1002, 2424, 5857, 14145, 34154, 82460, 199081, 480629, 1160346, 2801328, 6763009, 16327353, 39417722, 95162804, 229743337, 554649485, 1339042314, 3232734120, 7804510561, 18841755249, 45488021066, 109817797388, 265123615849

Offset: 0

Barry E. Williams

			a(n)=[ {(8+(9/2)*sqrt(2))(1+sqrt(2))^n -(8-(9/2)*sqrt(2))(1-sqrt(2))^n}/ 2*sqrt(2) ]-7/2.

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

Cf. A001333, A000129, A048694, A048695.

Table[6*Fibonacci[n, 2] + Fibonacci[n+1, 2], {n, 0, 22}] // Accumulate (* Jean-François Alcover, Mar 25 2013 *)
Accumulate[LinearRecurrence[{2,1},{1,8},40]] (* or *) LinearRecurrence[ {3,-1,-1},{1,9,26},40] (* Harvey P. Dale, May 01 2013 *)

a(n)=2*a(n-1)+a(n-1)+7; a(0)=1, a(1)=9.
G.f. ( 1+6*x ) / ( (x-1)*(x^2+2*x-1) ). a(n)=A048739(n)+6*A048739(n-1). - R. J. Mathar, Nov 08 2012
a(0)=1, a(1)=9, a(2)=26, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, May 01 2013

More terms from Harvey P. Dale, May 01 2013

cp's OEIS Frontend

A048770 Partial sums of A048694.

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A163864 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

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A163613 a(n) = ((1 + 3*sqrt(2))*(2 + sqrt(2))^n + (1 - 3*sqrt(2))*(2 - sqrt(2))^n)/2.

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A048772 Partial sums of A048696.

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A048771 Partial sums of A048695.

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A163613 a(n) = ((1 + 3sqrt(2))(2 + sqrt(2))^n + (1 - 3sqrt(2))(2 - sqrt(2))^n)/2.