A048697 -id:A048697 - cp's OEIS frontend

A048773 Partial sums of A048697.

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

A153593 a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).

Original entry on oeis.org

1, 18, 245, 2988, 34429, 383670, 4186169, 45041112, 480032665, 5082340122, 53559541661, 562566880260, 5895000053461, 61667217421758, 644304909368225, 6725778192309168, 70163919621475249, 731614075994130210

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

nonn

Preceded by zero, this is the eighth binomial transform of the Pell sequence A000129. - Sergio Falcon, Sep 21 2009; edited by Klaus Brockhaus, Oct 11 2009
Eighth binomial transform of A048697.
First differences are in A164600.
lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(2) = 10.4142135623....

Vincenzo Librandi, Table of n, a(n) for n = 1..100
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (18,-79).

Cf. A000129 (Pell numbers), A007070, A081185, A081184, A081183, A081182, A081180, A081179. - Sergio Falcon, Sep 21 2009

Cf. A002193 (decimal expansion of sqrt(2)), A048697, A164600.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

Join[{a=1,b=18},Table[c=18*b-79*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18,-79},{1,18},25] (* G. C. Greubel, Aug 22 2016 *)

a(n) = 18*a(n-1) - 79*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
G.f.: x/(1 - 18*x + 79*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = Sum[Binomial[n - 1 - i, i] (-1)^i * 18^(n - 1 - 2 i) * 79^i, {i, 0, Floor[(n - 1)/2]}]. - Sergio Falcon, Sep 21 2009
E.g.f.: exp(9*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

cp's OEIS Frontend

A048773 Partial sums of A048697.

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