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A048776 First partial sums of A048739; second partial sums of A000129. 9

%I

%S 1,4,12,32,81,200,488,1184,2865,6924,16724,40384,97505,235408,568336,

%T 1372096,3312545,7997204,19306972,46611168,112529329,271669848,

%U 655869048,1583407968,3822685009,9228778012,22280241060,53789260160,129858761409,313506783008

%N First partial sums of A048739; second partial sums of A000129.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,0,1).

%F a(n) = 2*a(n-1) + a(n-2) + n + 1; a(0)=1, a(1)=4.

%F a(n) = (((7/2 + (5/2)*sqrt(2))*(1+sqrt(2))^n - (7/2 - (5/2)*sqrt(2))*(1-sqrt(2))^n)/2*sqrt(2)) - (n+3)/2.

%F a(n) = (A000129(n+3) - (n+3))/2 = Sum_{j} A047662(n-j+1, j+1). - _Henry Bottomley_, Jul 09 2001

%F From _R. J. Mathar_, Feb 06 2010: (Start)

%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).

%F G.f.: -1/((x^2+2*x-1) * (x-1)^2). (End)

%F Define an array with m(n,1)=1 and m(1,k) = k*(k+1)/2 for n=1,2,3,... The interior terms are m(n,k) = m(n,k-1) + m(n-1,k-1) + m(n-1,k). The sum of the terms in each antidiagonal=a(n). - _J. M. Bergot_, Dec 01 2012 [This is A154948 without the first column. The diagonal is m(n,n) = A161731(n-1). _R. J. Mathar_, Dec 06 2012]

%p with(combinat):seq((fibonacci(n,2)-n)/2,n=3..25); # _Zerinvary Lajos_, Jun 02 2008

%t a=b=0;Table[c=2*b+a+n;a=b;b=c,{n,1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2011*)

%t LinearRecurrence[{4,-4,0,1},{1,4,12,32},30] (* _Harvey P. Dale_, Aug 27 2014 *)

%Y Cf. A001333, A000129, A048739.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_

%E More terms from _Harvey P. Dale_, Aug 27 2014

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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)