A048776 First partial sums of A048739; second partial sums of A000129.
1, 4, 12, 32, 81, 200, 488, 1184, 2865, 6924, 16724, 40384, 97505, 235408, 568336, 1372096, 3312545, 7997204, 19306972, 46611168, 112529329, 271669848, 655869048, 1583407968, 3822685009, 9228778012, 22280241060, 53789260160, 129858761409, 313506783008
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).
Programs
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Maple
with(combinat):seq((fibonacci(n+3, 2)-n-3)/2, n=0..25); # Zerinvary Lajos, Jun 02 2008
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Mathematica
a=b=0;Table[c=2*b+a+n;a=b;b=c,{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*) LinearRecurrence[{4,-4,0,1},{1,4,12,32},30] (* Harvey P. Dale, Aug 27 2014 *)
Formula
a(n) = 2*a(n-1) + a(n-2) + n + 1; a(0)=1, a(1)=4.
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.
From R. J. Mathar, Feb 06 2010: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
G.f.: -1/((x^2+2*x-1) * (x-1)^2). (End)
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]
E.g.f.: exp(x)*(10*cosh(sqrt(2)*x) + 7*sqrt(2)*sinh(sqrt(2)*x) - 2*(3 + x))/4. - Stefano Spezia, May 13 2023
Extensions
More terms from Harvey P. Dale, Aug 27 2014