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%I #28 Aug 26 2024 19:35:12
%S 1,2,7,14,27,48,83,140,233,384,629,1026,1669,2710,4395,7122,11535,
%T 18676,30231,48928,79181,128132,207337,335494,542857,878378,1421263,
%U 2299670,3720963,6020664,9741659,15762356,25504049,41266440,66770525,108037002,174807565
%N a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=1, a(1)=2.
%H Bruno Berselli, <a href="/A210728/b210728.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F G.f.: (1-x+3*x^2-2*x^3)/((1-x)^2*(1-x-x^2)). - _Bruno Berselli_, Jun 27 2012
%F a(n) = ((5+sqrt(5))*(1+sqrt(5))^(n+1)-(5-sqrt(5))*(1-sqrt(5))^(n+1))/(2^(n+1)*sqrt(5))-n-5. - _Bruno Berselli_, Jun 27 2012
%F a(n) = -n-5+A022112(n+1). _R. J. Mathar_, Jul 03 2012
%t RecurrenceTable[{a[0] == 1, a[1] == 2, a[n] == a[n - 1] + a[n - 2] + n + 2}, a, {n, 36}] (* _Bruno Berselli_, Jun 27 2012 *)
%t nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,2},40][[;;,2]] (* _Harvey P. Dale_, Aug 26 2024 *)
%Y Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
%Y Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
%Y Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
%Y Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
%Y Cf. A104161: a(n)=a(n-1)+a(n-2)+n, a(0)=1,a(1)=2 (except the first term).
%Y Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
%Y Cf. A210729: a(n)=a(n-1)+a(n-2)+n+3, a(0)=1,a(1)=2.
%K nonn,easy
%O 0,2
%A _Alex Ratushnyak_, May 10 2012