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%I #58 Jan 13 2023 11:12:21
%S 1,2,6,15,37,90,218,527,1273,3074,7422,17919,43261,104442,252146,
%T 608735,1469617,3547970,8565558,20679087,49923733,120526554,290976842,
%U 702480239,1695937321,4094354882,9884647086,23863649055,57611945197
%N a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.
%C Previous name was: a(n) = A048739(n) - A000129(n).
%C Partial sums of Pell numbers A000129 except omit next-to-last Pell number. E.g., 37 = 0+1+2+5+12+29 - 12.
%D M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.
%H Vincenzo Librandi, <a href="/A098790/b098790.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/bicknell.pdf">A Primer on the Pell Sequence and related sequences</a>, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434.
%H Hermann Stamm-Wilbrandt, <a href="/A098790/a098790.svg">4 interlaced bisections</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -1, -1).
%F a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.
%F G.f.: (x^2-x+1)/((1-x)(1-2x-x^2)).
%F a(n+1) = - A024537(n+1) + 2*A048739(n+1) - 2*A048739(n).
%F a(n) = - A024537(n) + A052542(n+1).
%F Partial sums of A074323. - _Paul Barry_, Mar 11 2007
%F a(n) = (sqrt(2)+1)^n*(3/4+sqrt(2)/4)+(sqrt(2)-1)^n*(3/4-sqrt(2)/4)*(-1)^n-1/2; - _Paul Barry_, Mar 11 2007
%F a(0)=1, a(1)=2, a(2)=6, a(n)=3*a(n-1)-a(n-2)-a(n-3). [_Harvey P. Dale_, Oct 15 2011]
%F a(2*n) = A124124(2*n+1). - _Hermann Stamm-Wilbrandt_, Aug 03 2014
%F a(2*n+1) = A006451(2*n+1). - _Hermann Stamm-Wilbrandt_, Aug 26 2014
%F a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>5. - _Hermann Stamm-Wilbrandt_, Aug 26 2014
%F 2*a(n) = A135532(n+1)-1. - _R. J. Mathar_, Jan 13 2023
%t a[0] = 1; a[1] = 2; a[n_] := a[n] = 2a[n - 1] + a[n - 2] + 1; Table[ a[n], {n, 0, 28}] (* _Robert G. Wilson v_, Nov 04 2004 *)
%t LinearRecurrence[{3,-1,-1},{1,2,6},31] (* _Harvey P. Dale_, Oct 15 2011 *)
%t CoefficientList[Series[(x^2 - x + 1)/((1 - x) (1 - 2 x - x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 14 2014 *)
%Y Cf. A006451, A000129, A048739, A124124, A024537, A074323, A052542.
%K nonn
%O 0,2
%A _Creighton Dement_, Oct 30 2004
%E More terms from _Robert G. Wilson v_, Nov 04 2004
%E Definition edited by _N. J. A. Sloane_, Aug 03 2014
%E New name from existing formula by _Joerg Arndt_, Aug 13 2014
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