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A048778 First partial sums of A048745; second partial sums of A048654.

%I A048778 #21 Aug 10 2022 02:58:58
%S A048778 1,6,20,56,145,362,888,2160,5233,12654,30572,73832,178273,430418,
%T A048778 1039152,2508768,6056737,14622294,35301380,85225112,205751665,
%U A048778 496728506,1199208744,2895146064,6989500945,16874148030,40737797084,98349742280,237437281729,573224305826,1383885893472
%N A048778 First partial sums of A048745; second partial sums of A048654.
%C A048778 Define a triangle T by T(n,0) = n*(n+1) + 1, T(n,n) = (n+1)*(n+2)/2, and T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). Then a(n) is the sum of row n. - _J. M. Bergot_, Mar 06 2013
%H A048778 G. C. Greubel, <a href="/A048778/b048778.txt">Table of n, a(n) for n = 0..1000</a>
%H A048778 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,0,1).
%F A048778 a(n) = 2*a(n-1) + a(n-2) + 3*n + 1, with a(0)=1, a(1)=6.
%F A048778 a(n) =  ( ((13 + 9*sqrt(2))/2)*(1 + sqrt(2))^n - ((13 - 9*sqrt(2))/2)*(1 -sqrt(2))^n )/2*sqrt(2) - (3*n + 7)/2.
%F A048778 From _R. J. Mathar_, Nov 08 2012: (Start)
%F A048778 G.f.: (1 + 2*x) / ( (1-x-x^2)*(1-x)^2 ).
%F A048778 a(n) = A048776(n) + 2*A048776(n-1). (End)
%F A048778 a(n) = (Pell(n+3) + 2*Pell(n+2) - 3*n - 7)/2, where Pell(n) = A000129(n). - _G. C. Greubel_, Aug 09 2022
%t A048778 Table[(Fibonacci[n+3,2] +2*Fibonacci[n+2,2] -(3*n+7))/2, {n, 0, 40}] (* _G. C. Greubel_, Aug 09 2022 *)
%o A048778 (PARI)
%o A048778 N=66;  x='x+O('x^N);
%o A048778 gf= ( -1-2*x ) / ( (x^2+2*x-1)*(x-1)^2 );  Vec(Ser(gf))
%o A048778 /* _Joerg Arndt_, Mar 07 2013 */
%o A048778 (Magma) I:=[1, 6, 20, 56]; [n le 4 select I[n] else 4*Self(n-1) - 4*Self(n-2) + Self(n-4): n in [1..41]]; // _G. C. Greubel_, Aug 09 2022
%o A048778 (SageMath) [(lucas_number1(n+3, 2, -1) + 2*lucas_number1(n+2, 2, -1) -3*n-7)/2 for n in (0..40)] # _G. C. Greubel_, Aug 09 2022
%Y A048778 Cf. A000129, A001333, A048745, A048776, A048654.
%K A048778 easy,nonn
%O A048778 0,2
%A A048778 _Barry E. Williams_
%E A048778 Corrected by _T. D. Noe_, Nov 08 2006