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A346375 a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.

%I A346375 #53 Aug 30 2021 06:42:02
%S A346375 6,16,37,92,263,858,3069,11584,44995,177350,704201,2806476,11205327,
%T A346375 44780242,179038933,715991768,2863639259,11453901534,45814295265,
%U A346375 183254559460,733012994791,2932041493226,11728145001197,46912538061552,187650068359923,750600105667318,3002400087124729
%N A346375 a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.
%H A346375 Roger B. Nelson, <a href="https://www.jstor.org/stable/30044199">Proof without Words: A Triangular Sum</a>, Mathematics Magazine Vol. 78, No. 5 (December 2005), p. 395.
%H A346375 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,22,-8).
%F A346375 a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.
%F A346375 a(n) = (2^(n+1) + 7) * (2^(n+1) + 8)/6 - 9 + 3*n.
%F A346375 More generally: let f(n, b) = Sum_{k=0..n} (2^k + b) * (2^k + b + 1)/2 then f(n, b) = (2^(n+1) + 3*b + 1) * (2^(n+1) + 3*b + 2) / 6 - (b + 1)^2 + b*(b + 1)*n/2.
%F A346375 G.f.: ((b^2+3*b+2)/2 - (3*b^2+8*b+4)*x + (4*b^2+8*b+3)*x^2) / ((4*x-1) * (2*x-1) * (x-1)^2).
%F A346375 E.g.f.: exp(x)*((6*b+3)*exp(x) + 2*exp(3*x) + 3*(b^2+b)*x/2 +(3*b^2-3*b-4) / 2) / 3.
%F A346375 Then b = -1 gives A006095, b = 0 gives A076024, b = 1 gives A346295, b = 2 gives A346375.
%F A346375 a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) with n > 3.
%F A346375 This recurrence is valid for all sequences f(n, b).
%F A346375 G.f.: (35*x^2 - 32*x + 6) / ((4*x - 1) * (2*x - 1) * (x - 1)^2).
%F A346375 E.g.f.: exp(x) * (1 + 15*exp(x) + 2*exp(3*x) + 9*x)/3. - _Stefano Spezia_, Aug 15 2021
%p A346375 a:= proc(n) option remember:
%p A346375      if n=0 then 6 else procname(n-1)+(2^n+3)*(2^n+2)/2 fi:
%p A346375     end proc:
%p A346375 seq(a(n), n=0..26);
%t A346375 a[n_]:=Sum[(2^k+2)*(2^k+3)/2,{k,0,n}];Array[a,30,0] (* _Giorgos Kalogeropoulos_, Jul 27 2021 *)
%o A346375 (PARI) a(n) = sum(k=0, n, (2^k+2)*(2^k+3)/2); \\ _Michel Marcus_, Jul 28 2021
%Y A346375 Cf. A006095, A076024.
%K A346375 nonn,easy
%O A346375 0,1
%A A346375 _Paul Weisenhorn_, Jul 14 2021