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A132925 a(n) = 2^n - 1 + n*(n-1)/2.

%I A132925 #36 Feb 01 2025 23:19:01
%S A132925 1,4,10,21,41,78,148,283,547,1068,2102,4161,8269,16474,32872,65655,
%T A132925 131207,262296,524458,1048765,2097361,4194534,8388860,16777491,
%U A132925 33554731,67109188,134218078,268435833,536871317,1073742258,2147484112
%N A132925 a(n) = 2^n - 1 + n*(n-1)/2.
%C A132925 Row sums of triangle A132924. n-th Mersenne number + (n-1)-th triangular number.
%C A132925 Partial sums of A006127. - _Jaroslav Krizek_, Oct 16 2009
%H A132925 Vincenzo Librandi, <a href="/A132925/b132925.txt">Table of n, a(n) for n = 1..1000</a>
%H A132925 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F A132925 Binomial transform of [1, 3, 3, 2, 2, 2, 2, ...].
%F A132925 a(n) = A000225(n) + A000217(n-1). - _Jaroslav Krizek_, Oct 16 2009
%F A132925 From _Harvey P. Dale_, Jun 19 2011: (Start)
%F A132925 a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4); a(1)=1, a(2)=4, a(3)=10, a(4)=21.
%F A132925 G.f.: -x*(x^2+x-1)/((x-1)^3*(2*x-1)). (End)
%F A132925 E.g.f.: exp(x)*(2*exp(x) + x^2 - 2)/2. - _Stefano Spezia_, Feb 01 2025
%e A132925 a(4) = 21 = sum of row 4 terms of triangle A132924: (4 + 4 + 5 + 8).
%e A132925 a(4) = 21 = (1, 3, 3, 1) dot (1, 3, 3, 2) = (1 + 9 + 9 + 2).
%p A132925 A132925 := proc(n) 2^n-1+n*(n-1)/2 ; end proc; # _R. J. Mathar_, Oct 23 2009
%t A132925 Table[2^n-1+n (n-1)/2,{n,40}] (* or *) LinearRecurrence[{5,-9,7,-2},{1,4,10,21},40] (* _Harvey P. Dale_, Jun 19 2011 *)
%o A132925 (PARI) a(n)=2^n+binomial(n,2)-1 \\ _Charles R Greathouse IV_, Jun 20 2011
%o A132925 (Magma) [2^n - 1 + n*(n-1)/2: n in [1..40]]; // _Vincenzo Librandi_, Jun 21 2011
%Y A132925 Cf. A006127, A132924.
%Y A132925 Cf. A000225, A000217.
%K A132925 nonn,easy
%O A132925 1,2
%A A132925 _Gary W. Adamson_, Sep 05 2007