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A132045 Row sums of triangle A132044.

%I A132045 #37 Jan 29 2025 07:56:47
%S A132045 1,2,3,6,13,28,59,122,249,504,1015,2038,4085,8180,16371,32754,65521,
%T A132045 131056,262127,524270,1048557,2097132,4194283,8388586,16777193,
%U A132045 33554408,67108839,134217702,268435429,536870884,1073741795,2147483618,4294967265,8589934560
%N A132045 Row sums of triangle A132044.
%C A132045 Apart from initial terms, and with a change of offset, same as A095768. - _Jon E. Schoenfield_, Jun 15 2017
%H A132045 G. C. Greubel, <a href="/A132045/b132045.txt">Table of n, a(n) for n = 0..1000</a>
%H A132045 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A132045 Binomial transform of (1, 1, 0, 2, 0, 2, 0, 2, 0, 2, ...).
%F A132045 For n>=1, a(n) = 2^n - n + 1 = A000325(n) + 1. - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 17 2009. (Corrected by _Franklin T. Adams-Watters_, Jan 17 2009)
%F A132045 E.g.f.: U(0) - 1, where U(k) = 1 - x/(2^k + 2^k/(x - 1 - x^2*2^(k+1)/(x*2^(k+1) + (k+1)/U(k+1)))). - _Sergei N. Gladkovskii_, Dec 01 2012
%F A132045 From _Colin Barker_, Mar 14 2014: (Start)
%F A132045 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>3.
%F A132045 G.f.: (1-2*x+2*x^3) / ((1-x)^2*(1-2*x)). (End)
%e A132045 a(4) = 13 = sum of row 4 terms of triangle A132044: (1 + 3 + 5 + 3 + 1).
%e A132045 a(4) = 13 = (1, 4, 6, 4, 1) dot (1, 1, 0, 2, 0) = (1 + 4 + 0 + 8 + 0).
%t A132045 Table[2^n -(n-1) -Boole[n==0], {n, 0, 35}] (* _G. C. Greubel_, Feb 12 2021 *)
%o A132045 (PARI) Vec((1-2*x+2*x^3)/((1-x)^2*(1-2*x)) + O(x^100)) \\ _Colin Barker_, Mar 14 2014
%o A132045 (Sage) [1]+[2^n -n +1 for n in (1..35)] # _G. C. Greubel_, Feb 12 2021
%o A132045 (Magma) [1] cat [2^n -n +1: n in [1..35]]; // _G. C. Greubel_, Feb 12 2021
%Y A132045 Cf. A000325, A095768, A132044.
%K A132045 nonn,easy
%O A132045 0,2
%A A132045 _Gary W. Adamson_, Aug 08 2007