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A125128 a(n) = 2^(n+1) - n - 2, or partial sums of main diagonal of array A125127 of k-step Lucas numbers.

%I A125128 #69 Jan 24 2025 12:15:40
%S A125128 1,4,11,26,57,120,247,502,1013,2036,4083,8178,16369,32752,65519,
%T A125128 131054,262125,524268,1048555,2097130,4194281,8388584,16777191,
%U A125128 33554406,67108837,134217700,268435427,536870882,1073741793,2147483616,4294967263,8589934558
%N A125128 a(n) = 2^(n+1) - n - 2, or partial sums of main diagonal of array A125127 of k-step Lucas numbers.
%C A125128 Essentially a duplicate of A000295: a(n) = A000295(n+1).
%C A125128 Partial sums of main diagonal of array A125127 = L(k,n): k-step Lucas numbers, read by antidiagonals.
%C A125128 Equals row sums of triangle A130128. - _Gary W. Adamson_, May 11 2007
%C A125128 Row sums of triangle A130330 which is composed of (1,3,7,15,...) in every column, thus: row sums of (1; 3,1; 7,3,1; ...). - _Gary W. Adamson_, May 24 2007
%C A125128 Row sums of triangle A131768. - _Gary W. Adamson_, Jul 13 2007
%C A125128 Convolution A130321 * (1, 2, 3, ...). Binomial transform of (1, 3, 4, 4, 4, ...). - _Gary W. Adamson_, Jul 27 2007
%C A125128 Row sums of triangle A131816. - _Gary W. Adamson_, Jul 30 2007
%C A125128 A000975 convolved with [1, 2, 2, 2, ...]. - _Gary W. Adamson_, Jun 02 2009
%C A125128 The eigensequence of a triangle with the triangular series as the left border and the rest 1's. - _Gary W. Adamson_, Jul 24 2010
%H A125128 Vincenzo Librandi, <a href="/A125128/b125128.txt">Table of n, a(n) for n = 1..1000</a>
%H A125128 Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, <a href="https://arxiv.org/abs/2501.06675">Chip-Firing on Infinite k-ary Trees</a>, arXiv:2501.06675 [math.CO], 2025. See p. 18.
%H A125128 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A125128 a(n) = A000295(n+1) = 2^(n+1) - n - 2 = Sum_{i=1..n} A125127(i,i) = Sum_{i=1..n} ((2^i)-1). [Edited by _M. F. Hasler_, Jul 30 2015]
%F A125128 From _Colin Barker_, Jun 17 2012: (Start)
%F A125128 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
%F A125128 G.f.: x/((1-x)^2*(1-2*x)). (End)
%F A125128 a(n) = A000225(n) + A000325(n) - 1. - _Miquel Cerda_, Aug 07 2016
%F A125128 a(n) = A095151(n) - A000225(n). - _Miquel Cerda_, Aug 12 2016
%F A125128 E.g.f.: 2*exp(2*x) - (2+x)*exp(x). - _G. C. Greubel_, Jul 26 2019
%e A125128 a(1) = 1 because "1-step Lucas number"(1) = 1.
%e A125128 a(2) = 4 = a(1) + [2-step] Lucas number(2) = 1 + 3.
%e A125128 a(3) = 11 = a(2) + 3-step Lucas number(3) = 1 + 3 + 7.
%e A125128 a(4) = 26 = a(3) + 4-step Lucas number(4) = 1 + 3 + 7 + 15.
%e A125128 a(5) = 57 = a(4) + 5-step Lucas number(5) = 1 + 3 + 7 + 15 + 31.
%e A125128 a(6) = 120 = a(5) + 6-step Lucas number(6) = 1 + 3 + 7 + 15 + 31 + 63.
%e A125128 G.f. = x + 4*x^2 + 11*x^3 + 26*x^4 + 57*x^5 + 120*x^6 + 247*x^7 + 502*x^8 + ...
%t A125128 CoefficientList[Series[1/((1-x)^2*(1-2*x)),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)
%t A125128 LinearRecurrence[{4,-5,2},{1,4,11},40] (* _Harvey P. Dale_, Nov 16 2014 *)
%t A125128 a[ n_] := With[{m = n + 1}, If[ m < 0, 0, 2^m - (1 + m)]]; (* _Michael Somos_, Aug 17 2015 *)
%o A125128 (Magma) I:=[1, 4, 11]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jun 28 2012
%o A125128 (PARI) A125128(n)=2<<n-n-2 \\ _M. F. Hasler_, Jul 30 2015
%o A125128 (PARI) {a(n) = n++; if( n<0, 0, 2^n - (1+n))}; /* _Michael Somos_, Aug 17 2015 */
%o A125128 (Sage) [2^(n+1) -n-2 for n in (1..40)] # _G. C. Greubel_, Jul 26 2019
%o A125128 (GAP) List([1..40], n-> 2^(n+1) -n-2); # _G. C. Greubel_, Jul 26 2019
%Y A125128 Cf. A000295, A125127, A125129, A130103.
%K A125128 easy,nonn
%O A125128 1,2
%A A125128 _Jonathan Vos Post_, Nov 22 2006
%E A125128 Edited by _M. F. Hasler_, Jul 30 2015