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A129696 Antidiagonal sums of triangular array T defined in A014430: T(j,k) = binomial(j+1, k) - 1 for 1 <= k <= j.

%I A129696 #75 Jan 30 2025 05:09:30
%S A129696 1,2,5,9,17,29,50,83,138,226,370,602,979,1588,2575,4171,6755,10935,
%T A129696 17700,28645,46356,75012,121380,196404,317797,514214,832025,1346253,
%U A129696 2178293,3524561,5702870,9227447,14930334,24157798,39088150,63245966
%N A129696 Antidiagonal sums of triangular array T defined in A014430: T(j,k) = binomial(j+1, k) - 1 for 1 <= k <= j.
%C A129696 If T is construed as a lower triangular matrix M over the rational field, the inverse M^-1 is a lower triangular matrix containing fractions. Its row sums are the Bernoulli numbers. First column of M^-1 is 1, -1, 2/3, -1/4, -1/30, 1/12, 1/42, -1/12, ... . Multiplied by j! this gives 1, -2, 4, -6, -4, 60, 120, -3660, ... .
%C A129696 The Kn22 sums, see A180662 for the definition of these sums, of the 'Races with Ties' triangle A035317 lead to this sequence. - _Johannes W. Meijer_, Jul 20 2011
%C A129696 This sequence is the convolution of (1,1,2,3,5,8,13,21,...) and (1,1,2,2,3,3,4,4,5,5,...), i.e., the Fibonacci numbers (A000045) and A008619. - _Clark Kimberling_, May 28 2012
%C A129696 a(n) is the sum of the first summands over all Arndt compositions of n (see the Checa link). - _Daniel Checa_, Jan 01 2024
%D A129696 Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969.
%H A129696 Vincenzo Librandi, <a href="/A129696/b129696.txt">Table of n, a(n) for n = 1..200</a>
%H A129696 Daniel F. Checa, <a href="https://github.com/dfcheca/Arndt-Compositions">Arndt Compositions: Connections with Fibonacci Numbers, Statistics, and Generalizations</a>, 2023, p. 31.
%H A129696 Larry Ericksen, <a href="https://www.fq.math.ca/Papers1/57-5/problems.pdf">Problem 7</a>, Problem Section, The Fibonacci Quarterly, Vol. 57, No. 5 (2019), pp. 172, 175-183.
%H A129696 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,0,1).
%F A129696 From _Paul Barry_, Jan 18 2009: (Start)
%F A129696 a(n) = Sum_{k=0..floor(n/2)} A000071(n-2*k+3).
%F A129696 a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-2*k} Fibonacci(j+1)). (End)
%F A129696 a(n+1) = a(n-1) + a(n) + 1 + floor(n/2) for n>1, a(0)=1, a(1)=2. - _Alex Ratushnyak_, Jul 30 2012
%F A129696 From _R. J. Mathar_, Jul 25 2013: (Start)
%F A129696 G.f.: x/((1 + x)*(1 - x)^2*(1 - x - x^2)).
%F A129696 a(n) + a(n+1) = A001924(n+1). (End)
%F A129696 a(n) = Fibonacci(n+3) - 2 - floor(n/2). - _Emeric Deutsch_, Nov 22 2014
%F A129696 a(n) = (-5/4 - (-1)^n/4 + (2^(-n)*((1 - t)^n*(-2 + t) + (1 + t)^n*(2 + t)))/t + (-1 - n)/2), where t=sqrt(5). - _Colin Barker_, Feb 09 2017
%F A129696 E.g.f.: (4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2)) - 5*(4 + x)*cosh(x) - 5*(3 + x)*sinh(x))/10. - _Stefano Spezia_, Apr 06 2024
%F A129696 a(n) = max_{k = 2^(n+1)..2^(n+2)-1} (A002487(k) - A000120(k)) (Ericksen, 2019). - _Amiram Eldar_, Jan 30 2025
%p A129696 with(combinat): a := proc (n) options operator, arrow: fibonacci(n+3)-2-floor((1/2)*n) end proc: seq(a(n), n = 1 .. 34); # _Emeric Deutsch_, Nov 22 2014
%t A129696 a[n_]:= a[n]= If[n<3, n, a[n-1] + a[n-2] + (n + Mod[n, 2])/2];
%t A129696 Table[a[n], {n,40}] (* _Jean-François Alcover_, Mar 04 2013 *)
%t A129696 Table[Fibonacci[n+3] -2 -Floor[n/2], {n, 100}] (* _Vincenzo Librandi_, Nov 23 2014 *)
%o A129696 (Magma) m:=36; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=1 to j do M[j, k]:=Binomial(j+1, k)-1; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // _Klaus Brockhaus_, Jun 11 2007
%o A129696 (Python)
%o A129696 prpr = 1
%o A129696 prev = 2
%o A129696 for n in range(2,100):
%o A129696     print(prpr, end=", ")
%o A129696     curr = prpr+prev + 1 + n//2
%o A129696     prpr = prev
%o A129696     prev = curr
%o A129696 # _Alex Ratushnyak_, Jul 30 2012
%o A129696 (Magma) [Fibonacci(n+3)-2-Floor(n/2): n in [1..40]]; // _Vincenzo Librandi_, Nov 23 2014
%o A129696 (SageMath) [fibonacci(n+3) -2 -(n//2) for n in range(1,41)] # _G. C. Greubel_, Mar 17 2023
%Y A129696 Cf. A000045, A000071, A001924, A008619, A014430, A027641, A035317, A052952 (first differences), A180662.
%Y A129696 Cf. A000120, A002487.
%K A129696 nonn,easy
%O A129696 1,2
%A A129696 _Paul Curtz_, Jun 01 2007
%E A129696 Edited and extended by _Klaus Brockhaus_, Jun 11 2007