cp's OEIS Frontend

A054452 Partial sums of A027941(n-1) with a(-1) = 0.

%I A054452 #55 Jul 02 2025 16:01:59
%S A054452 0,0,1,5,17,50,138,370,979,2575,6755,17700,46356,121380,317797,832025,
%T A054452 2178293,5702870,14930334,39088150,102334135,267914275,701408711,
%U A054452 1836311880,4807526952,12586269000,32951280073,86267571245,225851433689,591286729850
%N A054452 Partial sums of A027941(n-1) with a(-1) = 0.
%H A054452 Colin Barker, <a href="/A054452/b054452.txt">Table of n, a(n) for n = 0..1000</a>
%H A054452 László Németh, <a href="https://arxiv.org/abs/1701.06022">Pascal pyramid in the space H^2 x R</a>, arXiv:1701.06022 [math.CO], 2017 (5th line of Table 1 is a(n-2)).
%H A054452 A. Shriki and O. Liba, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnMay16No53.pdf">Polygons with Fibonacci Number Coordinates: Problem B-1167</a>, Fib. Quart. 54,2 May 2016, p. 180-181.
%H A054452 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,5,-1).
%F A054452 a(n) = +5*a(n-1) -8*a(n-2) +5*a(n-3) -1*a(n-4).
%F A054452 G.f.: x^2/((1-x)^2*(1-3*x+x^2)).
%F A054452 a(n) = Sum_{k=0..n} A027941(k-1) = F(2*n)-n = A054450(2*n-1, 2) = A054451(2*n-3).
%F A054452 G.f.: x^2*Fibe(x)/(1-x)^2, with Fibe(x) := 1/(1-3*x+x^2) = g.f. A001906(n+1) (Fibonacci numbers F(2(n+1))).
%F A054452 Fourth diagonal of array defined by T(i, 1) = T(1, j) = 1, T(i, j) = Max(T(i-1, j) + T(i-1, j-1); T(i-1, j-1) + T(i, j-1)). - _Benoit Cloitre_, Aug 05 2003
%F A054452 a(n) = Sum_{k=0..n-2} binomial(2*n-k-1, k). - _Johannes W. Meijer_, Aug 12 2013
%F A054452 a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} binomial(i+j, i-j). - _Wesley Ivan Hurt_, Mar 25 2015
%F A054452 a(n) = Sum_{k=0..n} (binomial(n+1,k+2)*Fibonacci(k)). - _Vladimir Kruchinin_, Oct 21 2016
%F A054452 a(n) = (-((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n)/sqrt(5) - n. - _Colin Barker_, Jan 28 2017
%p A054452 a[0]:=0: a[1]:=1: for n from 2 to 50 do a[n] := 3*a[n-1]-a[n-2] od: seq(a[n]-n, n=0..27); # _Zerinvary Lajos_, Mar 20 2008
%p A054452 with(combinat): seq(fibonacci(2*n)-n, n=0..27); # _Zerinvary Lajos_, Jun 19 2008
%p A054452 g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)-n), n=0..27); # _Zerinvary Lajos_, Mar 22 2009
%t A054452 CoefficientList[Series[x^2 / ((1 - x)^2 (1 - 3 x + x^2)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Mar 26 2015 *)
%o A054452 (Sage) [(lucas_number1(n, 3, 1)-lucas_number1(n, 2, 1)) for n in range(1, 28)]# _Zerinvary Lajos_, Mar 13 2009
%o A054452 (Magma) I:=[0,0,1,5]; [n le 4 select I[n] else 5*Self(n-1)-8*Self(n-2)+5*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Mar 26 2015
%o A054452 (Maxima)
%o A054452 makelist(sum(fib(k)*binomial(n+1,k+2),k,0,n),n,0,20); /* _Vladimir Kruchinin_, Oct 21 2016 */
%o A054452 (PARI) concat(vector(2), Vec(x^2/((1-x)^2*(1-3*x+x^2)) + O(x^40))) \\ _Colin Barker_, Jan 28 2017
%Y A054452 Cf. A027941, A054451, A001906, A052952.
%K A054452 easy,nonn
%O A054452 0,4
%A A054452 _Wolfdieter Lang_, Apr 27 2000
%E A054452 More terms from _James Sellers_, Apr 28 2000
%E A054452 a(0) added by _Arkadiusz Wesolowski_, Jun 07 2011