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A383527 Partial sums of A005773.

%I A383527 #23 Aug 25 2025 04:38:56
%S A383527 1,2,4,9,22,57,153,420,1170,3293,9339,26642,76363,219728,634312,
%T A383527 1836229,5328346,15494125,45137995,131712826,384900937,1126265986,
%U A383527 3299509114,9676690939,28407473191,83470059532,245465090758,722406781935,2127562036990,6270020029353
%N A383527 Partial sums of A005773.
%C A383527 For p prime of the form 4*k+3 (A002145), a(p) == 0 (mod p).
%C A383527 For p Pythagorean prime (A002144), a(p) - 2 == 0 (mod p).
%C A383527 a(n) (mod 2) = A010059(n).
%C A383527 a(A000069(n+1)) is even.
%C A383527 a(A001969(n+1)) is odd.
%H A383527 Paolo Xausa, <a href="/A383527/b383527.txt">Table of n, a(n) for n = 0..1000</a>
%F A383527 First differences of A211278.
%F A383527 a(n) = Sum_{k=0..n} A167630(n, k).
%F A383527 Binomial transform of A210736 (see Python program).
%F A383527 G.f.: (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)).
%F A383527 E.g.f.: (Integral_{x=-oo..oo} BesselI(0,2*x) dx + (1 + BesselI(0,2*x)) / 2)*exp(x).
%F A383527 Recurrence: n*a(n) = 3*n*a(n-1) - (6-n)*a(n-2) + 3*(2-n)*a(n-3). If n <= 2, a(n) = 2^n.
%F A383527 a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, May 02 2025
%F A383527 From _Mélika Tebni_, May 09 2025: (Start)
%F A383527 a(n) = A257520(n) + A097893(n-1) for n > 0.
%F A383527 a(n) = Sum_{j=0..n}(Sum_{k=0..j} A122896(j, k)).
%F A383527 a(n+2) - 3*a(n+1) + 2*a(n) = A005774(n).
%F A383527 a(n+2) - 4*a(n+1) + 4*a(n) - a(n-1) = A005775(n) for n >= 3. (End)
%p A383527 gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)):
%p A383527 a := n-> coeff(series(gf, x, n+1), x, n):
%p A383527 seq(a(n), n = 0 .. 29);
%p A383527 # Recurrence:
%p A383527 a:= proc(n) option remember; `if`(n<=2, 2^n, 3*a(n-1) - (6/n-1)*a(n-2) + (6/n-3)*a(n-3)) end:
%p A383527 seq(a(n), n = 0 .. 29);
%t A383527 Module[{a, n}, RecurrenceTable[{a[n] == 3*a[n-1] - (6-n)*a[n-2]/n + 3*(2-n)*a[n-3]/n, a[0] == 1, a[1] == 2, a[2] == 4}, a, {n, 0, 30}]] (* _Paolo Xausa_, May 05 2025 *)
%o A383527 (Python)
%o A383527 from math import comb as C
%o A383527 def a(n):
%o A383527   return sum(C(n, k)*abs(sum((-1)**j*C(k, j) for j in range(k//2 + 1))) for k in range(n + 1))
%o A383527 print([a(n) for n in range(30)])
%Y A383527 Cf. A000069, A001969, A002144, A002145, A005773, A005774, A005775, A010059, A097893, A122896, A167630, A210736, A211278, A257520.
%K A383527 nonn,changed
%O A383527 0,2
%A A383527 _Mélika Tebni_, Apr 29 2025