A383527 Partial sums of A005773.

1, 2, 4, 9, 22, 57, 153, 420, 1170, 3293, 9339, 26642, 76363, 219728, 634312, 1836229, 5328346, 15494125, 45137995, 131712826, 384900937, 1126265986, 3299509114, 9676690939, 28407473191, 83470059532, 245465090758, 722406781935, 2127562036990, 6270020029353

Offset: 0

Author

Mélika Tebni, Apr 29 2025

Keywords

• nonn
• changed

Comments

For p prime of the form 4*k+3 (A002145), a(p) == 0 (mod p).

For p Pythagorean prime (A002144), a(p) - 2 == 0 (mod p).

a(n) (mod 2) = A010059(n).

a(A000069(n+1)) is even.

a(A001969(n+1)) is odd.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000069, A001969, A002144, A002145, A005773, A005774, A005775, A010059, A097893, A122896, A167630, A210736, A211278, A257520.

Programs

Maple

gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)):
a := n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n = 0 .. 29);
# Recurrence:
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:
seq(a(n), n = 0 .. 29);

Mathematica

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 *)

Python

from math import comb as C
def a(n):
  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))
print([a(n) for n in range(30)])

Formula

First differences of A211278.

a(n) = Sum_{k=0..n} A167630(n, k).

Binomial transform of A210736 (see Python program).

G.f.: (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)).

E.g.f.: (Integral_{x=-oo..oo} BesselI(0,2*x) dx + (1 + BesselI(0,2*x)) / 2)*exp(x).

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.

a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, May 02 2025

From Mélika Tebni, May 09 2025: (Start)

a(n) = A257520(n) + A097893(n-1) for n > 0.

a(n) = Sum_{j=0..n}(Sum_{k=0..j} A122896(j, k)).

a(n+2) - 3*a(n+1) + 2*a(n) = A005774(n).

a(n+2) - 4*a(n+1) + 4*a(n) - a(n-1) = A005775(n) for n >= 3. (End)