A309878 The real part of b(n) where b(n) = (n + b(n-1)) * (1 + i) with b(-1)=0; i = sqrt(-1).
0, 1, 2, 1, -4, -13, -22, -23, -8, 23, 54, 53, -12, -141, -270, -271, -16, 495, 1006, 1005, -20, -2069, -4118, -4119, -24, 8167, 16358, 16357, -28, -32797, -65566, -65567, -32, 131039, 262110, 262109, -36, -524325, -1048614, -1048615, -40, 2097111, 4194262, 4194261, -44
Offset: 0
Examples
For n = -1; b(n) = 0 For n = 0; b(n) = (0+0)*(1+i) = 0 For n = 1; b(n) = (1+0)*(1+i) = 1+i ; a(1) = Re(1+i) = 1 For n = 2; b(n) = (2+1+i)*(1+i) = (3+i)*(1+i) = 3+i+3i-1 = 2+4i ; a(2) = Re(2+4i) = 2 For n = 3; b(n) = 1+9i ; a(3) = 1 For n = 4; b(n) = -4+14i ; a(4) = -4 For n = 5; b(n) = -13+15i ; a(5) = -13 For n = 6; b(n) = -22+8i ; a(6) = -22 For n = 7; b(n) = -23-7i ; a(7) = -23 ... For n = 31; b(n) = -65567-65503i ; a(31) = -65567 For n = 32; b(n) = (32-65567-65503i)*(1+i) = (-65535-65503i)*(1+i) = -65535-65503i-65535i+65503 = -32-131038i ; a(32) = -32 For n = 33; b(n) = 131039-131037i ; a(33) = 131039
Links
- Sela Fried, Sequence A309878, 2024.
- Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 9.
Programs
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PARI
b(n) = if (n==0, 0, (n + b(n-1)) * (1 + I)); for (n=0, 50, print1(real(b(n)), ", ")) \\ Michel Marcus, Aug 21 2019
Formula
Conjectures from Colin Barker, Aug 21 2019: (Start)
G.f.: x*(1 - 2*x) / ((1 - x)^2*(1 - 2*x + 2*x^2)).
a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 2*a(n-4) for n>3.
a(n) = i*((1-i)^n - (1+i)^n + i*n) where i=sqrt(-1).
(End)
E.g.f.: exp(x)*(-x + 2*sin(x)). - Conjectured by Stefano Spezia, Aug 21 2019 after Colin Barker
Conjecture: a(n) = 2*A009545(n)-n. - R. J. Mathar, Mar 06 2022
All conjectures stated above hold true. See links. - Sela Fried, Jul 27 2024.
Comments