The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A007759

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A007759

Knopfmacher expansion of sqrt(2): a(2n) = 2*(a(2n-1) + 1)^2 - 1, a(2n+1) = 2*(a(2n)^2 - 1).
(history; published version)

#19 by Charles R Greathouse IV at Thu Sep 08 08:44:35 EDT 2022

PROG

(MAGMAMagma)

Discussion

Thu Sep 08

08:44

OEIS Server: https://oeis.org/edit/global/2944

#18 by Giovanni Resta at Wed Mar 04 08:55:22 EST 2020

STATUS

reviewed

approved

#17 by Michel Marcus at Wed Mar 04 03:13:25 EST 2020

STATUS

proposed

reviewed

#16 by G. C. Greubel at Wed Mar 04 03:08:22 EST 2020

STATUS

editing

proposed

Discussion

Wed Mar 04

03:13

Michel Marcus: ok

#15 by G. C. Greubel at Wed Mar 04 03:07:48 EST 2020

MAPLE

seq(a(n), n = 1..159); # G. C. Greubel, Mar 04 2020

MATHEMATICA

a[n_]:= a[n]= If[n==1, 2, If[EvenQ[n], 2*(a[n-1] +1)^2 -1, 2*a[n-1]^2 -2]]; Table[a[n], {n, 159}] (* G. C. Greubel, Mar 04 2020 *)

PROG

[a(n): n in [1..159]]; // G. C. Greubel, Mar 04 2020

[a(n) for n in (1..159)] # G. C. Greubel, Mar 04 2020

STATUS

proposed

editing

Discussion

Wed Mar 04

03:08

G. C. Greubel: 9 is one more term than the data section.

#14 by G. C. Greubel at Wed Mar 04 02:44:06 EST 2020

STATUS

editing

proposed

Discussion

Wed Mar 04

02:53

Michel Marcus: I would replace the 15's with 7's

#13 by G. C. Greubel at Wed Mar 04 02:43:15 EST 2020

NAME

Knopfmacher expansion of sqrt(2): a(2n) = 2*(a(2n-1) + 1)^2 - 1, a(2n+1) =2a 2*(a(2n)^2 - 21).

LINKS

G. C. Greubel, <a href="/A007759/b007759.txt">Table of n, a(n) for n = 1..11</a>

MAPLE

a:= proc(n) option remember;

if n=1 then 2

elif `mod`(n, 2) = 0 then 2*(a(n-1) +1)^2 -1

else 2*(a(n-1)^2 -1)

end if; end proc;

seq(a(n), n = 1..15); # G. C. Greubel, Mar 04 2020

MATHEMATICA

a[n_]:= a[n]= If[n==1, 2, If[EvenQ[n], 2*(a[n-1] +1)^2 -1, 2*a[n-1]^2 -2]]; Table[a[n], {n, 15}] (* G. C. Greubel, Mar 04 2020 *)

PROG

(MAGMA)

function a(n)

if n eq 1 then return 2;

elif n mod 2 eq 0 then return 2*(a(n-1) +1)^2 -1;

else return 2*(a(n-1)^2 -1);

end if; return a; end function;

[a(n): n in [1..15]]; // G. C. Greubel, Mar 04 2020

(Sage)

@CachedFunction

def a(n):

if (n==1): return 2

elif (n%2==0): return 2*(a(n-1) +1)^2 -1

else: return 2*(a(n-1)^2 -1)

[a(n) for n in (1..15)] # G. C. Greubel, Mar 04 2020

STATUS

approved

editing

#12 by Bruno Berselli at Wed Feb 20 05:39:08 EST 2019

STATUS

proposed

approved

#11 by Michel Marcus at Wed Feb 20 05:36:35 EST 2019

STATUS

editing

proposed

#10 by Michel Marcus at Wed Feb 20 05:36:28 EST 2019

REFERENCES

A. Knopfmacher and J. Knopfmacher, "An alternating product representation for real numbers", in Applications of Fibonacci numbers, Vol. 3 (Kluwer 1990), pp. 209-216.

LINKS

A. Knopfmacher and J. Knopfmacher, <a href="https://doi.org/10.1007/978-94-009-1910-5_24">An alternating product representation for real numbers</a>, in Applications of Fibonacci numbers, Vol. 3 (Kluwer 1990), pp. 209-216.

PROG

(PARI) a(n) = if (n==1, 2, if (n % 2, 2*a(n-1)^2 - 2, 2*(a(n-1)+1)^2 - 1)); \\ Michel Marcus, Feb 20 2019

CROSSREFS

Cf. A002193 (sqrt(2)), A001601.

STATUS

approved

editing