The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A048899 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-1). Here the 3 (mod 5) case (except for n=0). 31
 0, 3, 18, 68, 443, 1068, 1068, 32318, 110443, 1672943, 3626068, 23157318, 120813568, 1097376068, 1097376068, 19407922943, 49925501068, 355101282318, 355101282318, 15613890344818, 15613890344818, 110981321985443 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is the root congruent to 3 (mod 5) for n>0. The other case with the 2 (mod 5) numbers (except for n=0) is given in A048898. - Wolfdieter Lang, Feb 19 2016 REFERENCES J. H. Conway, The Sensual Quadratic Form, p. 118, Mathematical Association of America, 1997, The Carus Mathematical Monographs, Number 26. K. Mahler, Introduction to p-Adic Numbers and Their Functions, Cambridge, 1973, p. 35. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1431 FORMULA a(n) = 5^n - A048898(n), n>=1. a(n) = A066601(5^n), n>=0. 0 <= a(n) < 5^n. 5^n divides a(n)^2 + 1. From Wolfdieter Lang, Apr 28 2012: (Start) Recurrence: a(n) = a(n-1)^5 (mod 5^n), a(1) = 3, n>=2. See the Pari program below, and the J.- F. Alcover Mathematica program for A048898. a(n) = 3^(5^(n-1)) (mod 5^n), n>=1. Compare with the above given formula involving A066601. a(n)*a(n-1) + 1 == 0 (mod 5^(n-1)), n>=1. (a(n)^2 + 1)/5^n = A210849(n), n>=0. (End) Another recurrence: a(n) = modp(a(n-1) + 4*(a(n-1)^2 + 1), 5^n), n >= 2, a(1) = 3. Here modp(a, m) is the representative from {0, 1, ... ,|m|-1} of the residue class a modulo m. Note that a(n) is in the residue class of a(n-1) modulo 5^(n-1) (see Hensel lifting). - Wolfdieter Lang, Feb 28 2016 EXAMPLE a(2) = 18 because the two roots of x^2 + 1 == 0 (mod 5^2) are 7 and 18 and 18 == 3 (mod 5). For 7 see A048898(2). MATHEMATICA Join[{0}, RecurrenceTable[{a[1] == 3, a[n] == Mod[a[n-1]^5, 5^n]}, a, {n, 25}]] (* Vincenzo Librandi, Feb 29 2016 * ) PROG (PARI) {a(n) = if( n<2, 3, a(n - 1)^5) % 5^n} (MAGMA) [n le 2 select 3*(n-1) else Self(n-1)^5 mod 5^(n-1): n in [1..30]]; // Vincenzo Librandi, Feb 29 2016 CROSSREFS Cf. A048898, A066601, A210851. Sequence in context: A110689 A027333 A026576 * A107583 A157535 A098522 Adjacent sequences:  A048896 A048897 A048898 * A048900 A048901 A048902 KEYWORD nonn,easy AUTHOR Michael Somos, Jul 26 1999 EXTENSIONS Example corrected by Wolfdieter Lang, Apr 28 2012 Name clarified by Wolfdieter Lang, Feb 19 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 26 05:10 EST 2020. Contains 331273 sequences. (Running on oeis4.)