A000221 Take sum of squares of digits of previous term; start with 5.

5, 25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy
• nice

Comments

Essentially the same as A080709, cf. formula. - M. F. Hasler, May 24 2009

As the orbit of 5 under A003132, this could as well start with index 0. - M. F. Hasler, Apr 27 2018

References

• R. Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 83.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 25.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Crossrefs

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4), A008460 (starting with 6), A008462 (starting with 8), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

Programs

Haskell

a000221 n = a000221_list !! (n-1)
a000221_list = iterate a003132 5
-- Reinhard Zumkeller, Mar 04 2013

Magma

[5, 25, 29, 85] cat &cat[[89, 145, 42, 20, 4, 16, 37, 58]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

Mathematica

NestList[Plus @@ IntegerDigits[ # ]^2 &, 5, 50]
PadRight[{5,25,29,85},120,{4,16,37,58,89,145,42,20}] (* Harvey P. Dale, Jan 14 2022 *)

PARI

A000221(n)=[20,4,16,37,58,89,145,42,5,25,29,85][n%8+8^(n<5)] \\ M. F. Hasler, May 24 2009, edited Apr 27 2018

Formula

Ultimately periodic with period 8.

a(n) = A080709(n) for n >= 5. - M. F. Hasler, May 24 2009

a(n+1) = A003132(a(n)). - Reinhard Zumkeller, Dec 19 2011