A188381 -id:A188381 - cp's OEIS frontend

A020714 a(n) = 5 * 2^n.

5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

David W. Wilson

Same as Pisot sequences E(5,10), L(5,10), P(5,10), T(5,10). See A008776 for definitions of Pisot sequences.
The first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini, Sep 07 2005
5 times powers of 2. - Omar E. Pol, Dec 16 2008
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
With the addition of "2, 3," at the beginning, this sequence gives terms (n + 3) through the first term greater than 2^n, for n odd, of the negabinary Keith sequence for 2^n, thus proving that with the exception of 2 itself, no odd-indexed power of 2 is a negabinary Keith number (see A188381). - Alonso del Arte, Feb 02 2012
Let b(0) = 5 and b(n+1) = the smallest number not in the sequence such that b(n+1) - Product_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n+2) = a(n) for n > 0. - Derek Orr, Jan 15 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..238
John Elias, Illustration: Alternating Tetrahedrons of Tetrahedrons
Tanya Khovanova, Recursive Sequences.
Petro Kosobutskyy, Anastasiia Yedyharova, and Taras Slobodzyan, From Newton's binomial and Pascal's triangle to Collatz's problem, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, 121-127.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1003.
Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (2).

Row sums of (4, 1)-Pascal triangle A093561.
Row sums of (9, 1)-Pascal triangle A093644.
Row sums of (1, 4)-Pascal triangle A095666 (with leading 4).
Cf. A000079, A001045, A008776, A051916, A118416, A173786, A188381.

[5*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

Table[5*2^n, {n, 0, 31}] (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
NestList[2#&,5,40] (* Harvey P. Dale, Mar 13 2022 *)

a(n)=5<Charles R Greathouse IV, Sep 24 2015

a(n) = 5*2^n. a(n) = 2*a(n-1).
G.f.: 5/(1-2*x).
If m is a term greater than 5 of this sequence then m = 5*phi(phi(m)). - Farideh Firoozbakht, Aug 16 2005
a(n) = A118416(n+1,3) for n>2. - Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n)*5. - Omar E. Pol, Dec 16 2008
a(n) = A173786(n+2,n) for n > 1. - Reinhard Zumkeller, Feb 28 2010
a(n) = A001045(n+4) - A001045(n). - Paul Curtz, Nov 08 2012
Sum_{n>=1} 1/a(n) = 2/5. - Amiram Eldar, Oct 28 2020
E.g.f.: 5*exp(2*x). - Stefano Spezia, May 15 2021

A007629 Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).

Original entry on oeis.org

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171, 33445755, 44121607

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Numbers k > 9 with following property: form a sequence {b(i)} whose initial terms are the t digits of k, later terms given by rule that b(i) = sum of t previous terms; then k itself appears in the sequence.
Called Sep-Numbers by Baumann (2004). - N. J. A. Sloane, Mar 02 2014
Sometimes named after the American mathematician, software engineer and author Mike Keith (b. 1955), who introduced them in 1987 as "repfigit numbers". - Amiram Eldar, Jun 27 2021

			197 is a term since sequence {b(i)} (see Comments) is A186830 = { 1, 9, 7, 17, 33, 57, 107, 197, ... }, which contains 197.

Charles Ashbacher, J. Rec. Math., Vol. 21, No. 4 (1989), p. 310.
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 197, p. 59, Ellipses, Paris 2008.
Mike Keith, Repfigit Numbers, J. Recreational Math., Vol. 19, No. 2 (1987), pp. 41-42.
Clifford A. Pickover, All Known Replicating Fibonacci Digits Less Than One Billion, J. Recreational Math., Vol. 22, No. 3, p. 176, 1990.
Clifford A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 229.
Clifford A. Pickover, Wonders of Numbers, "Looping Replicating Fibonacci digits", pp. 174-5, OUP 2000.
K. Sherriff, Computing Replicating Fibonacci Digits, J. Recreational Math., Vol. 26, No. 3, p. 191, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, see p. 71.

N. J. A. Sloane, Table of n, a(n) for n = 1..94 [Taken from first Keith link below.]
Rüdeger Baumann, Sep-Zahlen or Sep-Numbers, DERIVE Newsletter, #53 (2004), p. 33.
Jhon J. Bravo, Sergio Guzmán, and Florian Luca, Repdigit Keith numbers, Lithuanian Mathematical Journal, Vol. 53, No. 2 (April 2013), pp. 143-148.
Edmund Copeland and Brady Haran, Keith Numbers, Numberphile video (2012).
Mike Keith, Keith numbers.
Mike Keith, Determination of All Keith Numbers Up to 10^19.
Mike Keith, Power-sum numbers, J. Recreational Mathematics, Vol. 18, No. 4 (1986), pp. 275-278. (Annotated scanned copy)
Martin Klazar and Florian Luca, Counting Keith numbers, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.2.
Madras Math's Amazing Number Facts, Repfigits.
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Eric Weisstein's World of Mathematics, Keith Number.
Wikipedia, Keith number.

Cf. A006576, A048970, A050235, A186830. See A130010 for another version.
Cf. A162724, A187713, A188195-A188200 (base 2, 5, 3-4, 6-9).
Cf. A188380 (balanced ternary), A188381 (base -2).
Cf. A188201 (least base-n Keith number >= n).
Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920, A281921 (starting with n^k, 2<=k<=10).

import Data.Char (digitToInt)
a007629 n = a007629_list !! (n-1)
a007629_list = filter isKeith [10..] where
   isKeith n = repfigit $ reverse $ map digitToInt $ show n where
      repfigit ns = s == n || s < n && (repfigit $ s : init ns) where
         s = sum ns
-- Reinhard Zumkeller, Nov 04 2010, Mar 31 2011

isA007629 := proc(n)
    local L,t,a ;
    if n < 10 then
        return false;
    end if;
    L := ListTools[Reverse](convert(n,base,10)) ;
    t :=  nops(L) ;
    while true do
        a := add(op(-i,L),i=1..t) ;
        L := [op(L),a] ;
        if a > n then
            return false;
        elif a = n then
            return true;
        end if;
    end do:
end proc:
for n from 1 do
    if isA007629(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016

keithQ[n_Integer] := Module[{b = IntegerDigits[n], s, k = 0}, s = Total[b]; While[s < n, AppendTo[b, s]; k++; s = 2*s - b[[k]]]; s == n]; Select[Range[10, 100000], keithQ] (* T. D. Noe, Mar 15 2011 *)
nxt[n_]:=Rest[Flatten[Join[{n,Total[n]}]]]; repfigitQ[m_]:=MemberQ[ NestWhileList[ nxt,IntegerDigits[m],Max[#]<=m&][[All,-1]],m]; Select[ Range[10,45*10^6],repfigitQ] (* Harvey P. Dale, Sep 02 2016 *)
keithQ[n_, e_] := Last[NestWhile[Rest[Append[#, Apply[Plus, #]]]&, IntegerDigits[n^e], Last[#]9
a007629[n_] := Select[Range[10, n], keithQ[#, 1]&]
a007629[45*10^6] (* Hartmut F. W. Hoft, Jun 02 2021 *)

is(n)=if(n<14,return(0));my(v=digits(n),t=#v);while(v[#v]Charles R Greathouse IV, Feb 01 2013

A007629_list = []
for n in range(10,10**9):
    x = [int(d) for d in str(n)]
    y = sum(x)
    while y < n:
        x, y = x[1:]+[y], 2*y-x[0]
    if y == n:
        A007629_list.append(n) # Chai Wah Wu, Sep 12 2014

12th term corrected from 2508 to 2580, Aug 15 1997
More terms from Mike Keith, Feb 15 1999
Keith's old links fixed and C. Ashbacher's name added by Christopher Carl Heckman, Nov 18 2010

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A020714 a(n) = 5 * 2^n.

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A007629 Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).

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