Revision History for A332131
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing all changes.
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#6 by M. F. Hasler at Tue Feb 11 08:04:45 EST 2020
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#5 by M. F. Hasler at Tue Feb 11 08:04:24 EST 2020
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| MATHEMATICA
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Array[3 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]
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| STATUS
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approved
editing
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#4 by M. F. Hasler at Sun Feb 09 09:23:43 EST 2020
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#3 by M. F. Hasler at Sun Feb 09 09:23:03 EST 2020
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| NAME
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a(n) = 3*() = (10^(2n+1)-1)/93 - 2*10^n.
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| MAPLE
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A332131 := n -> 3*( -> (10^(2n2*n+1)-1)/93-2*10^n;
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| PROG
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(PARI) apply( {A332131(n)=10^(n*2+1)\9*3-2*10^n}, [0..15])
(Python) def A332131(n): return 10**(n*2+1)//9*3-2*10**n
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| KEYWORD
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nonn,base,easy,changed
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#2 by M. F. Hasler at Sun Feb 09 08:33:51 EST 2020
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| NAME
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allocated for M. F. Hasler
a(n) = 3*(10^(2n+1)-1)/9 - 2*10^n.
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| DATA
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1, 313, 33133, 3331333, 333313333, 33333133333, 3333331333333, 333333313333333, 33333333133333333, 3333333331333333333, 333333333313333333333, 33333333333133333333333, 3333333333331333333333333, 333333333333313333333333333, 33333333333333133333333333333, 3333333333333331333333333333333
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| OFFSET
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0,2
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| COMMENTS
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See A183174 = {1, 3, 7, 61, 90, 92, 269, ...} for the indices of primes.
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| LINKS
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Brady Haran and Simon Pampena, <a href="https://youtu.be/HPfAnX5blO0">Glitch Primes and Cyclops Numbers</a>, Numberphile video (2015).
Patrick De Geest, <a href="http://www.worldofnumbers.com/wing.htm#pwp313">Palindromic Wing Primes: (3)1(3)</a>, updated: June 25, 2017.
Makoto Kamada, <a href="https://stdkmd.net/nrr/3/33133.htm">Factorization of 33...33133...33</a>, updated Dec 11 2018.
<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
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| FORMULA
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a(n) = 3*A138148(n) + 1*10^n = A002277(2n+1) - 2*10^n.
G.f.: (1 + 202*x - 500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
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| MAPLE
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A332131 := n -> 3*(10^(2n+1)-1)/9-2*10^n;
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| MATHEMATICA
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Array[3 (10^(2 # + 1)-1)/9 - 2*10^# &, 15]
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| PROG
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(PARI) apply( {A332131(n)=10^(n*2+1)\9*3-2*10^n}, [0..15])
(Python) def A332131(n): return 10**(n*2+1)//9*3-2*10**n
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| CROSSREFS
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Cf. (A077775-1)/2 = A183174: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).
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| KEYWORD
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allocated
nonn,base,easy
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| AUTHOR
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M. F. Hasler, Feb 09 2020
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| STATUS
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approved
editing
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#1 by M. F. Hasler at Thu Feb 06 20:54:39 EST 2020
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| NAME
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allocated for M. F. Hasler
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| KEYWORD
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allocated
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| STATUS
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approved
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