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Search: a332121 -id:a332121
Displaying 1-10 of 11 results found.
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A332120
a(n) = 2*(10^(2n+1)-1)/9 - 2*10^n.
+10
16
0, 202, 22022, 2220222, 222202222, 22222022222, 2222220222222, 222222202222222, 22222222022222222, 2222222220222222222, 222222222202222222222, 22222222222022222222222, 2222222222220222222222222, 222222222222202222222222222, 22222222222222022222222222222, 2222222222222220222222222222222
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 2*A138148(n) = A002276(2n+1) - 2*10^n.
G.f.: 2*x*(101 - 200*x)/((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.
E.g.f.: 2*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024
MAPLE
A332120 := n -> 2*((10^(2*n+1)-1)/9-10^n);
MATHEMATICA
Array[2 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
PROG
(PARI) apply( {A332120(n)=(10^(n*2+1)\9-10^n)*2}, [0..15])
(Python) def A332120(n): return (10**(n*2+1)//9-10**n)*2
CROSSREFS
Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332130 .. A332190 (variants with different repeated digit 3, ..., 9).
Cf. A332121 .. A332129 (variants with different middle digit 1, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 09 2020
STATUS
approved
A332171
a(n) = 7*(10^(2n+1)-1)/9 - 6*10^n.
+10
9
1, 717, 77177, 7771777, 777717777, 77777177777, 7777771777777, 777777717777777, 77777777177777777, 7777777771777777777, 777777777717777777777, 77777777777177777777777, 7777777777771777777777777, 777777777777717777777777777, 77777777777777177777777777777, 7777777777777771777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
For n == 0 or n == 2 (mod 6), there is no obvious divisibility pattern.
According to M. Kamada, n = 116 is the only index of a prime up to n = 10^5.
LINKS
Table of n, a(n) for n=0..15.
Makoto Kamada, Factorization of 77...77177...77.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 10^n.
For n == 1 (mod 3), 3 | a(n) and a(n)/3 = 259*(10^(2n+1)-1)/999 - 2*10^n;
for n == 3 or 5 (mod 6), 13 | a(n) and a(n)/13 = (A(n)-1)*10^n + B(n), where A(n) (resp. B(n)) are the n leftmost (resp. rightmost) digits of 59829*(10^(ceiling(n/6)*6)-1)/(10^6-1).
From Colin Barker, Feb 07 2020: (Start)
G.f.: (1 + 606*x - 1300*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.
(End)
E.g.f.: (1/9)*exp(x)*(70*exp(99*x) - 54*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020
MATHEMATICA
Array[7 (10^(2 # + 1) - 1)/9 - 6*10^# &, 15, 0] (* or *)
CoefficientList[Series[(1 + 606 x - 1300 x^2)/((1 - x) (1 - 10 x) (1 - 100 x)), {x, 0, 15}], x] (* Michael De Vlieger, Feb 08 2020 *)
Table[FromDigits[Join[PadRight[{}, n, 7], {1}, PadRight[{}, n, 7]]], {n, 0, 20}] (* or *) LinearRecurrence[ {111, -1110, 1000}, {1, 717, 77177}, 20] (* Harvey P. Dale, Apr 04 2024 *)
PROG
(PARI) apply( {A332171(n)=10^(n*2+1)\9*7-6*10^n}, [0..15])
(PARI) Vec((1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^15)) \\ Colin Barker, Feb 07 2020
(Python) def A332171(n): return 10**(n*2+1)//9*7-6*10^n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332170 .. A332179 (variants with different middle digit 2, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 06 2020
STATUS
approved
A332191
a(n) = 10^(2n+1) - 1 - 8*10^n.
+10
9
1, 919, 99199, 9991999, 999919999, 99999199999, 9999991999999, 999999919999999, 99999999199999999, 9999999991999999999, 999999999919999999999, 99999999999199999999999, 9999999999991999999999999, 999999999999919999999999999, 99999999999999199999999999999, 9999999999999991999999999999999
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
See A183184 = {1, 5, 13, 43, 169, 181, ...} for the indices of primes.
LINKS
Table of n, a(n) for n=0..15.
Patrick De Geest, Palindromic Wing Primes: (9)1(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 9999199...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 9*A138148(n) + 10^n.
G.f.: (1 + 808*x - 1700*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.
MAPLE
A332191 := n -> 10^(n*2+1)-1-8*10^n;
MATHEMATICA
Array[ 10^(2 # + 1)-1-8*10^# &, 15, 0]
PROG
(PARI) apply( {A332191(n)=10^(n*2+1)-1-8*10^n}, [0..15])
(Python) def A332191(n): return 10**(n*2+1)-1-8*10^n
CROSSREFS
Cf. (A077776-1)/2 = A183184: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332121 .. A332181 (variants with different repeated digit 2, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
A332131
a(n) = (10^(2n+1)-1)/3 - 2*10^n.
+10
4
1, 313, 33133, 3331333, 333313333, 33333133333, 3333331333333, 333333313333333, 33333333133333333, 3333333331333333333, 333333333313333333333, 33333333333133333333333, 3333333333331333333333333, 333333333333313333333333333, 33333333333333133333333333333, 3333333333333331333333333333333
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
See A183174 = {1, 3, 7, 61, 90, 92, 269, ...} for the indices of primes.
LINKS
Table of n, a(n) for n=0..15.
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)1(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33133...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
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.
MAPLE
A332131 := n -> (10^(2*n+1)-1)/3-2*10^n;
MATHEMATICA
Array[3 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]
PROG
(PARI) apply( {A332131(n)=10^(n*2+1)\3-2*10^n}, [0..15])
(Python) def A332131(n): return 10**(n*2+1)//3-2*10**n
CROSSREFS
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).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 09 2020
STATUS
approved
A332181
a(n) = 8*(10^(2n+1)-1)/9 - 7*10^n.
+10
3
1, 818, 88188, 8881888, 888818888, 88888188888, 8888881888888, 888888818888888, 88888888188888888, 8888888881888888888, 888888888818888888888, 88888888888188888888888, 8888888888881888888888888, 888888888888818888888888888, 88888888888888188888888888888, 8888888888888881888888888888888
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 8*A138148(n) + 10^n = A002282(2n+1) - 7*10^n.
G.f.: (1 + 707*x - 1500*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.
MAPLE
A332181 := n -> 8*(10^(2*n+1)-1)/9-7*10^n;
MATHEMATICA
Array[8 (10^(2 # + 1)-1)/9 - 7*10^# &, 15, 0]
PROG
(PARI) apply( {A332181(n)=10^(n*2+1)\9*8-7*10^n}, [0..15])
(Python) def A332181(n): return 10**(n*2+1)//9*8-7*10**n
CROSSREFS
Cf. (A077776-1)/2 = A183184: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
A332141
a(n) = 4*(10^(2*n+1)-1)/9 - 3*10^n.
+10
2
1, 414, 44144, 4441444, 444414444, 44444144444, 4444441444444, 444444414444444, 44444444144444444, 4444444441444444444, 444444444414444444444, 44444444444144444444444, 4444444444441444444444444, 444444444444414444444444444, 44444444444444144444444444444, 4444444444444441444444444444444
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 4*A138148(n) + 1*10^n = A002278(2n+1) - 3*10^n.
G.f.: (1 + 303*x - 700*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.
MAPLE
A332141 := n -> 4*(10^(2*n+1)-1)/9-3*10^n;
MATHEMATICA
Array[4 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
LinearRecurrence[{111, -1110, 1000}, {1, 414, 44144}, 20] (* or *) Table[ FromDigits[Join[PadRight[{}, n, 4], {1}, PadRight[{}, n, 4]]], {n, 0, 20}](* Harvey P. Dale, Aug 17 2020 *)
PROG
(PARI) apply( {A332141(n)=10^(n*2+1)\9*4-3*10^n}, [0..15])
(Python) def A332141(n): return 10**(n*2+1)//9*4-3*10**n
CROSSREFS
Cf. A002275 (repunits R_n = (10^n-1)/9), A002278 (4*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. A332140 .. A332149 (variants with different middle digit 0, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 09 2020
STATUS
approved
A332151
a(n) = 5*(10^(2*n+1)-1)/9 - 4*10^n.
+10
2
1, 515, 55155, 5551555, 555515555, 55555155555, 5555551555555, 555555515555555, 55555555155555555, 5555555551555555555, 555555555515555555555, 55555555555155555555555, 5555555555551555555555555, 555555555555515555555555555, 55555555555555155555555555555, 5555555555555551555555555555555
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..499
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 5*A138148(n) + 10^n = A002279(2n+1) - 4*10^n.
G.f.: (1 + 404*x - 900*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.
MAPLE
A332151 := n -> 5*(10^(2*n+1)-1)/9-4*10^n;
MATHEMATICA
Array[5 (10^(2 # + 1)-1)/9 - 4*10^# &, 15, 0]
Table[With[{c=PadRight[{}, n, 5]}, FromDigits[Join[c, {1}, c]]], {n, 0, 20}] (* Harvey P. Dale, Mar 16 2021 *)
PROG
(PARI) apply( {A332151(n)=10^(n*2+1)\9*5-4*10^n}, [0..15])
(Python) def A332151(n): return 10**(n*2+1)//9*5-4*10**n
CROSSREFS
Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*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. A332150 .. A332159 (variants with different middle digit 0, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 09 2020
STATUS
approved
A332161
a(n) = 6*(10^(2*n+1)-1)/9 - 5*10^n.
+10
2
1, 616, 66166, 6661666, 666616666, 66666166666, 6666661666666, 666666616666666, 66666666166666666, 6666666661666666666, 666666666616666666666, 66666666666166666666666, 6666666666661666666666666, 666666666666616666666666666, 66666666666666166666666666666, 6666666666666661666666666666666
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 6*A138148(n) + 1*10^n = A002280(2n+1) - 5*10^n.
G.f.: (1 + 505*x - 1100*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.
MAPLE
A332161 := n -> 6*(10^(2*n+1)-1)/9-5*10^n;
MATHEMATICA
Array[6 (10^(2 # + 1)-1)/9 - 5*10^# &, 15, 0]
PROG
(PARI) apply( {A332161(n)=10^(n*2+1)\9*6-5*10^n}, [0..15])
(Python) def A332161(n): return 10**(n*2+1)//9*6-5*10**n
CROSSREFS
Cf. A002275 (repunits R_n = (10^n-1)/9), A002280 (6*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. A332160 .. A332169 (variants with different middle digit 0, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 09 2020
STATUS
approved
A332184
a(n) = 8*(10^(2n+1)-1)/9 - 4*10^n.
+10
2
4, 848, 88488, 8884888, 888848888, 88888488888, 8888884888888, 888888848888888, 88888888488888888, 8888888884888888888, 888888888848888888888, 88888888888488888888888, 8888888888884888888888888, 888888888888848888888888888, 88888888888888488888888888888, 8888888888888884888888888888888
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 8*A138148(n) + 4*10^n = A002282(2n+1)- 4*10^n = 4*A332121(n).
G.f.: (4 + 404*x - 1200*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.
MAPLE
A332184 := n -> 8*(10^(2*n+1)-1)/9-4*10^n;
MATHEMATICA
Array[8 (10^(2 # + 1)-1)/9- 4*10^# &, 15, 0]
PROG
(PARI) apply( {A332184(n)=10^(n*2+1)\9*8-4*10^n}, [0..15])
(Python) def A332184(n): return 10**(n*2+1)//9*8-4*10**n
CROSSREFS
Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
A332142
a(n) = 4*(10^(2*n+1)-1)/9 - 2*10^n.
+10
1
2, 424, 44244, 4442444, 444424444, 44444244444, 4444442444444, 444444424444444, 44444444244444444, 4444444442444444444, 444444444424444444444, 44444444444244444444444, 4444444444442444444444444, 444444444444424444444444444, 44444444444444244444444444444, 4444444444444442444444444444444
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 4*A138148(n) + 2*10^n = A002278(2n+1) - 2*10^n = 2*A332121(n).
G.f.: (2 + 202*x - 600*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.
MAPLE
A332142 := n -> 4*(10^(2*n+1)-1)/9-2*10^n;
MATHEMATICA
Array[4 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]
PROG
(PARI) apply( {A332142(n)=10^(n*2+1)\9*4-2*10^n}, [0..15])
(Python) def A332142(n): return 10**(n*2+1)//9*4-2*10**n
CROSSREFS
Cf. A002275 (repunits R_n = (10^n-1)/9), A002278 (4*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 09 2020
STATUS
approved
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