A332181 -id:A332181

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A332180

a(n) = 8*(10^(2n+1)-1)/9 - 8*10^n.

+10
11

0, 808, 88088, 8880888, 888808888, 88888088888, 8888880888888, 888888808888888, 88888888088888888, 8888888880888888888, 888888888808888888888, 88888888888088888888888, 8888888888880888888888888, 888888888888808888888888888, 88888888888888088888888888888, 8888888888888880888888888888888

(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) = A002282(2n+1) - 8*10^n.

G.f.: 8*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.: 8*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

MAPLE

A332180 := n -> 8*((10^(2*n+1)-1)/9-10^n);

MATHEMATICA

Array[8 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]

PROG

(PARI) apply( {A332180(n)=(10^(n*2+1)\9-10^n)*8}, [0..15])

(Python) def A332180(n): return (10**(n*2+1)//9-10**n)*8

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), A002113 (palindromes).

Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).

Cf. A332181 .. A332189 (variants with different middle digit 1, ..., 9).

Subsequence of A006072 (numbers with mirror symmetry about middle), A153806 (strobogrammatic cyclops numbers), and A204095 (numbers whose decimal digits are in {0,8}).

KEYWORD

nonn,base,easy

AUTHOR

M. F. Hasler, Feb 08 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).