A165155
a(n) = 100*a(n-1) + 11^(n-1) for n>0, a(0)=0.
Original entry on oeis.org
0, 1, 111, 11221, 1123431, 112357741, 11235935151, 1123595286661, 112359548153271, 11235955029685981, 1123595505326545791, 112359550558592003701, 11235955056144512040711, 1123595505617589632447821, 112359550561793485956926031, 11235955056179728345526186341
Offset: 0
From _Mark Dols_, Jan 28 2010: (Start)
As triangle:
........... 1
......... 1 1 1
....... 1 1 2 2 1
..... 1 1 2 3 4 3 1
... 1 1 2 3 5 7 7 4 1
. 1 1 2 3 5 9 3 5 1 5 1
1 1 2 3 5 9 5 2 8 6 6 6 1
(Mirrored version of A162741) (End)
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[(1/89)*(100^n-11^n): n in [0..40]] // Vincenzo Librandi, Dec 05 2010
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RecurrenceTable[{a[0]==0,a[n]==100a[n-1]+11^(n-1)},a,{n,40}] (* Harvey P. Dale, Feb 20 2016 *)
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[(10^(2*n) - 11^n)/89 for n in range(41)] # G. C. Greubel, Feb 09 2023
A165154
a(n) = 100*a(n-1) + (-9)^(n-1) for n>0, a(0)=0.
Original entry on oeis.org
0, 1, 91, 9181, 917371, 91743661, 9174307051, 917431236541, 91743118871131, 9174311930159821, 917431192628561611, 91743119266342945501, 9174311926602913490491, 917431192660573778585581, 91743119266054835992729771, 9174311926605506476065432061
Offset: 0
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[(1/109)*(100^n-(-9)^n): n in [0..20]]; // Vincenzo Librandi, Jun 10 2011
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LinearRecurrence[{91,900}, {0,1}, 40] (* G. C. Greubel, Feb 09 2023 *)
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Vec(x/((1+9*x)*(1-100*x)) + O(x^20)) \\ Colin Barker, Oct 02 2015
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[(100^n-(-9)^n)/109 for n in range(41)] # G. C. Greubel, Feb 09 2023
A164915
Inverse of binomial matrix (10^n,1) A164899. (See A164899 for companion sequence.)
Original entry on oeis.org
1, 1, 10, 1, 9, 100, 1, 8, 90, 1000, 1, 7, 81, 900, 10000, 1, 6, 73, 810, 9000, 100000, 1, 5, 66, 729, 8100, 90000, 1000000, 1, 4, 60, 656, 7290, 81000, 900000, 10000000, 1, 3, 55, 590, 6561, 72900, 810000, 9000000, 100000000
Offset: 1
Matrix array, A(n, k), begins:
1, 10, 100, 1000, 10000, 100000, ...
1, 9, 90, 900, 9000, 90000, ...
1, 8, 81, 810, 8100, 81000, ...
1, 7, 73, 729, 7290, 72900, ...
1, 6, 66, 656, 6561, 65610, ...
1, 5, 60, 590, 5905, 59049, ...
1, 4, 55, 530, 5315, 53144, ...
Antidiagonal triangle, T(n, k), begins as:
1;
1, 10;
1, 9, 100;
1, 8, 90, 1000;
1, 7, 81, 900, 10000;
1, 6, 73, 810, 9000, 100000;
1, 5, 66, 729, 8100, 90000, 1000000;
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function T(n,k) // T = A164915
if k eq n then return 10^(n-1);
elif k eq 1 then return 1;
else return T(n-1,k) - T(n-2,k-1);
end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Feb 10 2023
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T[n_, k_]:= T[n, k]= If[k==n, 10^(n-1), If[k==1, 1, T[n-1,k] - T[n-2, k -1]]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Feb 10 2023 *)
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def T(n,k): # T = A164915
if (k==n): return 10^(n-1)
elif (k==1): return 1
else: return T(n-1,k) - T(n-2,k-1)
flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Feb 10 2023
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