A009977 -id:A009977 - cp's OEIS frontend

A218736 a(n) = (33^n - 1)/32.

0, 1, 34, 1123, 37060, 1222981, 40358374, 1331826343, 43950269320, 1450358887561, 47861843289514, 1579440828553963, 52121547342280780, 1720011062295265741, 56760365055743769454, 1873092046839544391983, 61812037545704964935440, 2039797239008263842869521

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 33 (A009977).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (34,-33).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218737-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 34*Self(n-1)-33*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{34, -33}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218736(n):=(33^n-1)/32$
makelist(A218736(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218736(n)=33^n>>5
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 33*x)).
a(n) = 34*a(n-1) - 33*a(n-2).
a(n) = floor(33^n/32). (End)
E.g.f.: exp(x)*(exp(32*x) - 1)/32. - Stefano Spezia, Mar 24 2023

A100403 Digital root of 6^n.

Original entry on oeis.org

1, 6, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Cino Hilliard, Dec 31 2004

Also the digital root of k^n for any k == 6 (mod 9). - Timothy L. Tiffin, Dec 02 2023

			For n=8, the digits of 6^8 = 1679616 sum to 36, whose digits sum to 9. So, a(8) = 9. - _Timothy L. Tiffin_, Dec 01 2023

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000400, A001024, A007953, A009968, A009977, A009986, A010888, A100401, A159991.

PadRight[{1, 6}, 100, 9] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n) = if( n<2, [1,6][n+1], 9); \\ Joerg Arndt, Dec 03 2023

From Timothy L. Tiffin, Dec 01 2023: (Start)
a(n) = 9 for n >= 2.
G.f.: (1+5x+3x^2)/(1-x).
a(n) = A100401(n) for n <> 1.
a(n) = A010888(A000400(n)) = A010888(A001024(n)) = A010888(A009968(n)) = A010888(A009977(n)) = A010888(A009986(n)) = A010888(A159991(n)). (End)
E.g.f.: 9*exp(x) - 3*x - 8. - Elmo R. Oliveira, Aug 09 2024
a(n) = A007953(6*a(n-1)) = A010888(6*a(n-1)). - Stefano Spezia, Mar 20 2025

A264871 Array read by antidiagonals: T(n,m) = (1+2^n)^m; n,m>=0.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 9, 1, 32, 81, 125, 81, 17, 1, 64, 243, 625, 729, 289, 33, 1, 128, 729, 3125, 6561, 4913, 1089, 65, 1, 256, 2187, 15625, 59049, 83521, 35937, 4225, 129, 1, 512, 6561, 78125, 531441, 1419857, 1185921, 274625, 16641, 257

Offset: 0

R. J. Mathar, Nov 27 2015

			       1,       2,       4,       8,      16,      32,
       1,       3,       9,      27,      81,     243,
       1,       5,      25,     125,     625,    3125,
       1,       9,      81,     729,    6561,   59049,
       1,      17,     289,    4913,   83521, 1419857,
       1,      33,    1089,   35937, 1185921,39135393,

Cf. A000079 (row 0), A000244 (row 1), A000351 (row 2), A001019 (row 3), A001026 (row 4), A009977 (row 5), A000051 (column 1), A028400 (column 2), A136516 (main diagonal), A165327 (upper subdiagonal).

Reverse /@ Table[(1 + 2^(n - m))^m, {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Nov 27 2015 *)

G.f. for row n: 1/(1-(1+2^n)*x). - R. J. Mathar, Dec 15 2015

A165854 Totally multiplicative sequence with a(p) = 33.

Original entry on oeis.org

1, 33, 33, 1089, 33, 1089, 33, 35937, 1089, 1089, 33, 35937, 33, 1089, 1089, 1185921, 33, 35937, 33, 35937, 1089, 1089, 33, 1185921, 1089, 1089, 35937, 35937, 33, 35937, 33, 39135393, 1089, 1089, 1089, 1185921, 33, 1089, 1089, 1185921, 33, 35937, 33

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

33^PrimeOmega[Range[50]] (* Harvey P. Dale, Oct 02 2015 *)

a(n) = 33^bigomega(n); \\ Altug Alkan, Apr 15 2016

a(n) = A009977(A001222(n)) = 33^bigomega(n) = 33^A001222(n).

cp's OEIS Frontend

A218736 a(n) = (33^n - 1)/32.

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A100403 Digital root of 6^n.

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A264871 Array read by antidiagonals: T(n,m) = (1+2^n)^m; n,m>=0.

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A165854 Totally multiplicative sequence with a(p) = 33.

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Mathematica

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