%I #38 Mar 23 2023 17:17:23

%S 0,1,33,1057,33825,1082401,34636833,1108378657,35468117025,

%T 1134979744801,36319351833633,1162219258676257,37191016277640225,

%U 1190112520884487201,38083600668303590433,1218675221385714893857,38997607084342876603425,1247923426698972051309601

%N a(n) = (2^(5*n) - 1)/31.

%C Partial sums of powers of 32 (A009976), a.k.a. q-numbers for q=32. - _M. F. Hasler_, Nov 05 2012

%D A. K. Devaraj, "Minimum Universal Exponent Generalisation of Fermat's Theorem", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.

%H Vincenzo Librandi, <a href="/A132469/b132469.txt">Table of n, a(n) for n = 0..600</a>

%H Quynh Nguyen, Jean Pedersen, and Hien T. Vu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Pedersen/pedersen2.html">New Integer Sequences Arising From 3-Period Folding Numbers</a>, Vol. 19 (2016), Article 16.3.1. See Table 1.

%H <a href="/index/Par#partial">Index entries related to partial sums</a>.

%H <a href="/index/Q#q-numbers">Index entries related to q-numbers</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (33,-32).

%F a(n) = (32^n - 1)/31 = floor(32^n/31) = Sum_{k=0..n} 32^k. - _M. F. Hasler_, Nov 05 2012

%F G.f.: x/((1 - x)*(1 - 32*x)). - _Bruno Berselli_, Nov 06 2012

%F E.g.f.: exp(x)*(exp(31*x) - 1)/31. - _Stefano Spezia_, Mar 23 2023

%t Table[(2^(5 n) - 1)/31, {n, 16}] (* _Robert G. Wilson v_ *)

%t LinearRecurrence[{33, -32}, {0, 1}, 30] (* _Vincenzo Librandi_, Nov 07 2012 *)

%o (Sage) [gaussian_binomial(5*n,1,2)/31 for n in range(1,17)] # _Zerinvary Lajos_, May 28 2009

%o (Magma) [n le 2 select n-1 else 33*Self(n-1) - 32*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 07 2012

%o (PARI) A132469(n)=32^n\31 \\ _M. F. Hasler_, Nov 07 2012

%o (Maxima) A132469(n):=(32^n-1)/31$

%o makelist(A132469(n),n,0,30); /* _Martin Ettl_, Nov 07 2012 */

%Y 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, A218736-A218753, A133853, A094028, A218723.

%K nonn,easy

%O 0,3

%A _A.K. Devaraj_, Aug 22 2007

%E Edited and extended by _Robert G. Wilson v_, Aug 22 2007

%E Edited and extended to offset 0 by _M. F. Hasler_, Nov 05 2012