%I A003463 M4209
%S A003463 0,1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,
%T A003463 305175781,1525878906,7629394531,38146972656,190734863281,953674316406,
%U A003463 4768371582031,23841857910156,119209289550781,596046447753906,2980232238769531
%N A003463 (5^n - 1)/4.
%C A003463 5^a(n) is the highest power of 5 dividing (5^n)! - Benoit Cloitre (benoit7848c(AT)orange.fr),
Feb 04 2002
%C A003463 n such that A002294(n) is not divisible by 5 - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jan 14 2003
%C A003463 Numbers n such that a(n) is prime are listed in A004061(n) = {3,7,11,
13,47,127,149,181,619,929,...}. Corresponding prime a(n) are listed
in A086122(n) = {31,19531,12207031,305175781,177635683940025046467781066894531,
...}. 3^(m+1) divides a(2*3^m*k). 31 divides a(3k). 13 divides a(4k).
11 divides a(5k). 71 divides a(5k). 7 divides a(6k). 19531 divides
a(7k). 313 divides a(8k). 19 divides a(9k). 829 divides a(9k). 71
divides a(10k). 521 divides a(10k). 17 divides a(16k). p divides
a(p-1) for all prime p except p = {2,5}. p^(m+1) divides a(p^m*(p-1))
for all prime p except p = {2,5}. p divides a((p-1)/2) for prime
p = {11,19,29,31,41,59,61,71,79,89,101,109,...} = A045468 Primes
congruent to {1, 4} mod 5. p divides a((p-1)/3) for prime p = {13,
67,127,163,181,199,211,241,313,337,367,379,409,457,...}. p divides
a((p-1)/4) for prime p = {101,109,149,181,269,389,401,409,449,461,
521,541,...} = A107219 Primes of the form x^2+100y^2. p divides a((p-1)/
5) for prime p = {31,191,251,271,601,641,761,1091,1861,...}. p divides
a((p-1)/6) for prime p = {181,199,211,241,379,409,631,691,739,769,
1039,...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 23
2007
%C A003463 Starting with 1 = convolution square of A026375: (1, 3, 11, 45, 195,
873,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]
%C A003463 Except for the first term, a(n)=5*a(n-1)+1 (with a(1)=1) [From Vincenzo
Librandi (vincenzo.librandi(AT)tin.it), Oct 29 2009]
%D A003463 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A003463 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques
Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A003463 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A003463 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A003463 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=374">
Encyclopedia of Combinatorial Structures 374</a>
%H A003463 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Repunit.html">Link to a section of The World of Mathematics.</a>
%F A003463 Second binomial transform of A015518; binomial transform of A000302 (preceded
by 0). - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A003463 a(n)=sum{k=1..n, C(n, k)4^(k-1) } - Paul Barry (pbarry(AT)wit.ie), Mar
28 2003
%F A003463 Without leading zero, this is (5*5^n-1)/4, the binomial transform of
A003947. - Paul Barry (pbarry(AT)wit.ie), May 19 2003
%F A003463 a(n) = (-1)^n times the (i, j)-th element of M^n (for all i and j such
that i is not equal to j), where M = ((1, -1, 1, -2), (-1, 1, -2,
1), (1, -2, 1, -1), (-2, 1, -1, 1)). - Simone Severini (ss54(AT)york.ac.uk),
Nov 25 2004
%F A003463 a(n) = A125118(n,4) for n>3. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Nov 21 2006
%F A003463 ((3+sqrt4)^n-(3-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=31. [From
Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008]
%F A003463 a(n)=6*a(n-1)-5*a(n-2), n>1 ; a(0)=0, a(1)=1 . [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Jan 01 2009]
%e A003463 Base 5...........decimal (comment from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jan 14 2007):
%e A003463 0......................0
%e A003463 1......................1
%e A003463 11.....................6
%e A003463 111...................31
%e A003463 1111.................156
%e A003463 11111................781
%e A003463 111111..............3906
%e A003463 1111111............19531
%e A003463 11111111...........97656
%e A003463 111111111.........488281
%e A003463 1111111111.......2441406
%e A003463 etc. ...............etc.
%p A003463 a:=n->sum(5^(n-j),j=1..n): seq(a(n), n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jan 04 2007
%p A003463 A003463:=1/(5*z-1)/(z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
%p A003463 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=6*a[n-1]-5*a[n-2]od: seq(a[n],
n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 21
2008
%t A003463 lst={};Do[p=(5^n-1)/4;AppendTo[lst, p], {n, 0, 5!}];lst [From Vladimir
Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008]
%o A003463 (Other) sage: [lucas_number1(n,6,5) for n in xrange(0, 24)]# [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A003463 (Other) sage: [gaussian_binomial(n,1,5) for n in xrange(0,24)] # [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
%Y A003463 Cf. A004061, A086122, A045468, A107219, A074479.
%Y A003463 A026375 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]
%Y A003463 Sequence in context: A056015 A128740 A026705 this_sequence A026771 A065096
A077352
%Y A003463 Adjacent sequences: A003460 A003461 A003462 this_sequence A003464 A003465
A003466
%K A003463 nonn
%O A003463 0,3
%A A003463 N. J. A. Sloane (njas(AT)research.att.com).
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