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A003463
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(5^n - 1)/4.
(Formerly M4209)
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38
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0, 1, 6, 31, 156, 781, 3906, 19531, 97656, 488281, 2441406, 12207031, 61035156, 305175781, 1525878906, 7629394531, 38146972656, 190734863281, 953674316406, 4768371582031, 23841857910156, 119209289550781, 596046447753906, 2980232238769531
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 5^a(n) is the highest power of 5 dividing (5^n)! - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 04 2002
n such that A002294(n) is not divisible by 5 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 14 2003
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
Starting with 1 = convolution square of A026375: (1, 3, 11, 45, 195, 873,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). [From Milan R. Janjic (agnus(AT)blic.net), Jan 27 2010]
This is the sequence A(0,1;4,5;2) = A(0,1;6,-5;0) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 18 2010]
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 374
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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Repunit
W. Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 18 2010]
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FORMULA
| Second binomial transform of A015518; binomial transform of A000302 (preceded by 0). - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
a(n)=sum{k=1..n, C(n, k)4^(k-1) } - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
Without leading zero, this is (5*5^n-1)/4, the binomial transform of A003947. - Paul Barry (pbarry(AT)wit.ie), May 19 2003
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 (simoseve(AT)gmail.com), Nov 25 2004
a(n) = A125118(n,4) for n>3. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2006
((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]
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]
Contribution from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 18 2010: (Start)
O.g.f.: x/((1-5*x)*(1-x)).
a(n) = 4*a(n-1) + 5*a(n-2) + 2, a(0)=0, a(1)=1.
a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3), a(0)=0, a(1)=1, a(2)=6. Observation by G. Detlefs. See the W. Lang comment and link, (End)
a(n) = 5*a(n-1)+1 with a(0)=0. [From Vincenzo Librandi, Nov 17 2010]
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EXAMPLE
| Base 5...........decimal (comment from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 14 2007):
0......................0
1......................1
11.....................6
111...................31
1111.................156
11111................781
111111..............3906
1111111............19531
11111111...........97656
111111111.........488281
1111111111.......2441406
etc. ...............etc.
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MAPLE
| a:=n->sum(5^(n-j), j=1..n): seq(a(n), n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007
A003463:=1/(5*z-1)/(z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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
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MATHEMATICA
| lst={}; Do[p=(5^n-1)/4; AppendTo[lst, p], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008]
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PROG
| (Other) sage: [lucas_number1(n, 6, 5) for n in xrange(0, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(Other) sage: [gaussian_binomial(n, 1, 5) for n in xrange(0, 24)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
(PARI) a(n)=5^n\4 \\ Charles R Greathouse IV, Jul 15 2011
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CROSSREFS
| Cf. A004061, A086122, A045468, A107219, A074479.
A026375 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]
Sequence in context: A056015 A128740 A026705 * A026771 A065096 A077352
Adjacent sequences: A003460 A003461 A003462 * A003464 A003465 A003466
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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