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A091030
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Partial sums of powers of 13 (A001022).
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3
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1, 14, 183, 2380, 30941, 402234, 5229043, 67977560, 883708281, 11488207654, 149346699503, 1941507093540, 25239592216021, 328114698808274, 4265491084507563, 55451384098598320, 720867993281778161
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 13^a(n) is highest power of 13 dividing (13^n)!.
Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1,A[i,i]:=13, (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), Feb 21 2010]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=14, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n)*charpoly(A,1). [From Milan R. Janjic (agnus(AT)blic.net), Feb 21 2010]
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (14,-13).
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FORMULA
| a(n)= sum(13^k, k=0..n-1) = (13^n-1)/12.
G.f.: x/((1-13*x)*(1-x))= (1/(1-13*x) - 1/(1-x))/12.
For analogues with primes 2, 3, 5, 7 and 11 see A000225, A003462, A003463, A023000 and A016123 respectively.
a(1)=1, a(n)=13*a(n-1)+1. [From Vincenzo Librandi, Feb 05 2011]
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MAPLE
| a:=n->sum(13^(n-j), j=1..n): seq(a(n), n=1..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007
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PROG
| (Other) sage: [gaussian_binomial(n, 1, 13) for n in xrange(1, 18)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
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CROSSREFS
| Cf. A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125.
Sequence in context: A170733 A186229 A181237 * A179090 A165152 A198703
Adjacent sequences: A091027 A091028 A091029 * A091031 A091032 A091033
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 23 2004
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