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A131865
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Partial sums of powers of 16.
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43
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1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057
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OFFSET
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0,2
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COMMENTS
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a(n) = if n=0 then 1 else a(n-1)+A001025(n);
for n>0: A131851(a(n))=n and ABS(A131851(m))<n for m<a(n);
a(n) = A098704(n+2)/2.
16=2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008
Bisection of A115451. - Paul Curtz, May 20 2008
Second quadrisection of A115451. - Paul Curtz, May 21 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=16, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det(A). [From Milan Janjic, Feb 21 2010]
Partial sums are in A014899. Also, the sequence is related to A014931 by A014931(n+1) = (n+1)*a(n)-sum(a(i), i=1..n). [Bruno Berselli, Nov 07 2012]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries related to partial sums.
Index entries related to q-numbers.
Index to sequences with linear recurrences with constant coefficients, signature (17,-16).
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FORMULA
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a(n) = (A001025(n+1)-1)/15.
a(n) = 16*a(n-1)+1. - Paul Curtz, May 20 2008
G.f. 1 / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011
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EXAMPLE
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a(3) = 1+16+256+4096 = 4369 = in binary: 1000100010001.
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MATHEMATICA
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Table[(2^(4 n) - 1)/15, {n, 16}] (* Robert G. Wilson v, Aug 22 2007 *)
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PROG
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(Sage) [gaussian_binomial(n, 1, 16) for n in xrange(1, 18)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
(MAGMA) [(16^(n+1)-1)/15: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011
(Maxima)
a[0]:0$
a[n]:=16*a[n-1]+1$
A131865(n):=a[n]$
makelist(A131865(n), n, 1, 30); /* Martin Ettl, Nov 05 2012 */
(PARI) A131865(n)=16^n\15 \\ - M. F. Hasler, Nov 05 2012
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CROSSREFS
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Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723. - M. F. Hasler, Nov 05 2012
Sequence in context: A170650 A170698 A170736 * A179093 A031417 A029811
Adjacent sequences: A131862 A131863 A131864 * A131866 A131867 A131868
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KEYWORD
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nonn,easy
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AUTHOR
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Reinhard Zumkeller, Jul 22 2007
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STATUS
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approved
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