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A130482
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a(n) = Sum_{k=0..n} (k mod 4) (Partial sums of A010873).
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28
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0, 1, 3, 6, 6, 7, 9, 12, 12, 13, 15, 18, 18, 19, 21, 24, 24, 25, 27, 30, 30, 31, 33, 36, 36, 37, 39, 42, 42, 43, 45, 48, 48, 49, 51, 54, 54, 55, 57, 60, 60, 61, 63, 66, 66, 67, 69, 72, 72, 73, 75, 78, 78, 79, 81, 84, 84, 85, 87, 90, 90, 91, 93, 96, 96, 97, 99, 102, 102, 103, 105
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OFFSET
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0,3
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COMMENTS
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Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 4, A[i,i]:=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 24 2010
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
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FORMULA
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a(n) = 6*floor(n/4) + A010873(n)*(A010873(n)+1)/2.
G.f.: x*(1 + 2*x + 3*x^2)/((1-x^4)*(1-x)).
a(n) = (1 - (-1)^n - (2*i)*(-i)^n + (2*i)*i^n + 6*n) / 4 where i = sqrt(-1). - Colin Barker, Oct 15 2015
a(n) = 3*n/2 + (n mod 2)* ( (n-1) mod 4 ) - (n mod 2)/2. - Ammar Khatab, Aug 27 2020
E.g.f.: (3*x*exp(x) - 2*sin(x) + sinh(x))/2. - Stefano Spezia, Apr 22 2021
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MAPLE
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a:=n->add(chrem( [n, j], [1, 4] ), j=1..n):seq(a(n), n=0..70); # Zerinvary Lajos, Apr 07 2009
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MATHEMATICA
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Table[(6*n +(1-(-1)^n)*(1+2*I^(n+1)))/4, {n, 0, 70}] (* G. C. Greubel, Aug 31 2019 *)
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PROG
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(PARI) a(n) = (1 - (-1)^n - (2*I)*(-I)^n + (2*I)*I^n + 6*n) / 4 \\ Colin Barker, Oct 15 2015
(MAGMA) I:=[0, 1, 3, 6, 6]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..71]]; // G. C. Greubel, Aug 31 2019
(Sage)
def A130482_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(x*(1+2*x+3*x^2)/((1-x^4)*(1-x))).list()
A130482_list(70) # G. C. Greubel, Aug 31 2019
(GAP) a:=[0, 1, 3, 6, 6];; for n in [6..71] do a[n]:=a[n-1]+a[n-4]-a[n-5]; od; a; # G. C. Greubel, Aug 31 2019
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CROSSREFS
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Cf. A010872, A010874, A010875, A010876, A010877. A130481, A130483, A130484, A130485.
Sequence in context: A153035 A296216 A072910 * A239318 A177783 A228945
Adjacent sequences: A130479 A130480 A130481 * A130483 A130484 A130485
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KEYWORD
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nonn,easy
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AUTHOR
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Hieronymus Fischer, May 29 2007
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STATUS
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approved
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