A186423 Partial sums of A186421.
0, 1, 3, 4, 8, 11, 17, 20, 28, 33, 43, 48, 60, 67, 81, 88, 104, 113, 131, 140, 160, 171, 193, 204, 228, 241, 267, 280, 308, 323, 353, 368, 400, 417, 451, 468, 504, 523, 561, 580, 620, 641, 683, 704, 748, 771, 817, 840, 888, 913, 963, 988, 1040, 1067, 1121, 1148, 1204, 1233, 1291, 1320
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).
Programs
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GAP
List([0..65], n-> (6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16); # G. C. Greubel, Oct 09 2019
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Haskell
a186423 n = a186423_list !! n a186423_list = scanl1 (+) a186421_list
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Magma
[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16: n in [0..65]]; // G. C. Greubel, Oct 09 2019
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Maple
A087960 := proc(n) op((n mod 4)+1,[1,-1,-1,1]) ; end proc: A186423 := proc(n) 3*n*(n+1)/8 +3/16 +(-1)^n*(2*n+1)/16 -A087960(n)/4 ; end proc: # R. J. Mathar, Feb 28 2011
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Mathematica
CoefficientList[Series[x(1+2x+2x^3+x^4)/((1-x)^3(1+x)^2(1+x^2)),{x, 0, 65}],x] (* Harvey P. Dale, Mar 13 2011 *) Table[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial[n+1, 2])/16, {n, 0, 65}] (* G. C. Greubel, Oct 09 2019 *) -
PARI
vector(66, n, my(m=n-1); (6*m^2 +6*m +3 +(-1)^m*(2*m+1) -4*(-1)^binomial(m+1, 2))/16) \\ G. C. Greubel, Oct 09 2019
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Python
def A186423(n): return (6*n*(n+1)+3+(-2*n-1 if n&1 else 2*n+1)+(4 if n+1&2 else -4))>>4 # Chai Wah Wu, Jan 31 2023
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Sage
[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^binomial(n+1, 2))/16 for n in (0..65)] # G. C. Greubel, Oct 09 2019
Formula
From R. J. Mathar, Feb 28 2011: (Start)
G.f.: x*(1 + 2*x + 2*x^3 + x^4)/( (1+x^2)*(1+x)^2*(1-x)^3 ).
a(n) = (6*n*(n+1) + 3 + (-1)^n*(2*n+1) - 4*A087960(n))/16. (End)
E.g.f.: ((2 + 5*x + 3*x^2)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x) + 2*sin(x) - 2*cos(x))/8. - G. C. Greubel, Oct 09 2019
Extensions
More terms added by G. C. Greubel, Oct 09 2019