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(Magma)
A047999:= func< n, k | BitwiseAnd(n-k, k) eq 0 select 1 else 0 >;
[A047999(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 03 2024
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Triangle read by rows, Sierpinski's gasket, T(n, k) = 2^k * A047999 * (1,2,4,8,...n, k) diagonalized.
Former name: Triangle read by rows, Sierpinski's gasket, A047999 * (1,2,4,8,...) diagonalized. - G. C. Greubel, Dec 02 2024
New name by G. C. Greubel, Dec 02 2024
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CoefficientList[Series[(1+x)*(1-x^12)/(1 - 14*x + 104*x^12 - 91*x^13), {t, 0, 50}], t](* G. C. Greubel, May 17 2016 ; Dec 03 2024 *)
<a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, -91).
From G. C. Greubel, Dec 03 2024: (Start)
a(n) = 13*Sum_{j=1..11} a(n-j) - 91*a(n-12).
G.f.: (1+x)*(1-x^12)/(1 - 14*x + 104*x^12 - 91*x^13). (End)
CoefficientList[Series[(t1+x)*(1-x^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + )/(1)/(91 - 14*x + 104*tx^12 - 13*t^11 - 13*t^10 - 13*t^9 - 13*t^8 - 13*t^7 - 13*t^6 - 13*t^5 - 13*t^4 - 13*t^3 - 1391*tx^2 - 13*t + 1), {t, 0, 50}], t](* G. C. Greubel, May 17 2016 *)
coxG[{12, 91, -13}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 03 2024 *)
(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)*(1-x^12)/(1-14*x+104*x^12-91*x^13) )); // G. C. Greubel, Dec 03 2024
(SageMath)
def A166583_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)*(1-x^12)/(1-14*x+104*x^12-91*x^13) ).list()
A166583_list(40) # G. C. Greubel, Dec 03 2024
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<a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, -78).
From G. C. Greubel, Dec 03 2024: (Start)
a(n) = 12*Sum_{j=1..11} a(n-j) - 78*a(n-12).
G.f.: (1+x)*(1-x^12)/(1 - 13*x + 90*x^12 - 78*x^13). (End)
CoefficientList[Series[(t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 1+ 2*t^2 + 2)*t + (1)/(78*t^12 - 12*t^11 - 12*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 )/(1- 1213*t^5 - 12+90*t^4 - 12*t^3 - 1278*t^2 - 12*t + 113), {t, 0, 50}], t] (* G. C. Greubel, May 17 2016 ; Dec 03 2024 *)
(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)*(1-x^12)/(1-13*x+90*x^12-78*x^13) )); // G. C. Greubel, Dec 03 2024
(SageMath)
def A166568_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)*(1-x^12)/(1-13*x+90*x^12-78*x^13) ).list()
A166568_list(40) # G. C. Greubel, Dec 03 2024
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