%I #51 Sep 08 2022 08:46:20

%S 1,10,61,358,2089,12178,70981,413710,2411281,14053978,81912589,

%T 477421558,2782616761,16218279010,94527057301,550944064798,

%U 3211137331489,18715879924138,109084142213341,635788973355910,3705649697922121,21598109214176818,125883005587138789,733699924308655918

%N Expansion of (1 + 3*x - 2*x^2)/(1 - 7*x + 7*x^2 - x^3).

%C y solutions to A000217(x-1) + A000217(x) = A000290(y-1) + A000290(y+2). The corresponding x values are listed in A075841.

%C y solutions to A000217(x-1) + A000217(x) = A000290(y-1) + A000290(y+1) are in A002315, and A075870 gives the x values.

%C y solutions to A000217(x-1) + A000217(x) = A000290(y-1) + A000290(y) are in A046090, and A001653 gives the x values.

%C Also, indices y for which 4*A000217(y) + 5 is a square. The next integers k such that k*A000217(y) + 5 is a square for infinitely many y values are 11, 20, 22, 29, 31, ...

%C First differences are in A106329.

%H Robert Israel, <a href="/A301383/b301383.txt">Table of n, a(n) for n = 0..1304</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).

%F O.g.f.: (1 + 3*x - 2*x^2)/((1 - x)*(1 - 6*x + x^2)).

%F a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) = 6*a(n-1) - a(n-2) + 2.

%F a(n) = (3/4)*((1 + sqrt(2))^(2*n + 1) + (1 - sqrt(2))^(2*n + 1)) - 1/2.

%F a(n) = A033539(2*n+2) = A241976(n+1) + 1 = 3*A001652(n) + 1 = 3*A046090(n) - 2.

%F a(n) = A053142(n+1) + 3*A053142(n) - 2*A053142(n-1), n>0.

%F 2*a(n) = 3*A002315(n) - 1.

%F 4*a(n) = 3*A077444(n+1) - 2.

%F E.g.f.: (3*exp(3*x)*(cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)) - cosh(x) - sinh(x))/2. - _Stefano Spezia_, Mar 06 2020

%F Let T(n) be the n-th triangular number, A000217(n). Then T(a(n)-3) + 2*T(a(n)-2) + 3*T(a(n)-1) + 4*T(a(n)) + 3*T(a(n)+1) + 2*T(a(n)+2) + T(a(n)+3) = (A001653(n) + A001653(n+2))^2. - _Charlie Marion_, Mar 16 2021

%p f:= gfun:-rectoproc({a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3), a(0)=1,a(1)=10,a(2)=61},a(n),remember):

%p map(f, [$0..50]); # _Robert Israel_, Mar 21 2018

%t CoefficientList[Series[(1 + 3 x - 2 x^2)/(1 - 7 x + 7 x^2 - x^3), {x, 0, 30}], x]

%o (PARI) Vec((1+3*x-2*x^2)/(1-7*x+7*x^2-x^3)+O(x^30))

%o (Maxima) makelist(coeff(taylor((1+3*x-2*x^2)/(1-7*x+7*x^2-x^3), x, 0, n), x, n), n, 0, 30);

%o (Sage) m=30; L.<x> = PowerSeriesRing(ZZ, m); f=(1+3*x-2*x^2)/(1-7*x+7*x^2-x^3); print(f.coefficients())

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x-2*x^2)/(1-7*x+7*x^2-x^3)));

%o (Julia)

%o using Nemo

%o function A301383List(len)

%o R, x = PowerSeriesRing(ZZ, len+2, "x")

%o f = divexact(1+3*x-2*x^2, 1-7*x+7*x^2-x^3)

%o [coeff(f, k) for k in 0:len]

%o end

%o A301383List(23) |> println # _Peter Luschny_, Mar 21 2018

%Y Subsequence of A301451.

%Y Cf. A000217, A000290, A001652, A002315, A033539, A046090, A053142, A075841, A077444, A106329, A241976.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Mar 20 2018