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A053142 a(n) = A053141(n)/2.

%I A053142 #63 May 31 2025 05:18:22
%S A053142 0,1,7,42,246,1435,8365,48756,284172,1656277,9653491,56264670,
%T A053142 327934530,1911342511,11140120537,64929380712,378436163736,
%U A053142 2205687601705,12855689446495,74928449077266,436715005017102,2545361581025347,14835454481134981,86467365305784540
%N A053142 a(n) = A053141(n)/2.
%C A053142 Partial sums of A001109. - Barry Williams, May 03 2000.
%C A053142 Number m such that 16*m*(2*m+1)+1 is a square. - _Bruno Berselli_, Oct 19 2012
%C A053142 From _Robert K. Moniot_, Sep 21 2020: (Start)
%C A053142 Consecutive terms (a(n-1),a(n))=(u,v) give all points on the hyperbola u^2-u+v^2-v-6*u*v=0 in quadrant 1 with both coordinates an integer.
%C A053142 Let T(n) denote the n-th triangular number. If i, j are any two successive elements of the above sequence then (T(i-1) + T(j-1))/T(i+j-1) = 3/4.
%C A053142 (End)
%H A053142 G. C. Greubel, <a href="/A053142/b053142.txt">Table of n, a(n) for n = 0..1000</a>
%H A053142 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/58-2/alperin09212019.pdf">A nonlinear recurrence and its relations to Chebyshev polynomials</a>, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
%H A053142 Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.
%H A053142 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A053142 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A053142 a(n) = (A001653(n)-1)/4.
%F A053142 a(n) = 6*a(n-1)-a(n-2)+1, a(0)=0, a(1)=1.
%F A053142 G.f.: x/((1-x)*(1-6*x+x^2)).
%F A053142 From _Paul Barry_, Nov 14 2003: (Start)
%F A053142 a(n+1) = Sum_{k=0..n} S(k, 6) = Sum_{k=0..n} U(n, 3), Chebyshev polynomials of 2nd kind, A049310.
%F A053142 a(n+1) = (sqrt(2)-1)^(2*n)(5/8-7*sqrt(2)/16)+(sqrt(2)+1)^(2*n)*(7*sqrt(2)/16 + 5/8)-1/4. (End)
%F A053142 From _Antonio Alberto Olivares_, Jan 13 2004: (Start)
%F A053142 a(n) = 7*a(n-1)-7*a(n-2)+a(n-3).
%F A053142 a(n) = -(1/4) + (1-sqrt(2))/(-8*sqrt(2))*(3-2*sqrt(2))^n + (1+sqrt(2))/(8*sqrt(2))*(3+2*sqrt(2))^n. (End)
%F A053142 a(n) = Sum_{k=0..n} Sum_{j=0..2*k} (-1)^(j+1)*A000129(j)*A000129(2*k-j). _Paul Barry_, Oct 23 2009
%F A053142 a(2*k) = A001109(k)*(A001109(k) + A001109(k-1)) and a(2*k-1) = A001109(k)*(A001109(k) + A001109(k+1)). _Kenneth J Ramsey_, Sep 10 2010
%F A053142 a(n)*a(n-2) = a(n-1)*(a(n-1) - 1) for n>1. - _Robert K. Moniot_, Sep 21 2020
%F A053142 E.g.f.: (exp(3*x)*(2*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)) - 2*exp(x))/8. - _Stefano Spezia_, Mar 16 2024
%t A053142 Join[{a=0,b=1}, Table[c=6*b-a+1; a=b; b=c, {n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)
%t A053142 Table[(Fibonacci[2n + 1, 2] - 1)/4, {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)
%t A053142 LinearRecurrence[{7, -7, 1}, {0, 1, 7}, 30] (* _G. C. Greubel_, Jul 15 2018 *)
%o A053142 (PARI) {a=1+sqrt(2); b=1-sqrt(2); P(n) = (a^n - b^n)/(a-b)};
%o A053142 for(n=0, 30, print1(round((P(2*n+1) - 1)/4), ", ")) \\ _G. C. Greubel_, Jul 15 2018
%o A053142 (PARI) x='x+O('x^30); Vec(x/((1-x)*(1-6*x+x^2))) \\ _G. C. Greubel_, Jul 15 2018
%o A053142 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/((1-x)*(1-6*x+x^2)))); // _G. C. Greubel_, Jul 15 2018
%Y A053142 Cf. A000129, A001109, A001653, A053141, A001652, A046090, A049310.
%Y A053142 Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).
%K A053142 nonn,easy
%O A053142 0,3
%A A053142 _Wolfdieter Lang_