OFFSET
0,3
COMMENTS
Partial sums of A001109. - Barry Williams, May 03 2000.
Number m such that 16*m*(2*m+1)+1 is a square. - Bruno Berselli, Oct 19 2012
From Robert K. Moniot, Sep 21 2020: (Start)
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.
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.
(End)
REFERENCES
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).
FORMULA
a(n) = (A001653(n)-1)/4.
a(n) = 6*a(n-1)-a(n-2)+1, a(0)=0, a(1)=1.
G.f.: x/((1-x)*(1-6*x+x^2)).
From Paul Barry, Nov 14 2003: (Start)
a(n+1) = Sum_{k=0..n} S(k, 6) = Sum_{k=0..n} U(n, 3), Chebyshev polynomials of 2nd kind, A049310.
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)
From Antonio Alberto Olivares, Jan 13 2004: (Start)
a(n) = 7*a(n-1)-7*a(n-2)+a(n-3).
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)
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
a(n)*a(n-2) = a(n-1)*(a(n-1) - 1) for n>1. - Robert K. Moniot, Sep 21 2020
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
MATHEMATICA
Join[{a=0, b=1}, Table[c=6*b-a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
Table[(Fibonacci[2n + 1, 2] - 1)/4, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
LinearRecurrence[{7, -7, 1}, {0, 1, 7}, 30] (* G. C. Greubel, Jul 15 2018 *)
PROG
(PARI) {a=1+sqrt(2); b=1-sqrt(2); P(n) = (a^n - b^n)/(a-b)};
for(n=0, 30, print1(round((P(2*n+1) - 1)/4), ", ")) \\ G. C. Greubel, Jul 15 2018
(PARI) x='x+O('x^30); Vec(x/((1-x)*(1-6*x+x^2))) \\ G. C. Greubel, Jul 15 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/((1-x)*(1-6*x+x^2)))); // G. C. Greubel, Jul 15 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved