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

0, 1, 7, 42, 246, 1435, 8365, 48756, 284172, 1656277, 9653491, 56264670, 327934530, 1911342511, 11140120537, 64929380712, 378436163736, 2205687601705, 12855689446495, 74928449077266, 436715005017102, 2545361581025347, 14835454481134981, 86467365305784540

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 sequences related to Chebyshev polynomials.

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(n) = Sum_{k=0..n} Sum_{j=0..2*k} (-1)^(j+1)*A000129(j)*A000129(2*k-j). Paul Barry, Oct 23 2009

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

Cf. A000129, A001109, A001653, A053141, A001652, A046090, A049310.

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

Sequence in context: A030240 A054890 A102594 * A214941 A162941 A094168

Adjacent sequences: A053139 A053140 A053141 * A053143 A053144 A053145

Keyword

nonn,easy

Author

Wolfdieter Lang

Status

approved