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A053142 One half of A053141. 18
0, 1, 7, 42, 246, 1435, 8365, 48756, 284172, 1656277, 9653491, 56264670, 327934530, 1911342511, 11140120537, 64929380712, 378436163736, 2205687601705, 12855689446495, 74928449077266, 436715005017102 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partial sums of A001109. - Barry Williams, May 03 2000.

Number m such that 16m(2m+1)+1 is a square. - Bruno Berselli, Oct 19 2012

LINKS

Table of n, a(n) for n=0..20.

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)).

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)^(2n)(5/8-7sqrt(2)/16)+(sqrt(2)+1)^(2n)(7sqrt(2)/16 + 5/8)-1/4. - Paul Barry, Nov 14 2003

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. - Antonio Alberto Olivares, Jan 13 2004

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

Using b()=A001109(), a(2n) = b(n)*(b(n) + b(n-1)) and a(2n-1) = b(n)*(b(n) + b(n+1)). Kenneth J Ramsey, Sep 10 2010

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 *)

CROSSREFS

Cf. A001653, A053141, A001652, A046090.

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

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Last modified February 22 21:55 EST 2018. Contains 299469 sequences. (Running on oeis4.)