This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A053142 One half of A053141. 20
 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 16*m*(2*m+1)+1 is a square. - Bruno Berselli, Oct 19 2012 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 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 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:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/((1-x)*(1-6*x+x^2)))); // G. C. Greubel, Jul 15 2018 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 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 21 06:56 EST 2018. Contains 317428 sequences. (Running on oeis4.)