A007752 Odd bisection of A007750.

1, 24, 391, 6240, 99457, 1585080, 25261831, 402604224, 6416405761, 102259887960, 1629741801607, 25973608937760, 413948001202561, 6597194410303224, 105141162563649031, 1675661406608081280

Offset: 1

Author

John C. Hallyburton, Jr. (hallyb(AT)vmsdev.enet.dec.com)

Keywords

• nonn
• easy

References

• Mentioned in a problem on p. 334 of Two-Year College Math. Jnl., Vol. 25, 1994.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..825
• K. R. S. Sastry, Problem 533 The College Mathematics Journal, 25, issue 4, 1994, p. 334.
• K. R. S. Sastry, Square Products of Sums of Squares The College Mathematics Journal, 26, issue 4, 1995, p. 333.
• Index entries for linear recurrences with constant coefficients, signature (17,-17,-1).

Crossrefs

Cf. A007750, A007751.

Programs

GAP

a:=[1,24,391];; for n in [4..30] do a[n]:=17*a[n-1]-17*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Mar 04 2020

Magma

I:=[1,24,391]; [n le 3 select I[n] else 17*Self(n-1) -17*Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Mar 04 2020

Maple

seq( simplify( (4*ChebyshevU(n,8) - 53*ChebyshevU(n-1,8) -4)/7), n=1..20); # G. C. Greubel, Mar 04 2020

Mathematica

Table[(4*ChebyshevU[n, 8] -53*ChebyshevU[n-1, 8] -4)/7, {n,20}] (* G. C. Greubel, Mar 04 2020 *)

PARI

a(n)=local(w); w=8+3*quadgen(28); imag(1/w^n)+4*(real(1/w^n)-1)/7

PARI

vector(30, n, (4*polchebyshev(n,2,8) -53*polchebyshev(n-1,2,8) -4)/7 ) \\ G. C. Greubel, Mar 04 2020

Sage

[(4*chebyshev_U(n,8) -53*chebyshev_U(n-1,8) -4)/7 for n in (1..30)] # G. C. Greubel, Mar 04 2020

Formula

G.f.: x*(1+7*x)/((1-x)*(1-16*x+x^2)).

a(n) = 16*a(n-1) - a(n-2) + 8.

a(n) = (4*ChebyshevU(n, 8) -53*ChebyshevU(n-1, 8) -4)/7. - G. C. Greubel, Mar 04 2020

E.g.f.: (exp(8*x)*(4*cosh(3*sqrt(7)*x) - sqrt(7)*sinh(3*sqrt(7)*x)) - 4*exp(x))/7. - Stefano Spezia, Mar 14 2020

a(n) = A097830(n-1)+7*A097830(n-2). - R. J. Mathar, Jul 04 2024

Extensions

Edited by Michael Somos, Jul 27 2002