A175322 a(n) = A053141(n)*A001109(n+1) = Sum_{k=A053141(n)+1..A001109(n+1)-1} k.

12, 490, 17136, 584988, 19889100, 675741430, 22955884992, 779827644120, 26491203224556, 899921193951778, 30570830043692400, 1038508304094967860, 35278711531352926572, 1198437683891107427950, 40711602541519349266176, 1382996048732155862584368

Offset

1,1

Comments

Solution to a*b = (b*(b-1) - a*(a+1))/2 in A000027 with a,b >= 2.

Links

Colin Barker, Table of n, a(n) for n = 1..650

Index entries for linear recurrences with constant coefficients, signature (41,-246,246,-41,1).

Formula

From Colin Barker, Mar 31 2018: (Start)

G.f.: 2*x*(2 - x)*(3 + x) / ((1 - x)*(1 - 34*x + x^2)*(1 - 6*x + x^2)).

a(n) = 41*a(n-1) - 246*a(n-2) + 246*a(n-3) - 41*a(n-4) + a(n-5) for n>5.

(End)

a(n) = (1 - 8*U(n, 3) + 7*U(n, 17) + U(n-1, 17)) / 16 where U(n, x) is the Chebyshev polynomial of the second kind. - Michael Somos, Jul 18 2018

Example

A053141(1) = 2 and A001109(2) = 6, then 2*6 = 3+4+5 = 12, is a term.
A053141(2) = 14 and A001109(3) = 35, then 14*35 = 15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34 = 490, is a term.
G.f. = 12*x + 490*x^2 + 17136*x^3 + 584988*x^4 + 19889100*x^5 + ... - Michael Somos, Jul 18 2018

Mathematica

lst={}; k=1; j=0; s=1; Do[a=6*k-j; p=2*s*a; s=s+a; AppendTo[lst, p]; j=k; k=a, {n, 1, 16}]; lst
LinearRecurrence[{41, -246, 246, -41, 1}, {12, 490, 17136, 584988, 19889100}, 30] (* G. C. Greubel, Jul 15 2018 *)
a[ n_] := (1 - 8 ChebyshevU[n, 3] + 7 ChebyshevU[n, 17] + ChebyshevU[n - 1, 17]) / 16; (* Michael Somos, Jul 18 2018 *)

Prog

(PARI) Vec(2*x*(2 - x)*(3 + x) / ((1 - x)*(1 - 34*x + x^2)*(1 - 6*x + x^2)) + O(x^20)) \\ Colin Barker, Mar 31 2018
(PARI) {a(n) = if( n>0, polcoeff( 2*x * (2 - x)*(3 + x) / ((1 - x)*(1 - 34*x + x^2)*(1 - 6*x + x^2)) + x * O(x^n), n), n=-n; polcoeff( -2*x^2 * (1-2*x)*(1+3*x) / ((1 - x)*(1 - 34*x + x^2)*(1 - 6*x + x^2)) + x * O(x^n), n))}; /* Michael Somos, Jul 18 2018 */
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x*(2-x)*(3+x)/((1-x)*(1-34*x+x^2)*(1-6*x+x^2)))); // G. C. Greubel, Jul 15 2018

Crossrefs

Cf. A053141, A001109.

Sequence in context: A344113 A229634 A196704 * A297651 A296596 A196853

Adjacent sequences: A175319 A175320 A175321 * A175323 A175324 A175325

Keyword

nonn,easy

Author

Manuel Valdivia, Apr 03 2010

Extensions

Definition simplified by N. J. A. Sloane, Apr 05 2010

Status

approved