A122440 -id:A122440 - cp's OEIS frontend

A122441 Expansion of 2*(sqrt(1+8x)-3)/(sqrt(1+8x)-5).

1, -1, 1, -5, 25, -141, 849, -5349, 34825, -232445, 1582081, -10938709, 76616249, -542472685, 3876400305, -27919883205, 202480492905, -1477306676445, 10836099051105, -79861379898165, 591082795606425, -4391625157145805, 32742445489969425, -244889639907014565

Offset: 0

Paul Barry, Sep 05 2006

Robert Israel, Table of n, a(n) for n = 0..1108

Row sums of A122440.

Cf. A064311.

f:= gfun:-rectoproc({(4 + 8*n)*a(n) + (-10 - 23*n)*a(n + 1) + (-3*n - 6)*a(n + 2), a(0) = 1, a(1) = -1, a(2) = 1},a(n),remember):
map(f, [$0..40]); # Robert Israel, Aug 22 2025

def a(n):
    if n==0: return 1
    return -hypergeometric([1-n, n], [-n], -2).simplify()
[a(n) for n in range(21)] # Peter Luschny, Nov 30 2014

a(n) = -hypergeometric([1-n, n], [-n], -2) if n>0. - Peter Luschny, Nov 30 2014

(4 + 8*n)*a(n) + (-10 - 23*n)*a(n + 1) + (-3*n - 6)*a(n + 2) = 0. - Robert Israel, Aug 22 2025

A341032 Numbers k such that A124440(k) is a square.

Original entry on oeis.org

1, 2, 10, 17, 469, 646, 1542, 1601, 24939, 25090, 43690, 50925, 77577, 84002, 131087, 156817, 174755, 182106, 220974, 293930, 371307, 389130, 394290, 401573, 440819, 492886, 584326, 609301, 839590, 935685, 1727207, 1775622, 1939666, 1948705, 2235041, 2267650

Offset: 1

J. M. Bergot and Robert Israel, Feb 03 2021

nonn

Terms in common with A341031, i.e. numbers such that both A066840(k) and A124440(k) are squares, include 1, 2, 10, 17, and 25090.

			a(4) = 17 is a term because A124440(17) = 100 = 10^2.

Daniel Suteu, Table of n, a(n) for n = 1..100

Cf. A066840, A122440, A341031.

N:= 40000: # for terms <= N
G:= add(numtheory:-mobius(n)*n*x^(2*n)/((1-x^n)*(1-x^(2*n))^2), n=1..N/2):
S:= series(G, x, N+1):
A66840:= [seq(coeff(S, x, j), j=1..N)]:
f:= proc(n) n*numtheory:-phi(n)/2 - A66840[n] end proc:
f(1):= 1: f(2):= 1:
select(t -> issqr(f(t)), [$1..N]);

More terms from Daniel Suteu, Feb 03 2021

cp's OEIS Frontend

A122441 Expansion of 2*(sqrt(1+8x)-3)/(sqrt(1+8x)-5).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Sage

Formula