A192328 Numbers of the form 20*k+7 which are three times a square.

27, 147, 507, 867, 1587, 2187, 3267, 4107, 5547, 6627, 8427, 9747, 11907, 13467, 15987, 17787, 20667, 22707, 25947, 28227, 31827, 34347, 38307, 41067, 45387, 48387, 53067, 56307, 61347, 64827, 70227, 73947, 79707, 83667, 89787, 93987

Offset: 1

Author

Bruno Berselli, Jun 28 2011

Keywords

• nonn
• easy

Comments

Text of the theorem in the paper mentioned in References: The necessary and sufficient condition so that a number of the form 20*k+7 is three times a square is that k is of the form 3*h*(5*h+3)+1 or 3*h*(5*h+7)+7.

A119617 gives the values of k.

A080512*120 gives the first differences.

References

• "Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno) - Mar 1901 - p. 75 (Problem 286 and its generalization, G. Cardoso-Laynes).

Links

• Mohammed Yaseen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Bruno Berselli)
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A134153, A134154.

Cf. A020742, A047218.

Programs

Magma

[m: m in [7..10^5 by 20] | IsSquare(m/3)];

Maple

select(t -> issqr(t/3), [seq(20*i+7,i=1..10000,3)]); # Robert Israel, Apr 28 2023

Mathematica

Select[20 Range[5000] + 7, IntegerQ[Sqrt[#/3]] &] (* or *) LinearRecurrence[{1,2,-2,-1,1}, {27,147,507,867,1587}, 40] (* Harvey P. Dale, Jul 06 2011 *)
CoefficientList[Series[3 (9 + 40 x + 102 x^2 + 40 x^3 + 9 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 19 2013 *)

PARI

for(k=0, 5*10^3, m=20*k+7; if(issquare(m/3), print1(m",")));

PARI

a(n)=my(m=n--\4); 1200*m^2+[360*m+27, 840*m+147, 1560*m+507, 2040*m+867][n%4+1] \\ Charles R Greathouse IV, Jun 29 2011

Formula

G.f.: 3*x*(9 + 40*x + 102*x^2 + 40*x^3 + 9*x^4)/((1 + x)^2*(1 - x)^3).

a(n) = 3*((10*(n-1) + (-1)^(n-1) + 5)/2)^2.

a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

Sum_{i=1..n} a(i) = n*(50*(n-1)*(n+1) + 15*(-1)^(n-1) + 39)/2.

a(n) = 3*A020742(A047218(n))^2.

Extensions

Offset corrected by Mohammed Yaseen, Apr 27 2023