A245031 Numbers m such that 3*m+1 and 8*m+1 are both squares.

0, 1, 21, 120, 2080, 11781, 203841, 1154440, 19974360, 113123361, 1957283461, 11084934960, 191793804840, 1086210502741, 18793835590881, 106437544333680, 1841604094101520, 10429793134197921, 180458407386358101, 1022013289607062600

Offset

1,3

Comments

Naturally, all terms are triangular numbers.

Numbers m such that k*m+1 and 8*m+1 are both squares:

k=1: A006454;

k=3: this sequence;

k=4: A029549;

k=5: 0, 3, 231, 4560, 333336, 6575751, ...

k=6: A200999;

k=7: A157879.

Numbers m such that 3*m+1 and k*m+1 are both squares:

k=1: A045899;

k=2: A045502;

k=4: A059989;

k=5: A159683;

k=6: 8*A029546;

k=7: A160695;

k=8: this sequence.

Links

Bruno Berselli, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).

Formula

G.f.: x^2*(1 + 20*x + x^2)/((1 - x)*(1 - 10*x + x^2)*(1 + 10*x + x^2)).

a(n) = a(n-1) + 98*a(n-2) - 98*a(n-3) - a(n-4) + a(n-5).

G.f. of the quadrisections:

a(4k+1): 40*x*(52 + 3*x)/((1 - x)*(1 - 9602*x + x^2));

a(4k+2): (1 + 2178*x + 21*x^2)/((1 - x)*(1 - 9602*x + x^2));

a(4k+3): (21 + 2178*x + x^2)/((1 - x)*(1 - 9602*x + x^2));

a(4k+4): 40*(3 + 52*x)/((1 - x)*(1 - 9602*x + x^2)).

Mathematica

LinearRecurrence[{1, 98, -98, -1, 1}, {0, 1, 21, 120, 2080}, 20] (* or *) CoefficientList[Series[x (1 + 20 x + x^2)/((1 - x) (1 - 10 x + x^2) (1 + 10 x + x^2)), {x, 0, 20}], x]

Prog

(PARI) a=vector(20); a[1]=0; a[2]=1; a[3]=21; a[4]=120; a[5]=2080; for(i=6, #a, a[i]=a[i-1]+98*a[i-2]-98*a[i-3]-a[i-4]+a[i-5]); a
(Maxima) a[1]:0$ a[2]:1$ a[3]:21$ a[4]:120$ a[5]:2080$ a[n]:=a[n-1]+98*a[n-2]-98*a[n-3]-a[n-4]+a[n-5]$ makelist(a[n], n, 1, 20);
(Magma) I:=[0, 1, 21, 120, 2080]; [n le 5 select I[n] else Self(n-1)+98*Self(n-2)-98*Self(n-3)-Self(n-4)+Self(n-5): n in [1..20]];

Crossrefs

Cf. A000217.

Cf. A006454, A029546, A029549, A045502, A045899, A059989, A157879, A159683, A160695, A200999.

Sequence in context: A183318 A204214 A074088 * A316713 A044353 A044734

Adjacent sequences: A245028 A245029 A245030 * A245032 A245033 A245034

Keyword

nonn,easy

Author

Bruno Berselli, Jul 15 2014

Extensions

Changed offset from 0 to 1 and adapted formulas by Bruno Berselli, Mar 03 2016

Status

approved