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A245031 Numbers n such that 3*n+1 and 8*n+1 are both squares. 13
0, 1, 21, 120, 2080, 11781, 203841, 1154440, 19974360, 113123361, 1957283461, 11084934960, 191793804840, 1086210502741, 18793835590881, 106437544333680, 1841604094101520, 10429793134197921, 180458407386358101, 1022013289607062600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Numbers n such that k*n+1 and 8*n+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 n such that 3*n+1 and k*n+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. A006454, A029546, A029549, A045502, A045899, A059989, A157879, A159683, A160695, A200999.

Sequence in context: A183318 A204214 A074088 * A044353 A044734 A200888

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

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Last modified February 24 06:13 EST 2018. Contains 299597 sequences. (Running on oeis4.)