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%I #48 Sep 08 2022 08:46:08
%S 0,1,21,120,2080,11781,203841,1154440,19974360,113123361,1957283461,
%T 11084934960,191793804840,1086210502741,18793835590881,
%U 106437544333680,1841604094101520,10429793134197921,180458407386358101,1022013289607062600
%N Numbers m such that 3*m+1 and 8*m+1 are both squares.
%C Naturally, all terms are triangular numbers.
%C Numbers m such that k*m+1 and 8*m+1 are both squares:
%C k=1: A006454;
%C k=3: this sequence;
%C k=4: A029549;
%C k=5: 0, 3, 231, 4560, 333336, 6575751, ...
%C k=6: A200999;
%C k=7: A157879.
%C Numbers m such that 3*m+1 and k*m+1 are both squares:
%C k=1: A045899;
%C k=2: A045502;
%C k=4: A059989;
%C k=5: A159683;
%C k=6: 8*A029546;
%C k=7: A160695;
%C k=8: this sequence.
%H Bruno Berselli, <a href="/A245031/b245031.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,98,-98,-1,1).
%F G.f.: x^2*(1 + 20*x + x^2)/((1 - x)*(1 - 10*x + x^2)*(1 + 10*x + x^2)).
%F a(n) = a(n-1) + 98*a(n-2) - 98*a(n-3) - a(n-4) + a(n-5).
%F G.f. of the quadrisections:
%F a(4k+1): 40*x*(52 + 3*x)/((1 - x)*(1 - 9602*x + x^2));
%F a(4k+2): (1 + 2178*x + 21*x^2)/((1 - x)*(1 - 9602*x + x^2));
%F a(4k+3): (21 + 2178*x + x^2)/((1 - x)*(1 - 9602*x + x^2));
%F a(4k+4): 40*(3 + 52*x)/((1 - x)*(1 - 9602*x + x^2)).
%t 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]
%o (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
%o (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);
%o (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]];
%Y Cf. A000217.
%Y Cf. A006454, A029546, A029549, A045502, A045899, A059989, A157879, A159683, A160695, A200999.
%K nonn,easy
%O 1,3
%A _Bruno Berselli_, Jul 15 2014
%E Changed offset from 0 to 1 and adapted formulas by _Bruno Berselli_, Mar 03 2016
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