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A321383 Numbers k such that the concatenation k21 is a square.

%I A321383 #19 Sep 08 2022 08:46:23
%S A321383 1,15,37,79,123,193,259,357,445,571,681,835,967,1149,1303,1513,1689,
%T A321383 1927,2125,2391,2611,2905,3147,3469,3733,4083,4369,4747,5055,5461,
%U A321383 5791,6225,6577,7039,7413,7903,8299,8817,9235,9781,10221,10795,11257,11859,12343,12973,13479
%N A321383 Numbers k such that the concatenation k21 is a square.
%H A321383 Colin Barker, <a href="/A321383/b321383.txt">Table of n, a(n) for n = 1..1000</a>
%H A321383 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A321383 G.f.: x*(1 + 14*x + 20*x^2 + 14*x^3 + x^4)/((1 + x)^2*(1 - x)^3).
%F A321383 a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A321383 a(n) = 2*a(n-2) - a(n-4) + 50.
%F A321383 a(n) = (50*(n - 1)*n + 3*(2*n - 1)*(-1)^n + 11)/8. Therefore:
%F A321383 a(n) = (25*n^2 - 22*n + 4)/4 for even n;
%F A321383 a(n) = (25*n^2 - 28*n + 7)/4 for odd n.
%t A321383 Table[(50 (n - 1) n + 3 (2 n - 1) (-1)^n + 11)/8, {n, 1, 50}]
%o A321383 (PARI) vector(50, n, nn; (50*(n-1)*n+3*(2*n-1)*(-1)^n+11)/8)
%o A321383 (PARI) Vec(x*(1 + 14*x + 20*x^2 + 14*x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ _Colin Barker_, Nov 12 2018
%o A321383 (Sage) [(50*(n-1)*n+3*(2*n-1)*(-1)^n+11)/8 for n in (1..50)]
%o A321383 (Maxima) makelist((50*(n-1)*n+3*(2*n-1)*(-1)^n+11)/8, n, 1, 50);
%o A321383 (GAP) List([1..50], n -> (50*(n-1)*n+3*(2*n-1)*(-1)^n+11)/8);
%o A321383 (Magma) [(50*(n-1)*n+3*(2*n-1)*(-1)^n+11)/8: n in [1..50]];
%o A321383 (Python) [(50*(n-1)*n+3*(2*n-1)*(-1)**n+11)/8 for n in range(1, 50)]
%o A321383 (Julia) [div((50*(n-1)*n+3*(2*n-1)*(-1)^n+11), 8) for n in 1:50] |> println
%Y A321383 Cf. A008805.
%Y A321383 Numbers k such that the concatenation km is a square: A132356 (m = 1), A273365 (m = 4), A273366 (m = 5), A273367 (m = 6), A273368 (m = 9); missing sequence for m = 16; this sequence for m = 21; missing sequence for m = 24; A002378 (m = 25).
%K A321383 nonn,easy,base
%O A321383 1,2
%A A321383 _Bruno Berselli_, Nov 08 2018