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A274830 Numbers k such that 7*k+1 is a triangular number (A000217).

%I A274830 #32 Nov 27 2024 08:27:51
%S A274830 0,2,5,11,17,27,36,50,62,80,95,117,135,161,182,212,236,270,297,335,
%T A274830 365,407,440,486,522,572,611,665,707,765,810,872,920,986,1037,1107,
%U A274830 1161,1235,1292,1370,1430,1512,1575,1661,1727,1817,1886,1980,2052,2150,2225
%N A274830 Numbers k such that 7*k+1 is a triangular number (A000217).
%C A274830 From _Peter Bala_, Nov 21 2024: (Start)
%C A274830 Numbers of the form n*(7*n + 3)/2 for n in Z. Cf. A057570.
%C A274830 The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-2)) * (1 + x^(7*n-5)) = 1 + x^2 + x^5 + x^11 + x^17 + x^27 + x^36 + .... Cf. A363800. (End)
%H A274830 Colin Barker, <a href="/A274830/b274830.txt">Table of n, a(n) for n = 1..1000</a>
%H A274830 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A274830 G.f.: x^2*(2 + 3*x + 2*x^2) / ((1 - x)^3*(1 + x)^2).
%F A274830 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
%F A274830 a(n) = (14*(n - 1)*n + (2*n - 1)*(-1)^n + 1)/16. Therefore:
%F A274830 a(n) = n*(7*n - 6)/8 for n even,
%F A274830 a(n) = (n - 1)*(7*n - 1)/8 for n odd.
%F A274830 E.g.f.: (x*(7*x -1)*cosh(x) + (7*x^2 + x + 1)*sinh(x))/8. - _Stefano Spezia_, Nov 26 2024
%t A274830 Table[(14 (n - 1) n + (2 n - 1) (-1)^n + 1)/16, {n, 1, 60}] (* _Bruno Berselli_, Jul 08 2016 *)
%o A274830 (PARI) select(n->ispolygonal(7*n+1, 3), vector(3000, n, n-1))
%o A274830 (PARI) concat(0, Vec(x^2*(2+3*x+2*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))
%Y A274830 Cf. A000217, A057570, A363800.
%Y A274830 Cf. similar sequences where k*n+1 is a triangular number: A000096 (k=1), A074377 (k=2), A045943 (k=3), A274681 (k=4), A085787 (k=5), A274757 (k=6).
%K A274830 nonn,easy
%O A274830 1,2
%A A274830 _Colin Barker_, Jul 08 2016
%E A274830 Edited by _Bruno Berselli_, Jul 08 2016