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A266883 Numbers of the form m*(4*m+1)+1, where m = 0,-1,1,-2,2,-3,3,...

%I A266883 #42 May 11 2025 01:04:36
%S A266883 1,4,6,15,19,34,40,61,69,96,106,139,151,190,204,249,265,316,334,391,
%T A266883 411,474,496,565,589,664,690,771,799,886,916,1009,1041,1140,1174,1279,
%U A266883 1315,1426,1464,1581,1621,1744,1786,1915,1959,2094,2140,2281,2329,2476,2526
%N A266883 Numbers of the form m*(4*m+1)+1, where m = 0,-1,1,-2,2,-3,3,...
%C A266883 Also, numbers m such that 16*m-15 is a square. Therefore, the terms 1 and 4 are the only squares in this sequence.
%C A266883 Conjecture: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n-1} 1 - q^k) = q + q^4 + q^6 + q^15 + q^19 + q^34 + .... Cf. A174114. - _Peter Bala_, May 10 2025
%H A266883 Bruno Berselli, <a href="/A266883/b266883.txt">Table of n, a(n) for n = 0..1000</a>
%H A266883 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A266883 O.g.f.: (1 + 3*x + 3*x^3 + x^4)/((1 + x)^2*(1 - x)^3).
%F A266883 E.g.f.: (5 + 8*x + 4*x^2)*exp(x)/4 -(1 - 2*x)*exp(-x)/4.
%F A266883 a(n) = a(-n-1) = n*(n + 1) + 1 - ((2*n + 1)*(-1)^n - 1)/4 = (2*n + 1)*floor((n + 1)/2) + 1.
%F A266883 a(n) = A002061(n+1) + A001057(n) = A074378(n)+1.
%F A266883 a(n+1) + a(n+2) = A049486(n+3).
%t A266883 Table[n (n + 1) + 1 - ((2 n + 1) (-1)^n - 1)/4, {n, 0, 50}]
%t A266883 LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 6, 15, 19}, 60] (* _Vincenzo Librandi_, Jan 06 2016 *)
%o A266883 (PARI) vector(50, n, n--; n*(n+1)+1-((2*n+1)*(-1)^n-1)/4)
%o A266883 (PARI) Vec((1+3*x+3*x^3+x^4)/((1+x)^2*(1-x)^3) + O(x^100)) \\ _Altug Alkan_, Jan 06 2016
%o A266883 (Sage) [n*(n+1)+1-((2*n+1)*(-1)^n-1)/4 for n in range(50)]
%o A266883 (Python) [n*(n+1)+1-((2*n+1)*(-1)**n-1)/4 for n in range(60)]
%o A266883 (Magma) [n*(n+1)+1-((2*n+1)*(-1)^n-1)/4: n in [0..50]];
%o A266883 (Magma) I:=[1,4,6,15,19]; [n le 5 select I[n] else Self(n-1) + 2*Self(n-2) -2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // _Vincenzo Librandi_, Jan 06 2016
%Y A266883 Cf. A001057, A049486, A127365, A130472.
%Y A266883 Cf. A002061: m*(4*m+2)+1 for m = 0,0,-1,1,-2,2,-3,3, ...
%Y A266883 Cf. A174114: m*(4*m+3)+1 for m = 0,-1,1,-2,2,-3,3,-4,4, ...
%Y A266883 Cf. A054556: m*(4*m+1)+1 for nonpositive m.
%Y A266883 Cf. A054567: m*(4*m+1)+1 for nonnegative m.
%Y A266883 Cf. A074378: numbers m such that 16*m+1 is a square.
%K A266883 nonn,easy
%O A266883 0,2
%A A266883 _Bruno Berselli_, Jan 05 2016