This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A175110 #51 Jul 24 2025 05:01:07
%S A175110 1,41,313,1201,3281,7321,14281,25313,41761,65161,97241,139921,195313,
%T A175110 265721,353641,461761,592961,750313,937081,1156721,1412881,1709401,
%U A175110 2050313,2439841,2882401,3382601,3945241,4575313,5278001,6058681
%N A175110 a(n) = ((2*n+1)^4+1)/2.
%C A175110 Binomial transform of 1,40,232,384,192,0,0,.. (0 continued). Convolution of the finite sequence 1,36,118,36,1 with A000332, dropping zeros.
%C A175110 Hypotenuse of Pythagorean triangles with smallest side a square: A016754(n)^2 + (a(n)-1)^2 = a(n)^2. - _Martin Renner_, Nov 12 2011
%C A175110 a(n) is also the first integer in a sum of (2*n + 1)^4 consecutive integers that equal (2*n + 1)^8. See A016756 and A016760. - _Patrick J. McNab_, Dec 26 2016
%D A175110 Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 54.
%H A175110 Vincenzo Librandi, <a href="/A175110/b175110.txt">Table of n, a(n) for n = 0..1000</a>
%H A175110 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A175110 a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5).
%F A175110 G.f.: (1+36*x+118*x^2+36*x^3+x^4)/ (1-x)^5.
%F A175110 a(n)-a(n-1) = A117216(n).
%F A175110 a(n) = 8*A001844(n) * A000217(n) + 1 = 8*A219086(n) + 1. - _Bruce J. Nicholson_, Apr 13 2017
%p A175110 A175110:=n->((2*n+1)^4+1)/2: seq(A175110(n), n=0..50); # _Wesley Ivan Hurt_, Apr 13 2017
%t A175110 CoefficientList[Series[(1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 19 2012 *)
%t A175110 Table[((2 n + 1)^4 + 1)/2, {n, 0, 29}] (* _Michael De Vlieger_, Dec 26 2016 *)
%t A175110 LinearRecurrence[{5,-10,10,-5,1},{1,41,313,1201,3281},40] (* _Harvey P. Dale_, Jan 01 2022 *)
%o A175110 (Magma) I:=[1, 41, 313, 1201, 3281]; [n le 5 select I[n] else 5*Self(n-1) - 10*Self(n-2) + 10*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Dec 19 2012
%o A175110 (PARI) a(n)=((2*n+1)^4+1)/2 \\ _Charles R Greathouse IV_, Oct 16 2015
%Y A175110 Cf. A000332, A016756, A016760. Partial sums of A117216.
%K A175110 easy,nonn
%O A175110 0,2
%A A175110 _R. J. Mathar_, Feb 13 2010