A175110 a(n) = ((2*n+1)^4+1)/2.

1, 41, 313, 1201, 3281, 7321, 14281, 25313, 41761, 65161, 97241, 139921, 195313, 265721, 353641, 461761, 592961, 750313, 937081, 1156721, 1412881, 1709401, 2050313, 2439841, 2882401, 3382601, 3945241, 4575313, 5278001, 6058681

Offset

0,2

Comments

Partial sums of A117216. 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.

Hypotenuse of Pythagorean triangles with smallest side a square: A016754(n)^2 + (a(n)-1)^2 = a(n)^2. - Martin Renner, Nov 12 2011

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

References

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 54.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5).

G.f.: (1+36*x+118*x^2+36*x^3+x^4)/ (1-x)^5.

a(n) = 8*A001844(n) * A000217(n) + 1 = 8*A219086(n) + 1. - Bruce J. Nicholson, Apr 13 2017

Maple

A175110:=n->((2*n+1)^4+1)/2: seq(A175110(n), n=0..50); # Wesley Ivan Hurt, Apr 13 2017

Mathematica

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 *)
Table[((2 n + 1)^4 + 1)/2, {n, 0, 29}] (* Michael De Vlieger, Dec 26 2016 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 41, 313, 1201, 3281}, 40] (* Harvey P. Dale, Jan 01 2022 *)

Prog

(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
(PARI) a(n)=((2*n+1)^4+1)/2 \\ Charles R Greathouse IV, Oct 16 2015

Crossrefs

Cf. A000332, A016756, A016760, A117216.

Sequence in context: A201043 A364269 A002646 * A096170 A277201 A340465

Adjacent sequences: A175107 A175108 A175109 * A175111 A175112 A175113

Keyword

easy,nonn

Author

R. J. Mathar, Feb 13 2010

Status

approved