A016760 -id:A016760 - cp's OEIS frontend

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

R. J. Mathar, Feb 13 2010

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

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

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).

Cf. A000332, A016756, A016760. Partial sums of A117216.

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

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

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 *)

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

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)-a(n-1) = A117216(n).
a(n) = 8*A001844(n) * A000217(n) + 1 = 8*A219086(n) + 1. - Bruce J. Nicholson, Apr 13 2017

A016952 a(n) = (6*n + 3)^8.

Original entry on oeis.org

6561, 43046721, 2562890625, 37822859361, 282429536481, 1406408618241, 5352009260481, 16815125390625, 45767944570401, 111429157112001, 248155780267521, 513798374428641, 1001129150390625, 1853020188851841, 3282116715437121, 5595818096650401, 9227446944279201

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016760, A016945, A016946, A016947, A016948, A016949, A016950, A016951.

Subsequence of A001016.

[(6*n+3)^8: n in [0..40]]; // Vincenzo Librandi, May 05 2011

a[n_] := (6*n + 3)^8; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^8 = A016946(n)^4 = A016948(n)^2.
a(n) = 3^8*A016760(n).
Sum_{n>=0} 1/a(n) = 17*Pi^8/1058158080. (End)

A016832 a(n) = (4*n + 2)^8.

Original entry on oeis.org

256, 1679616, 100000000, 1475789056, 11019960576, 54875873536, 208827064576, 656100000000, 1785793904896, 4347792138496, 9682651996416, 20047612231936, 39062500000000, 72301961339136, 128063081718016, 218340105584896, 360040606269696, 576480100000000, 899194740203776

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016760, A016825.

[(4*n+2)^8: n in [0..30] ]; // Vincenzo Librandi, May 23 2011

(4Range[0,20]+2)^8 (* Harvey P. Dale, May 20 2011 *)

From Harvey P. Dale, May 20 2011: (Start)
a(0)=256, a(1)=1679616, a(2)=100000000, a(3)=1475789056, a(4)=11019960576, a(5)=54875873536, a(6)=208827064576, a(7)=656100000000, a(8)=1785793904896, a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
G.f.: -((256*(1 + 6552*x + 331612*x^2 + 2485288*x^3 + 4675014*x^4 + 2485288*x^5 + 331612*x^6 + 6552*x^7 + x^8))/(-1+x)^9). (End)
From Amiram Eldar, Apr 21 2023: (Start)
a(n) = A016825(n)^8.
a(n) = 2^8*A016760(n).
Sum_{n>=0} 1/a(n) = 17*Pi^8/41287680. (End)

A016748 a(n) = (2*n)^8.

Original entry on oeis.org

0, 256, 65536, 1679616, 16777216, 100000000, 429981696, 1475789056, 4294967296, 11019960576, 25600000000, 54875873536, 110075314176, 208827064576, 377801998336, 656100000000, 1099511627776, 1785793904896, 2821109907456, 4347792138496, 6553600000000, 9682651996416

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A001016, A005843, A016760.

[(2*n)^8: n in [0..20]]; // Vincenzo Librandi, Sep 05 2011

A016748:=n->(2*n)^8; seq(A016748(n), n=0..50); # Wesley Ivan Hurt, Nov 15 2013

Table[(2n)^8, {n,0,50}] (* Wesley Ivan Hurt, Nov 15 2013 *)
(2*Range[0,20])^8 (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36, -9, 1}, {0,256 ,65536, 1679616, 16777216, 100000000, 429981696, 1475789056, 4294967296}, 20] (* Harvey P. Dale, Jun 14 2016 *)

vector(30, n, n--; (2*n)^8) \\ G. C. Greubel, Sep 15 2018

a(n) = 256*A001016(n) = A001016(A005843(n)). - Michel Marcus, Nov 16 2013
G.f.: 256*x*(1+x)*(x^6 + 246*x^5 + 4047*x^4 + 11572*x^3 + 4047*x^2 + 246*x + 1) / (1-x)^9. - R. J. Mathar, May 01 2015
From Amiram Eldar, Oct 11 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi^8/2419200.
Sum_{n>=1} (-1)^(n+1)/a(n) = 127*Pi^8/309657600. (End)

A017120 a(n) = (8*n + 4)^8.

Original entry on oeis.org

65536, 429981696, 25600000000, 377801998336, 2821109907456, 14048223625216, 53459728531456, 167961600000000, 457163239653376, 1113034787454976, 2478758911082496, 5132188731375616, 10000000000000000, 18509302102818816, 32784148919812096, 55895067029733376

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016760, A016832, A017113.

[(8*n+4)^8: n in [0..15] ]; // Vincenzo Librandi, Jul 21 2011

Table[(8*n + 4)^8, {n, 0, 20}] (* Amiram Eldar, Apr 25 2023 *)

G.f.: -65536*(1 + 6552*x + 331612*x^2 + 2485288*x^3 + 4675014*x^4 + 2485288*x^5 + 331612*x^6 + 6552*x^7 + x^8)/(x-1)^9. - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^8.
a(n) = 2^8*A016832(n) = 2^16*A016760(n).
Sum_{n>=0} 1/a(n) = 17*Pi^8/10569646080. (End)

A017336 a(n) = (10*n + 5)^8.

Original entry on oeis.org

390625, 2562890625, 152587890625, 2251875390625, 16815125390625, 83733937890625, 318644812890625, 1001129150390625, 2724905250390625, 6634204312890625, 14774554437890625, 30590228625390625, 59604644775390625, 110324037687890625, 195408755062890625, 333160561500390625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016760, A017329.

[(10*n+5)^8: n in [0..20]]; // Vincenzo Librandi, Aug 02 2011

(10Range[0,20]+5)^8 (* Harvey P. Dale, Nov 02 2011 *)

G.f.: -((390625*(x^8 + 6552*x^7 + 331612*x^6 + 2485288*x^5 + 4675014*x^4 + 2485288*x^3 + 331612*x^2 + 6552*x + 1))/(x-1)^9). - Harvey P. Dale, Nov 02 2011
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^8.
a(n) = 5^8 * A016760(n).
Sum_{n>=0} 1/a(n) = 17*Pi^8/63000000000. (End)

cp's OEIS Frontend

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

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A016952 a(n) = (6*n + 3)^8.

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A016832 a(n) = (4*n + 2)^8.

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A016748 a(n) = (2*n)^8.

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A017120 a(n) = (8*n + 4)^8.

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A017336 a(n) = (10*n + 5)^8.

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