A016756 -id:A016756 - cp's OEIS frontend

A016948 a(n) = (6*n + 3)^4.

81, 6561, 50625, 194481, 531441, 1185921, 2313441, 4100625, 6765201, 10556001, 15752961, 22667121, 31640625, 43046721, 57289761, 74805201, 96059601, 121550625, 151807041, 187388721, 228886641, 276922881, 332150625, 395254161, 466948881, 547981281, 639128961

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 (5,-10,10,-5,1).

Cf. A000583, A016756, A016945, A016946, A016947.

[(6*n+3)^4: n in [0..50]]; // Vincenzo Librandi, May 05 2011

a[n_] := (6*n + 3)^4; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^4 = A016946(n)^2.
a(n) = 3^4*A016756(n).
Sum_{n>=0} 1/a(n) = Pi^4/7776. (End)

A016760 a(n) = (2*n+1)^8.

Original entry on oeis.org

1, 6561, 390625, 5764801, 43046721, 214358881, 815730721, 2562890625, 6975757441, 16983563041, 37822859361, 78310985281, 152587890625, 282429536481, 500246412961, 852891037441, 1406408618241, 2251875390625, 3512479453921, 5352009260481, 7984925229121, 11688200277601

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. A016756, A300710.

[(2*n+1)^8: n in [0..30]]; // Vincenzo Librandi, Sep 07 2011

Table[(2*n+1)^8, {n,0,30}] (* G. C. Greubel, Sep 15 2018 *)

A016760(n):=(2*n+1)^8$
makelist(A016760(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

vector(30, n, n--; (2*n+1)^8) \\ G. C. Greubel, Sep 15 2018

a(n) = A016756(n)^2. - Michel Marcus, Dec 26 2016
G.f.: -(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, Jul 07 2017
Sum_{n>=0} 1/a(n) = 17*Pi^8/161280 (A300710). - Amiram Eldar, Oct 11 2020
Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(Pi/2)*(cos(Pi/sqrt(2)) + cosh(Pi/sqrt(2)))/32. - Amiram Eldar, Jan 28 2021

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

Original entry on oeis.org

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

A016744 a(n) = (2*n)^4.

Original entry on oeis.org

0, 16, 256, 1296, 4096, 10000, 20736, 38416, 65536, 104976, 160000, 234256, 331776, 456976, 614656, 810000, 1048576, 1336336, 1679616, 2085136, 2560000, 3111696, 3748096, 4477456, 5308416, 6250000, 7311616, 8503056, 9834496, 11316496, 12960000, 14776336, 16777216

Offset: 0

N. J. A. Sloane

Suppose the vertices of a triangle are (S(n), S(n+j)), (S(n+2*j), S(n+3*j)) and (S(n+4*j), S(n+5*j)) where S(n) is the n-th square number, A000290(n). Then the area of this triangle will be a(j). A generalization follows: let P(k,n) = the n-th k-gonal number and suppose the vertices of a triangle are (P(k,n), P(k,n+j)), (P(k,n+2*j), P(k,n+3*j)) and (P(k,n+4*j), P(k,n+5*j)). Then the area of this triangle will be (2*k-4)^2*j^4. See also A141046 for k = 3. - Charlie Marion, Mar 26 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016756, A258945.

[(2*n)^4: n in [0..40]]; // Vincenzo Librandi, Sep 05 2011

A016744:=n->(2*n)^4: seq(A016744(n), n=0..50); # Wesley Ivan Hurt, Sep 15 2018

Table[(2*n)^4, {n,0,30}] (* G. C. Greubel, Sep 15 2018 *)

vector(30, n, n--; (2*n)^4) \\ G. C. Greubel, Sep 15 2018

G.f.: 16*x*(x+1)*(x^2+10*x+1)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
E.g.f.: 16*x*(1 + 7*x + 6*x^2 + x^3)*exp(x). - G. C. Greubel, Sep 15 2018
From Amiram Eldar, Oct 10 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi^4/1440 (conjecturally A258945).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/11520. (End)

A016828 a(n) = (4*n+2)^4.

Original entry on oeis.org

16, 1296, 10000, 38416, 104976, 234256, 456976, 810000, 1336336, 2085136, 3111696, 4477456, 6250000, 8503056, 11316496, 14776336, 18974736, 24010000, 29986576, 37015056, 45212176, 54700816, 65610000, 78074896, 92236816

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 (5,-10,10,-5,1)

[(4*n+2)^4: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Table[(4n+2)^4,{n,0,100}] (* Mohammad K. Azarian, Jun 21 2016 *)

a(n) = 16*A016756(n). O.g.f.: -16*(1+76*x+230*x^2+76*x^3+x^4)/(-1+x)^5. - R. J. Mathar, Mar 31 2008
From Ilya Gutkovskiy, Jun 21 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
Sum_{n>=0} 1/a(n) = Pi^4/1536. (End)

More terms from R. J. Mathar, Mar 31 2008

A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

Original entry on oeis.org

0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561, 9375, 14641, 19440, 28561, 36015, 50625, 61440, 83521, 98415, 130321, 150000, 194481, 219615, 279841, 311040, 390625, 428415, 531441, 576240, 707281, 759375, 923521, 983040, 1185921, 1252815, 1500625, 1574640, 1874161, 1954815

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A285989.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000583, A016756, A026741, A059377, A285989, A308422, A309335.

a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 15 x + 76 x^2 + 165 x^3 + 230 x^4 + 165 x^5 + 76 x^6 + 15 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]
Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]

G.f.: x * (1 + 15*x + 76*x^2 + 165*x^3 + 230*x^4 + 165*x^5 + 76*x^6 + 15*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 - x^(2*k)), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 1/2^s).
a(n) = n^4 * (31 - (-1)^n)/32.
a(n) = Sum_{d|n, n/d odd} J_4(d).
Sum_{n>=1} 1/a(n) = 241*Pi^4/21600 = 1.086832913851601267313987...
Multiplicative with a(2^e) = 15*2^(4*e-4), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020

A309338 a(n) = n^4 if n odd, 7*n^4/8 if n even.

Original entry on oeis.org

0, 1, 14, 81, 224, 625, 1134, 2401, 3584, 6561, 8750, 14641, 18144, 28561, 33614, 50625, 57344, 83521, 91854, 130321, 140000, 194481, 204974, 279841, 290304, 390625, 399854, 531441, 537824, 707281, 708750, 923521, 917504, 1185921, 1169294, 1500625, 1469664, 1874161, 1824494

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A284900.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000583, A016756, A059377, A129194, A193356, A284900, A309337.

a[n_] := If[OddQ[n], n^4, 7 n^4/8]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 14 x + 76 x^2 + 154 x^3 + 230 x^4 + 154 x^5 + 76 x^6 + 14 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 14, 81, 224, 625, 1134, 2401, 3584, 6561}, 39]
Table[n^4 (15 - (-1)^n)/16, {n, 0, 38}]

G.f.: x * (1 + 14*x + 76*x^2 + 154*x^3 + 230*x^4 + 154*x^5 + 76*x^6 + 14*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 + x^k), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 2^(1-s)).
a(n) = n^4 * (15 - (-1)^n)/16.
a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_4(d).
Sum_{n>=1} 1/a(n) = 113*Pi^4/10080 = 1.091986834012130496797...
Multiplicative with a(2^e) = 7*2^(4*e-3), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020

A017116 a(n) = (8*n + 4)^4.

Original entry on oeis.org

256, 20736, 160000, 614656, 1679616, 3748096, 7311616, 12960000, 21381376, 33362176, 49787136, 71639296, 100000000, 136048896, 181063936, 236421376, 303595776, 384160000, 479785216, 592240896, 723394816, 875213056, 1049760000, 1249198336, 1475789056, 1731891456

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 (5,-10,10,-5,1).

Cf. A016756, A016828, A017113.

[(8*n+4)^4: n in [0..35] ]; // Vincenzo Librandi, Jul 21 2011

(8*Range[0,20]+4)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1},{256,20736,160000,614656,1679616},20] (* Harvey P. Dale, Aug 05 2019 *)

G.f.: -256*(1 + 76*x + 230*x^2 + 76*x^3 + x^4)/(x-1)^5. - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^4.
a(n) = 2^4*A016828(n) = 2^8*A016756(n).
Sum_{n>=0} 1/a(n) = Pi^4/24576. (End)

A017332 a(n) = (10*n + 5)^4.

Original entry on oeis.org

625, 50625, 390625, 1500625, 4100625, 9150625, 17850625, 31640625, 52200625, 81450625, 121550625, 174900625, 244140625, 332150625, 442050625, 577200625, 741200625, 937890625, 1171350625, 1445900625, 1766100625, 2136750625, 2562890625, 3049800625, 3603000625

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 (5,-10,10,-5,1).

Cf. A016756, A017329.

[(10*n+5)^4: n in [0..35]]; // Vincenzo Librandi, Aug 02 2011

Table[(10*n + 5)^4, {n, 0, 30}] (* Amiram Eldar, Apr 18 2023 *)
LinearRecurrence[{5,-10,10,-5,1},{625,50625,390625,1500625,4100625},30] (* Harvey P. Dale, Aug 20 2024 *)

G.f.: -625*(x^4 + 76*x^3 + 230*x^2 + 76*x+1)/(x-1)^5. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^4.
a(n) = 5^4 * A016756(n).
Sum_{n>=0} 1/a(n) = Pi^4/60000. (End)

cp's OEIS Frontend

A016948 a(n) = (6*n + 3)^4.

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A016760 a(n) = (2*n+1)^8.

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

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A016744 a(n) = (2*n)^4.

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A016828 a(n) = (4*n+2)^4.

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A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

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A309338 a(n) = n^4 if n odd, 7*n^4/8 if n even.

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