A008456 -id:A008456 - cp's OEIS frontend

A016956 a(n) = (6*n + 3)^12.

531441, 282429536481, 129746337890625, 7355827511386641, 150094635296999121, 1667889514952984961, 12381557655576425121, 68952523554931640625, 309629344375621415601, 1176246293903439668001, 3909188328478827879681, 11646329922777311412561, 31676352024078369140625

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016764, A016945, A016946, A016947, A016948, A016949, A016950, A016951, A016952, A016953, A016954, A016955.

Subsequence of A008456.

[(6*n+3)^12: n in [0..20]]; // Vincenzo Librandi, May 06 2011

(6*Range[0,20]+3)^12 (* Harvey P. Dale, Jan 23 2013 *)

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016945(n)^12 = A016946(n)^6 = A016947(n)^4 = A016948(n)^3 = A016950(n)^2..
a(n) = 3^12*A016764(n).
Sum_{n>=0} 1/a(n) = 691*Pi^12/339414785740800. (End)

A016968 a(n) = (6*n + 4)^12.

Original entry on oeis.org

16777216, 1000000000000, 281474976710656, 12855002631049216, 232218265089212416, 2386420683693101056, 16777216000000000000, 89762301673555234816, 390877006486250192896, 1449225352009601191936, 4722366482869645213696, 13841287201000000000000, 37133262473195501387776

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016800, A016957, A016958, A016959, A016960, A016961, A016962, A016963, A016964, A016965, A016966, A016967.

Subsequence of A008456.

[(6*n+4)^12: n in [0..20]]; // Vincenzo Librandi, May 07 2011

(6Range[0,20]+4)^12  (* Harvey P. Dale, Apr 17 2011 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016957(n)^12 = A016958(n)^6 = A016959(n)^4 = A016960(n)^3 = A016962(n)^2.
a(n) = 2^12*A016800(n).
Sum_{n>=0} 1/a(n) = PolyGamma(11, 2/3)/86890185149644800. (End)

A016980 a(n) = (6*n + 5)^12.

Original entry on oeis.org

244140625, 3138428376721, 582622237229761, 21914624432020321, 353814783205469041, 3379220508056640625, 22563490300366186081, 116191483108948578241, 491258904256726154641, 1779197418239532716881, 5688009063105712890625, 16409682740640811134241, 43439888521963583647921

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974, A016975, A016976, A016977, A016978, A016979.

Subsequence of A008456 (n^12).

[(6*n+5)^12: n in [0..40]]; // Vincenzo Librandi, May 19 2011

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^12 = A016970(n)^6 = A016971(n)^4 = A016972(n)^3 = A016974(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(11, 5/6)/86890185149644800. (End)

A017220 a(n) = (9*n + 4)^12.

Original entry on oeis.org

16777216, 23298085122481, 12855002631049216, 787662783788549761, 16777216000000000000, 191581231380566414401, 1449225352009601191936, 8182718904632857144561, 37133262473195501387776, 142241757136172119140625

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 (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A008456 (n^12), A017209 (9*n+4).

[(9*n+4)^12: n in [0..15]]; // Vincenzo Librandi, Jul 24 2011

A017352 (10*n+6)^12.

Original entry on oeis.org

2176782336, 281474976710656, 95428956661682176, 4738381338321616896, 89762301673555234816, 951166013805414055936, 6831675453247426400256, 37133262473195501387776, 163674647745587512938496, 612709757329767363772416

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A008456 (12th Powers), A017341 (10n+6).

[(10*n+6)^12: n in [0..10]]; // Vincenzo Librandi, Aug 03 2011

A017352:=n->(10*n+6)^12: seq(A017352(n), n=0..10); # Wesley Ivan Hurt, Oct 28 2014

(10 Range[0, 10] + 6)^12 (* Wesley Ivan Hurt, Oct 28 2014 *)
CoefficientList[Series[4096 (531441 + 68712568003 x + 22404773377311 x^2 + 859316242027205 x^3 + 8673413722667370 x^4 + 30946876621062078 x^5 + 44108689210889694 x^6 + 25884027384156618 x^7 + 5972410776815445 x^8 + 467792550632655 x^9 + 8736164034131 x^10 + 13841233953 x^11 + 4096 x^12)/(1 - x)^13, {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 28 2014 *)

From Wesley Ivan Hurt, Oct 28 2014: (Start)
G.f.: 4096*(531441 + 68712568003*x + 22404773377311*x^2 + 859316242027205*x^3 + 8673413722667370*x^4 + 30946876621062078*x^5 + 44108689210889694*x^6 + 25884027384156618*x^7 + 5972410776815445*x^8 + 467792550632655*x^9 + 8736164034131*x^10 + 13841233953*x^11 + 4096*x^12) / (1-x)^13.
a(n) = 13*a(n-1)-78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+1716*a(n-7)-1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13).
a(n) = (10*n+6)^12 = A008456(A017341(n)). (End)

A024113 a(n) = 9^n-n^12.

Original entry on oeis.org

1, 8, -4015, -530712, -16770655, -244081576, -2176250895, -13836504232, -68676430015, -282042115992, -996513215599, -3107047317112, -8633670911775, -20756219294152, -33817119920335, 76144794204024, 1571545212141185, 16094559462436808, 148937803915572945

Offset: 0

N. J. A. Sloane, Jun 14 1998

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (22,-195,988,-3289,7722,-13299,17160,-16731,12298,-6721,2652,-715,118,-9).

Cf. A001019, A008456.

[9^n-n^12: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

A024113:=n->9^n - n^12; seq(A024113(n), n=0..20); # Wesley Ivan Hurt, May 25 2014

Table[9^n - n^12, {n, 0, 20}] (* Wesley Ivan Hurt, May 25 2014 *)

a(n) = A001019(n) - A008456(n). - Wesley Ivan Hurt, May 25 2014

G.f.: (8*x^13 +36759*x^12 +4300278*x^11 +91211180*x^10 +586677866*x^9 +1396293387*x^8 +1300096572*x^7 +434355696*x^6 +25369404*x^5 -5882531*x^4 -441810*x^3 -3996*x^2 -14*x +1) / ((x -1)^13*(9*x -1)). - Colin Barker, May 26 2014

A340048 Numbers that are the sum of a cube s and a fourth power t such that 0 < s <= t.

Original entry on oeis.org

2, 17, 24, 82, 89, 108, 145, 257, 264, 283, 320, 381, 472, 626, 633, 652, 689, 750, 841, 968, 1137, 1297, 1304, 1323, 1360, 1421, 1512, 1639, 1808, 2025, 2296, 2402, 2409, 2428, 2465, 2526, 2617, 2744, 2913, 3130, 3401, 3732, 4097, 4104, 4123, 4129, 4160, 4221, 4312

Offset: 1

Wesley Ivan Hurt, Dec 26 2020

nonn

Contains the entries of A340050 and numbers like 2, 8192, 1062882,.. which are 2 times 12th powers (A008456). - R. J. Mathar, Jan 05 2021

			24 is in the sequence since 2^3 + 2^4 = 8 + 16 = 24, where 0 < 8 <= 16.

Cf. A010057.

Cf. A008456, A340050.

isA340048 := proc(n)
    local t,s3 ;
    for t from 0 do
        s3 := n-t^4 ;
        if s3 <= 0 then
            return false ;
        elif s3 <= t^4 and isA000578(s3) then
            return true;
        end if;
    end do:
end proc:
for n from 1 do
    if isA340048(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Jan 05 2021

Table[If[Sum[(Floor[i^(1/3)] - Floor[(i - 1)^(1/3)]) (Floor[(n - i)^(1/4)] - Floor[(n - i - 1)^(1/4)]), {i, Floor[n/2]}] > 0, n, {}], {n, 1000}] // Flatten

def aupto(lim):
  cubes = [i**3 for i in range(1, int(lim**(1/3))+2)]
  fours = [i**4 for i in range(1, int(lim**(1/4))+2)]
  return sorted(s+t for s in cubes for t in fours if t >= s and s+t <= lim)
print(aupto(4312)) # Michael S. Branicky, Feb 17 2021

cp's OEIS Frontend

A016956 a(n) = (6*n + 3)^12.

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A016968 a(n) = (6*n + 4)^12.

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Magma

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A016980 a(n) = (6*n + 5)^12.

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Magma

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A017220 a(n) = (9*n + 4)^12.

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Magma

A017352 (10*n+6)^12.

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Magma

Maple

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A024113 a(n) = 9^n-n^12.

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Magma

Maple

Mathematica

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A340048 Numbers that are the sum of a cube s and a fourth power t such that 0 < s <= t.

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Maple

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