A000539 -id:A000539 - cp's OEIS frontend

A140150 a(1)=1, a(n)=a(n-1)+n^2 if n odd, a(n)=a(n-1)+ n^4 if n is even.

1, 17, 26, 282, 307, 1603, 1652, 5748, 5829, 15829, 15950, 36686, 36855, 75271, 75496, 141032, 141321, 246297, 246658, 406658, 407099, 641355, 641884, 973660, 974285, 1431261, 1431990, 2046646, 2047487, 2857487, 2858448, 3907024, 3908113

Offset: 1

Artur Jasinski, May 12 2008

nonn

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

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

a = {}; r = 2; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^2,a+(n+1)^4]}; NestList[nxt,{1,1},40][[All,2]] (* or *) LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1},{1,17,26,282,307,1603,1652,5748,5829,15829,15950},40] (* Harvey P. Dale, Aug 28 2017 *)

G.f.: x*(1+16*x+4*x^2+176*x^3-10*x^4+176*x^5+4*x^6+16*x^7+x^8)/((1+x)^5*(x-1)^6). [From R. J. Mathar, Feb 22 2009]

A140152 a(1)=1, a(n)=a(n-1)+n^3 if n odd, a(n)=a(n-1)+ n^0 if n is even.

Original entry on oeis.org

1, 2, 29, 30, 155, 156, 499, 500, 1229, 1230, 2561, 2562, 4759, 4760, 8135, 8136, 13049, 13050, 19909, 19910, 29171, 29172, 41339, 41340, 56965, 56966, 76649, 76650, 101039, 101040, 130831, 130832, 166769, 166770, 209645, 209646, 260299

Offset: 1

Artur Jasinski, May 12 2008

nonn

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

a = {}; r = 3; s = 0; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{a_,b_}]:={a+1,If[EvenQ[a],b+(a+1)^3,b+1]}; Transpose[NestList[nxt,{1,1},40]][[2]] (* Harvey P. Dale, Dec 13 2011 *)

a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9). G.f.: x*(-1-x-23*x^2+3*x^3-23*x^4-3*x^5-x^6+x^7)/((1+x)^4*(x-1)^5). [R. J. Mathar, Feb 22 2009]

A202107 a(n) = n^4*(n+1)^4/8.

Original entry on oeis.org

2, 162, 2592, 20000, 101250, 388962, 1229312, 3359232, 8201250, 18301250, 37949472, 74030112, 137149922, 243101250, 414720000, 684204032, 1095962562, 1710072162, 2606420000, 3889620000, 5694792642, 8194304162, 11605565952, 16200000000, 22313281250, 30356972802

Offset: 1

Martin Renner, Dec 11 2011

A relation between fourth powers and the sum of fifth and seventh powers. See the first formula, which is from Beiler.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 161.

Temple Rice Hollcroft, On sums of powers of n consecutive integers, Bulletin of the American Mathematical Society 59 (1953), nr. 6, p. 526 (574t).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000217, A000539, A000541, A002117, A059977.

A202107:=n->(n^4)*(n+1)^4/8; seq(A202107(n), n=1..100); # Wesley Ivan Hurt, Nov 12 2013

Table[n^4 (n+1)^4/8, {n, 100}] (* Wesley Ivan Hurt, Nov 12 2013 *)

a(n) = 2*(Sum_{k=1..n} k)^4 = Sum_{k=1..n} (k^5 + k^7).
a(n) = 2*A059977(n-1).
a(n) = A000539(n) + A000541(n).
G.f.: -2*x*(1+72*x+603*x^2+1168*x^3+603*x^4+72*x^5+x^6) / (x-1)^9. - R. J. Mathar, Dec 13 2011
a(n) = 2*(A000217(n)^4). - Zak Seidov, Jan 21 2012
From Amiram Eldar, Apr 09 2024: (Start)
Sum_{n>=1} 1/a(n) = 8*Pi^4/45 + 80*Pi^2/3 - 280.
Sum_{n>=1} (-1)^(n+1)/a(n) = 280 - 320*log(2) - 48*zeta(3). (End)

A263689 a(n) = (2n^6 - 6n^5 + 5*n^4 - n^2 + 12)/12.

Original entry on oeis.org

1, 1, 2, 34, 277, 1301, 4426, 12202, 29009, 61777, 120826, 220826, 381877, 630709, 1002002, 1539826, 2299201, 3347777, 4767634, 6657202, 9133301, 12333301, 16417402, 21571034, 28007377, 35970001, 45735626, 57617002, 71965909, 89176277, 109687426, 133987426, 162616577, 196171009, 235306402, 280741826

Offset: 0

Ilya Gutkovskiy, Nov 20 2015

			a(0) = 1,
a(1) = 0^5 + 1 = 1,
a(2) = 1^5 + 1 = 2,
a(3) = 2^5 + 2 = 34,
a(4) = 3^5 + 34 = 227,
a(5) = 4^5 + 227 = 1301, etc.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000124, A000539, A000584, A056520, A154323.

Table[(1/12) (12 + (-1 + n)^2 n^2 (-1 + 2 (-1 + n) n)), {n, 0, 35}]

first(m)=vector(m,n,n--;(2*n^6 - 6*n^5 + 5*n^4 - n^2 + 12)/12) \\ Anders Hellström, Nov 20 2015

G.f.: (1 - 6*x + 16*x^2 + 6*x^3 + 81*x^4 + 20*x^5 + 2*x^6)/(1 - x)^7.
a(n + 1) = a(n) + n^5, a(0) = 1.
a(n + 1) - a(n) = A000584(n).
a(n + 1) = A000539(n) + 1.
Sum_{n>0} 1/(a(n + 1) - a(n)) = zeta(5) = 1.036927755...

A265021 Sum of fifth powers of the first n even numbers.

Original entry on oeis.org

0, 32, 1056, 8832, 41600, 141600, 390432, 928256, 1976832, 3866400, 7066400, 12220032, 20182656, 32064032, 49274400, 73574400, 107128832, 152564256, 213030432, 292265600, 394665600, 525356832, 690273056, 896236032, 1151040000, 1463540000, 1843744032

Offset: 0

Assoul Abdelkarim, Nov 30 2015

nonn

			a(4) =  2^5 + 4^5 + 6^5 + 8^5 = 41600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Assoul Abdelkarim, The sums of powers of integers natural even, odd, RMSS-D-15-00105.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000539, A002594 (the same for odd numbers).

[(8/3)*n^2*(n+1)^2*(2*n^2+2*n-1): n in [0..30]]; // Vincenzo Librandi, Dec 01 2015

Accumulate[Range[0, 60, 2]^5] (* Michael De Vlieger, Nov 30 2015 *)
CoefficientList[Series[32 x (1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^7, {x, 0, 33}], x] (* Vincenzo Librandi, Dec 01 2015 *)

vector(100, n, n--; (8/3)*n^2*(n+1)^2*(2*n^2+2*n-1)) \\ Altug Alkan, Dec 01 2015

a(n) = 32 * Sum_{i=0..n} i^5 = (8/3)*n^2*(n+1)^2*(2*n^2+2*n-1).
a(n) = 32 * A000539(n).
G.f.: 32*x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^7. - Vincenzo Librandi, Dec 01 2015
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Vincenzo Librandi, Dec 01 2015

A351770 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^5.

Original entry on oeis.org

0, 1, 1057, 68125, 1399325, 15227450, 110102426, 597639882, 2621915850, 9756511275, 31839011275, 93340522951, 250280856007, 622316813300, 1450471654100, 3196426654100, 6706824221076, 13476181309557, 26055415288725, 48670370285425, 88136930285425, 155187254126926

Offset: 0

Roudy El Haddad, Feb 18 2022

a(n) is the sum of all products of two elements from the set {1^5, ..., n^5}.

Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A001296 (for power 1), A060493 (for squares), A346642 (for cubes), A351766 (for fourth powers).

Cf. A000584 (fifth powers), A000539 (sum of fifth powers).

seq(n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168,
n=0..30);# Robert Israel, Feb 18 2022

{a(n) = n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168};

a(n) = sum(j=1, n, sum(i=1, j, i^5*j^5));

a(n) = n*(n+1)*(n+2)*(44*n^9 + 276*n^8 + 492*n^7 - 48*n^6 - 609*n^5 + 207*n^4 + 487*n^3 - 291*n^2 - 90*n + 60)/3168.

G.f.: x*(1 + 1044*x + 54462*x^2 + 595860*x^3 + 2048388*x^4 + 2563644*x^5 + 1193226*x^6 + 188508*x^7 + 7635*x^8 + 32*x^9)/(1-x)^13. - Robert Israel, Feb 18 2022

A351805 a(n) = Sum_{1 <= i < j <= n} j^5*i^5.

Original entry on oeis.org

0, 0, 32, 8051, 290675, 4353175, 38761975, 243824182, 1194358326, 4842169350, 16924669350, 52488756425, 147511725257, 381689190701, 920589376525, 2089893985900, 4500779925100, 9254143113132, 18262909865676, 34746798604575, 63973358604575, 114343801467875

Offset: 0

Roudy El Haddad, Feb 19 2022

a(n) is the sum of all products of two distinct elements from the set {1^5, ..., n^5}.

Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares), A347107 (for cubes), (for fourth powers).

Cf. A000584 (fifth powers), A000539 (sum of fifth powers).

{a(n) = n*(n-1)*(n+1)*(44*n^9+120*n^8-132*n^7-540*n^6+99*n^5+912*n^4-11*n^3-672*n^2+120)/3168};

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^5*i^5.
a(n) = n*(n - 1)*(n + 1)*(44*n^9 + 120*n^8 - 132*n^7 - 540*n^6 + 99*n^5 + 912*n^4 - 11*n^3 - 672*n^2 + 120)/3168.
G.f.: -x^2*(x^9 +1044*x^8 +54462*x^7 +595860*x^6 +2048388*x^5 +2563644*x^4 +1193226*x^3 +188508*x^2 +7635*x +32)/(x-1)^13. - Alois P. Heinz, Feb 19 2022

A362946 Positive integers that cannot be expressed as 1^e_1 + 2^e_2 + 3^e_3 ... + k^e_k with each exponent positive.

Original entry on oeis.org

2, 4, 7, 11, 13, 19, 25, 31

Offset: 1

Robert C. Lyons, Jul 05 2023

I conjecture that this list is finite.

			1 is not in the sequence because it's equal to 1^1.
3 is not in the sequence because it's equal to 1^1 + 2^1.
20 is not in the sequence because it's equal to 1^1 + 2^4 + 3^1.
29 is not in the sequence because it's equal to 1^1 + 2^2 + 3^1 + 4^2 + 5^1.

Cf. A000217, A000330, A000537, A000538, A000539.

from itertools import product
import math
max_term = 250
seq_set = set(range(1, max_term+1))
# Use the quadratic formula to calculate the maximum value for k,
# such that 1^1 + 2^1 + 3^1 + ... + k^1 is less than max_term.
max_k = int((-1 + math.sqrt(1 + 8 * max_term))/2.0) + 1
for k in range(1, max_k+1):
    list_of_exponent_ranges = [range(1,2)]
    for i in range(2, k+1):
        max_exponent = int(math.log(max_term, i))
        list_of_exponent_ranges.append(range(1, max_exponent+1))
    for exponents in product(*list_of_exponent_ranges):
        total = 0
        for i in range(1, k+1):
            total += int(i**exponents[i-1])
            if total > max_term:
                total = 0
                break
        if total in seq_set:
            seq_set.remove(total)
print(sorted(seq_set))

cp's OEIS Frontend

A140150 a(1)=1, a(n)=a(n-1)+n^2 if n odd, a(n)=a(n-1)+ n^4 if n is even.

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A140152 a(1)=1, a(n)=a(n-1)+n^3 if n odd, a(n)=a(n-1)+ n^0 if n is even.

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A202107 a(n) = n^4*(n+1)^4/8.

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A263689 a(n) = (2*n^6 - 6*n^5 + 5*n^4 - n^2 + 12)/12.

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A265021 Sum of fifth powers of the first n even numbers.

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A351770 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^5.

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A351805 a(n) = Sum_{1 <= i < j <= n} j^5*i^5.

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A362946 Positive integers that cannot be expressed as 1^e_1 + 2^e_2 + 3^e_3 ... + k^e_k with each exponent positive.

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A263689 a(n) = (2n^6 - 6n^5 + 5*n^4 - n^2 + 12)/12.