A000539 -id:A000539 - cp's OEIS frontend

A140157 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^0 if n is even.

1, 2, 83, 84, 709, 710, 3111, 3112, 9673, 9674, 24315, 24316, 52877, 52878, 103503, 103504, 187025, 187026, 317347, 317348, 511829, 511830, 791671, 791672, 1182297, 1182298, 1713739, 1713740, 2421021, 2421022, 3344543, 3344544, 4530465

Offset: 1

Artur Jasinski, May 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.

[(1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018

a = {}; r = 4; 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 *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 2, 83, 84, 709, 710, 3111, 3112, 9673, 9674, 24315}, 50] (* or *) Table[(1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)

for(n=1,50, print1((1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

G.f.: x*(1 + x + 76*x^2 - 4*x^3 + 230*x^4 + 6*x^5 + 76*x^6 - 4*x^7 + x^8 + x^9)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140158 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^1 if n is even.

Original entry on oeis.org

1, 3, 84, 88, 713, 719, 3120, 3128, 9689, 9699, 24340, 24352, 52913, 52927, 103552, 103568, 187089, 187107, 317428, 317448, 511929, 511951, 791792, 791816, 1182441, 1182467, 1713908, 1713936, 2421217, 2421247, 3344768, 3344800, 4530721

Offset: 1

Artur Jasinski, May 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.

[(1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018

a = {}; r = 4; s = 1; 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 *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 3, 84, 88, 713, 719, 3120, 3128, 9689, 9699, 24340}, 50] (* or *) Table[(1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^4,a+n+1]}; NestList[nxt,{1,1},40][[;;,2]] (* Harvey P. Dale, Dec 28 2024 *)

for(n=1,50, print1((1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

G.f.: x*(1 + 2*x + 76*x^2 - 6*x^3 + 230*x^4 + 6*x^5 + 76*x^6 - 2*x^7 + x^8)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140159 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^2 if n is even.

Original entry on oeis.org

1, 5, 86, 102, 727, 763, 3164, 3228, 9789, 9889, 24530, 24674, 53235, 53431, 104056, 104312, 187833, 188157, 318478, 318878, 513359, 513843, 793684, 794260, 1184885, 1185561, 1717002, 1717786, 2425067, 2425967, 3349488, 3350512, 4536433

Offset: 1

Artur Jasinski, May 12 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
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.

[(1/60)*(n +n^2)* (4 + 30*(-1)^n + (11 -15*(-1)^n)*n + (9 -15*(-1)^n)*n^2 + 6*n^3): n in [1..50]]; // G. C. Greubel, Jul 05 2018

a = {}; r = 4; s = 2; 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)^4,a+(n+1)^2]}; NestList[nxt, {1, 1}, 40][[All, 2]] (* Harvey P. Dale, Sep 21 2016 *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 5, 86, 102, 727, 763, 3164, 3228, 9789, 9889, 24530}, 50] (* or *) Table[(1/60)*(n +n^2)* (4 + 30*(-1)^n + (11 -15*(-1)^n)*n + (9 -15*(-1)^n)*n^2 + 6*n^3), {n,1, 50}] (* G. C. Greubel, Jul 05 2018 *)

for(n=1,50, print1((1/60)*(n +n^2)* (4 + 30*(-1)^n + (11 -15*(-1)^n)*n + (9 -15*(-1)^n)*n^2 + 6*n^3), ", ")) \\ G. C. Greubel, Jul 05 2018

G.f.: x*(1+4*x+76*x^2-4*x^3+230*x^4-4*x^5+76*x^6+4*x^7+x^8)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140160 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^3 if n is even.

Original entry on oeis.org

1, 9, 90, 154, 779, 995, 3396, 3908, 10469, 11469, 26110, 27838, 56399, 59143, 109768, 113864, 197385, 203217, 333538, 341538, 536019, 546667, 826508, 840332, 1230957, 1248533, 1779974, 1801926, 2509207, 2536207, 3459728, 3492496

Offset: 1

Artur Jasinski, May 12 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
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.

[(1/240)*(15*(1 -(-1)^n) - 4*(1 - 15*(-1)^n)*n + 30*(1 + 3(-1)^n)*n^2 + 20*(5 - 3*(-1)^n)*n^3 + 30*(3 - 2*(-1)^n)*n^4 + 24*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018

a = {}; r = 4; s = 3; 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)^4,a+(n+1)^3]}; NestList[nxt,{1,1},40][[All,2]] (* or *) LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1},{1,9,90,154,779,995,3396,3908,10469,11469,26110},40] (* Harvey P. Dale, Oct 05 2016 *)
Table[(1/240)*(15*(1 -(-1)^n) - 4*(1 - 15*(-1)^n)*n + 30*(1 + 3(-1)^n)*n^2 + 20*(5 - 3*(-1)^n)*n^3 + 30*(3 - 2*(-1)^n)*n^4 + 24*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)

for(n=1,50, print1((1/240)*(15*(1 -(-1)^n) - 4*(1 - 15*(-1)^n)*n + 30*(1 + 3(-1)^n)*n^2 + 20*(5 - 3*(-1)^n)*n^3 + 30*(3 - 2*(-1)^n)*n^4 + 24*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

G.f.: x*(1 + 8*x + 76*x^2 + 24*x^3 + 230*x^4 - 24*x^5 + 76*x^6 - 8*x^7 + x^8)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140161 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^5 if n is even.

Original entry on oeis.org

1, 33, 114, 1138, 1763, 9539, 11940, 44708, 51269, 151269, 165910, 414742, 443303, 981127, 1031752, 2080328, 2163849, 4053417, 4183738, 7383738, 7578219, 12731851, 13011692, 20974316, 21364941, 33246317, 33777758, 50988126

Offset: 1

Artur Jasinski, May 12 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6, -1,1).

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

[(1/120)*(15*(-1 +(-1)^n) - 2*(1 -15*(-1)^n)*n - 5*(1 +15*(-1)^n)*n^2 + 20*(1 -3*(-1)^n)*n^3 + (55 + 45*(-1)^n)*n^4 + (42 +30*(-1)^n)*n^5 + 10*n^6): n in [1..50]]; // G. C. Greubel, Jul 05 2018

a = {}; r = 4; s = 5; 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 *)
next[{a_,b_}]:={a+1,If[OddQ[a+1],b+(a+1)^4,b+(a+1)^5]}; Transpose[ NestList[ next[#]&,{1,1},30]][[2]] (* Harvey P. Dale, Nov 23 2011 *)
Table[(1/120)*(15*(-1 +(-1)^n) - 2*(1 -15*(-1)^n)*n - 5*(1 +15*(-1)^n)*n^2 + 20*(1 -3*(-1)^n)*n^3 + (55 + 45*(-1)^n)*n^4 + (42 +30*(-1)^n)*n^5 + 10*n^6), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)

for(n=1, 50, print1((1/120)*(15*(-1 +(-1)^n) - 2*(1 -15*(-1)^n)*n - 5*(1 +15*(-1)^n)*n^2 + 20*(1 -3*(-1)^n)*n^3 + (55 + 45*(-1)^n)*n^4 + (42 +30*(-1)^n)*n^5 + 10*n^6), ", ")) \\ G. C. Greubel, Jul 05 2018

G.f.: x*(-1 - 32*x - 75*x^2 - 832*x^3 - 154*x^4 - 2112*x^5 + 154*x^6 - 832*x^7 + 75*x^8 - 32*x^9 + x^10)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009

A140162 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^0 if n is even.

Original entry on oeis.org

1, 2, 245, 246, 3371, 3372, 20179, 20180, 79229, 79230, 240281, 240282, 611575, 611576, 1370951, 1370952, 2790809, 2790810, 5266909, 5266910, 9351011, 9351012, 15787355, 15787356, 25552981, 25552982, 39901889, 39901890, 60413039

Offset: 1

Artur Jasinski, May 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1).

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

[(1/24)*(3*(-1 +(-1)^n) + 12*n + (-1 +15*(-1)^n)*n^2 + 5*(1 -3* (-1)^n)*n^4 - 6*(-1 +(-1)^n)*n^5 + 2*n^6): n in [1..50]]; // G. C. Greubel, Jul 05 2018

a = {}; r = 5; 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 *)
LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,2,245,246, 3371,3372,20179,20180,79229,79230,240281,240282,611575},40]  (* Harvey P. Dale, Apr 21 2011 *)

for(n=1,50, print1((1/24)*(3*(-1 +(-1)^n) + 12*n + (-1 +15*(-1)^n)*n^2 + 5*(1 -3* (-1)^n)*n^4 - 6*(-1 +(-1)^n)*n^5 + 2*n^6), ", ")) \\ G. C. Greubel, Jul 05 2018

G.f.: x*(-1 - x - 237*x^2 + 5*x^3 - 1682*x^4 - 10*x^5 - 1682*x^6 + 10*x^7 - 237*x^8 - 5*x^9 - x^10 + x^11)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009

A140163 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n if n is even.

Original entry on oeis.org

1, 3, 246, 250, 3375, 3381, 20188, 20196, 79245, 79255, 240306, 240318, 611611, 611625, 1371000, 1371016, 2790873, 2790891, 5266990, 5267010, 9351111, 9351133, 15787476, 15787500, 25553125, 25553151, 39902058, 39902086, 60413235

Offset: 1

Artur Jasinski, May 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6, -1,1).

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

[(1/24)*(n +n^2)*(6*(1 +(-1)^n) - (1-9*(-1)^n)*n + (1 -9*(-1)^n)*n^2 + (4 -6*(-1)^n)*n^3 + 2*n^4): n in [1..50]]; // G. C. Greubel, Jul 05 2018

a:=proc(n) option remember: if n=1 then 1 elif modp(n,2)<>0 then procname(n-1)+n^5 else procname(n-1)+n; fi: end; seq(a(n),n=1..30); # Muniru A Asiru, Jul 07 2018

Table[(1/24)*(n +n^2)*(6*(1 +(-1)^n) - (1-9*(-1)^n)*n + (1 -9*(-1)^n)*n^2 + (4 -6*(-1)^n)*n^3 + 2*n^4), {n, 1, 50}] (* or *) LinearRecurrence[{1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1}, {1, 3, 246, 250, 3375, 3381, 20188, 20196, 79245, 79255, 240306, 240318, 611611}, 60] (* G. C. Greubel, Jul 05 2018 *)

for(n=1,50, print1((1/24)*(n +n^2)*(6*(1 +(-1)^n) - (1-9*(-1)^n)*n + (1 -9*(-1)^n)*n^2 + (4 -6*(-1)^n)*n^3 + 2*n^4), ", ")) \\ G. C. Greubel, Jul 05 2018

G.f.: -x*(1 + 2*x + 237*x^2 - 8*x^3 + 1682*x^4 + 12*x^5 + 1682*x^6 - 8*x^7 + 237*x^8 + 2*x^9 + x^10)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009

a(n) = (1/24)*(n + n^2)*(6*(1 + (-1)^n) - (1 - 9*(-1)^n)*n + (1 - 9*(-1)^n)*n^2 + (4 - 6*(-1)^n)*n^3 + 2*n^4). - G. C. Greubel, Jul 05 2018

A145218 a(n) is the self-convolution series of the sum of 5th powers of the first n natural numbers.

Original entry on oeis.org

1, 64, 1510, 17600, 130835, 713216, 3098604, 11320320, 36074325, 102925120, 268038706, 646519744, 1460878055, 3120396800, 6346379480, 12363588096, 23184837609, 42023883840, 73881649150, 126362703040, 210792998011, 343726413824, 548946959300, 860095808000

Offset: 1

Abdullahi Umar, Oct 05 2008

			a(3) = 1510 because 1(3^5)+(2^5)(2^5)+(3^5)1= 1510

A. Umar, B. Yushau and B. M. Ghandi, (2006), "Patterns in convolution of two series", in Stewart, S. M., Olearski, J. E. and Thompson, D. (Eds), Proceedings of the Second Annual Conference for Middle East Teachers of Science, Mathematics and Computing (pp. 95-101). METSMaC: Abu Dhabi.
A. Umar, B. Yushau and B. M. Ghandi, "Convolution of two series", Australian Senior Maths. Journal, 21(2) (2007), 6-11.

T. D. Noe, Table of n, a(n) for n = 1..1000
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - _N. J. A. Sloane_, Mar 23 2014
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

a(n)=Conv(A000539, A000539)

[Binomial(n+2,3)*(n^8+8*n^7+29*n^6+62*n^5+86*n^4 +80*n^3+28*n^2-24*n+192)/462: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

f:=n->(n^11-22*n^5+231*n^3-210*n)/2772;
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 23 2014

CoefficientList[Series[(x^4 + 26 x^3 + 66 x^2 + 26 x + 1)^2/(x - 1)^12, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)

a(n) = C(n+2, 3)*(n^8 + 8*n^7 + 29*n^6 + 62*n^5 + 86*n^4 + 80*n^3 + 28*n^2 - 24*n + 192)/462.

G.f.: x*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1)^2/(x-1)^12. [Colin Barker, Jul 08 2012]

A257450 a(n) = 541(2^n - 1) - 5n^4 - 30n^3 - 130n^2 - 375*n.

Original entry on oeis.org

1, 33, 277, 1335, 4771, 14193, 37417, 90795, 207871, 456693, 974437, 2036655, 4195771, 8558073, 17337697, 34964595, 70300471, 141070653, 282727837, 566179575, 1133243251, 2267556033, 4536394777, 9074315835, 18150434671, 36302985093, 72608437717, 145219736895

Offset: 1

Luciano Ancora, Apr 23 2015

See the first comment of A257448.

			This sequence provides the antidiagonal sums of the array:
1, 32, 243, 1024,  3125,  7776, ...   A000584
1, 33, 276, 1300,  4425, 12201, ...   A000539
1, 34, 310, 1610,  6035, 18236, ...   A101092
1, 35, 345, 1955,  7990, 26226, ...   A101099
1, 36, 381, 2336, 10326, 36552, ...   A254644
1, 37, 418, 2754, 13080, 49632, ...   A254682
...
See also A254682 (Example field).

Index entries for linear recurrences with constant coefficients, signature (7,-20,30,-25,11,-2).

Cf. A000225, A000670, A050488, A208744, A257448, A257449.

[541*(2^n-1)-5*n^4-30*n^3-130*n^2-375*n: n in [1..30]]; // Vincenzo Librandi, Apr 24 2015

Table[541 (2^n - 1) - 5 n^4 - 30 n^3 - 130 n^2 - 375 n, {n, 30}]
LinearRecurrence[{7,-20,30,-25,11,-2},{1,33,277,1335,4771,14193},30] (* Harvey P. Dale, Dec 24 2018 *)

G.f.: x*(1+26*x+66*x^2+26*x^3+x^4)/(-1+x)^5*(-1+2*x).

a(n) = 7*a(n-1) -20*a(n-2) +30*a(n-3) -25*a(n-4) +11*a(n-5) -2*a(n-6) for n>6.

A140143 a(1)=1, a(n)=a(n-1)+n^0 if n odd, a(n)=a(n-1)+ n^5 if n is even.

Original entry on oeis.org

1, 33, 34, 1058, 1059, 8835, 8836, 41604, 41605, 141605, 141606, 390438, 390439, 928263, 928264, 1976840, 1976841, 3866409, 3866410, 7066410, 7066411, 12220043, 12220044, 20182668, 20182669, 32064045, 32064046

Offset: 1

Artur Jasinski, May 12 2008, corrected May 17 2008

nonn

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

a = {}; r = 0; s = 5; 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*)

a(n)=a(n-1)+6a(n-2)-6a(n-3)-15a(n-4)+15a(n-5)+20a(n-6)-20a(n-7)-15a(n-8)+15a(n-9)+6a(n-10)-6a(n-11)-a(n-12)+a(n-13). G.f.: x*(-1-32*x+5*x^2-832*x^3-10*x^4-2112*x^5+10*x^6-832*x^7-5*x^8-32*x^9+x^10 )/((1+x)^6*(x-1)^7). [From R. J. Mathar, Feb 22 2009]

cp's OEIS Frontend

A140157 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^0 if n is even.

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A140158 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^1 if n is even.

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A140159 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^2 if n is even.

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A140160 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^3 if n is even.

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A140161 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^5 if n is even.

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A140162 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^0 if n is even.

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Magma

Mathematica

PARI

Formula

A140163 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n if n is even.

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Magma

Maple

Mathematica

PARI

Formula

A145218 a(n) is the self-convolution series of the sum of 5th powers of the first n natural numbers.

A257450 a(n) = 541(2^n - 1) - 5n^4 - 30n^3 - 130n^2 - 375*n.