A000583 -id:A000583 - cp's OEIS frontend

A262955 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is a pentagonal number (A000326) or twice a pentagonal number.

1, 2, 3, 3, 2, 3, 3, 3, 2, 1, 4, 4, 3, 2, 3, 5, 4, 3, 3, 3, 4, 5, 5, 4, 3, 5, 6, 5, 5, 3, 6, 4, 4, 4, 1, 4, 5, 7, 6, 2, 6, 3, 3, 3, 5, 8, 5, 4, 3, 5, 4, 4, 4, 5, 5, 5, 7, 4, 3, 3, 7, 3, 3, 2, 2, 8, 5, 6, 2, 3, 5, 7, 6, 2, 1, 4, 4, 3, 6, 7, 6, 3, 5, 4, 3, 2, 6, 6, 6, 4, 6, 8

Offset: 1

Zhi-Wei Sun, Oct 05 2015

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 10, 35, 75, 134, 415, 515, 1465, 2365, 3515, 4140.

			a(1) = 1 since 1 = 0^4 + 1*2/2 + p_5(0), where p_5(n) denotes the pentagonal number n*(3*n-1)/2.
a(10) = 1 since 10 = 0^4 + 4*5/2 + p_5(0).
a(35) = 1 since 35 = 1^4 + 4*5/2 + 2*p_5(3).
a(75) = 1 since 75 = 2^4 + 5*6/2 + 2*p_5(4).
a(134) = 1 since 134 = 2^4 + 1*2/2 + p_5(9).
a(415) = 1 since 415 = 0^4 + 21*22/2 + 2*p_5(8).
a(515) = 1 since 515 = 0^4 + 6*7/2 + 2*p_5(13).
a(1465) = 1 since 1465 = 5^4 + 35*36/2 + p_5(12).
a(2365) = 1 since 2365 = 5^4 + 8*9/2 + 2*p_5(24).
a(3515) = 1 since 3515 = 5^4 + 51*52/2 + 2*p_5(23).
a(4140) = 1 since 4140 = 1^4 + 90*91/2 + 2*p_5(4).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.

Cf. A000217, A000326, A000583, A262941, A262944, A262945, A262954.

PenQ[n_]:=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0)
PQ[n_]:=PenQ[n]||PenQ[n/2]
Do[r=0;Do[If[PQ[n-x^4-y(y+1)/2],r=r+1],{x,0,n^(1/4)},{y,1,(Sqrt[8(n-x^4)+1]-1)/2}];Print[n," ",r];Continue,{n,1,100}]

A266153 Least positive integer y such that -n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.

Original entry on oeis.org

3, 3, 2, 6, 13, 2, 3, 5, 5, 3, 28, 4, 15, 4, 10, 33, 3, 7, 5, 238, 31, 3, 4, 5, 3, 11, 4, 5, 21, 11, 6, 4, 17, 11, 5, 98, 7, 4, 4, 5, 147, 19, 5, 4, 5, 6, 4, 29, 75, 1011, 7, 9, 7, 4, 8, 6, 59, 47, 4, 5, 71, 4, 17, 45, 13, 7, 18, 9, 175, 8

Offset: 1

Zhi-Wei Sun, Dec 22 2015

nonn

The conjecture in A266152 implies that a(n) > 0 for all n > 0.

It seems that a(n) < n*(n+4)/2 for all n > 1.

			a(1) = 3 since -1 = 1^4 - 3^3 + 5^2.
a(2) = 3 since -2 = 2^4 - 3^3 + 3^2.
a(11) = 28 since -11 = 5^4 - 28^3 + 146^2.
a(20) = 238 since -20 = 32^4 - 238^3 + 3526^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A000583, A262827, A266004, A266152.

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
Do[y=Floor[n^(1/3)]+1;Label[bb];Do[If[SQ[-n+y^3-x^4],Print[n," ",y];Goto[aa]],{x,1,(-n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Continue,{n,1,70}]

A038992 Number of sublattices of index n in generic 5-dimensional lattice.

Original entry on oeis.org

1, 31, 121, 651, 781, 3751, 2801, 11811, 11011, 24211, 16105, 78771, 30941, 86831, 94501, 200787, 88741, 341341, 137561, 508431, 338921, 499255, 292561, 1429131, 508431, 959171, 925771, 1823451, 732541, 2929531, 954305, 3309747, 1948705, 2750971, 2187581, 7168161, 1926221

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
Index entries for sequences related to sublattices.

Column 5 of A160870.

Cf. A001001, A038991, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

a[n_] := DivisorSum[n, #*DivisorSum[#, #*DivisorSum[#, #*DivisorSum[#, # &] &] &] &]; Array[a, 50] (* Jean-François Alcover, Dec 02 2015, after Joerg Arndt *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 4}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z * sumdiv(z,w, w ) ) ) ); /* Joerg Arndt, Oct 07 2012 */

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=5.
Multiplicative with a(p^e) = Product_{k=1..4} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3)*zeta(s-4). Dirichlet convolution of A038991 with A000583. - R. J. Mathar, Mar 31 2011
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^6*zeta(3)*zeta(5)/2700 = 0.443822... . - Amiram Eldar, Oct 19 2022

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Joerg Arndt, Oct 07 2012

A219086 a(n) = floor((n + 1/2)^4).

Original entry on oeis.org

0, 5, 39, 150, 410, 915, 1785, 3164, 5220, 8145, 12155, 17490, 24414, 33215, 44205, 57720, 74120, 93789, 117135, 144590, 176610, 213675, 256289, 304980, 360300, 422825, 493155, 571914, 659750, 757335, 865365, 984560, 1115664

Offset: 0

Clark Kimberling, Jan 01 2013

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/4 and { } = fractional part.  Equivalently, the jump sequence of f(x) = x^(1/4), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p).  For details and a guide to related sequences, see A219085.
-4*a(n) gives the real part of (n+n*i)*((n+1)+n*i)*(n+(n+1)*i)*((n+1)+(n+1)*i). The imaginary part is always zero. - Jon Perry, Feb 05 2014
Numbers k such that 16*k+1 is a fourth power. - Bruno Berselli, May 29 2018
The row sums of "Floyd's Triangle", which is a triangular array of natural numbers beginning with the number 1, produce the sequence A006003. A006003 can be bisected to get the Rhombic Dodecahedron Sequence A005917, whose n-th partial sum is n^4, and A317297, whose n-th partial sum is a(n). Interleave n^4 or A000583 back with {a(n)} to get A011863, whose first differences are A019298. Finally, A011863(n)-A011863(n-2) = A006003(n-1). - Bruce J. Nicholson, Dec 22 2019

			0^(1/4) = 0.000...; 1^(1/4) = 1.000...
5^(1/4) = 1.495...; 6^(1/4) = 1.565...
39^(1/4) = 2.499...; 40^(1/4) = 2.514...

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

Cf. A123865, A219085, A234459, A000217, A001844.

Cf. A317297, A006003, A005917, A000583, A011863, A019298.

A219086:=n->floor((n + (1/2))^4); seq(A219086(n), n=0..50); # Wesley Ivan Hurt, Apr 05 2014

Table[Floor[(n + 1/2)^4], {n, 0, 100}]
LinearRecurrence[{5,-10,10,-5,1},{0,5,39,150,410},40] (* Harvey P. Dale, Jan 15 2023 *)

a(n)=floor((n + 1/2)^4) \\ Charles R Greathouse IV, Apr 15 2014

G.f.: (5*x^3 + 14*x^2 + 5*x)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (2*n^4 + 4*n^3 + 3*n^2 + n)/2. - J. M. Bergot, Apr 05 2014
a(n) = Sum_{i=0..n} i*(4*i^2 + 1) = n*(n + 1)*(2*n^2 + 2*n + 1)/2. - Bruno Berselli, Feb 09 2017
a(n) = lcm((2*n + 1)^2 - 1, (2*n + 1)^2 + 1)/8 for n>=1. - Lechoslaw Ratajczak, Mar 26 2017
a(n) = A000217(n) * A001844(n). - Bruce J. Nicholson, May 14 2017
E.g.f.: (1/2)*exp(x)*x*(10 + 29*x + 16*x^2 + 2*x^3). - Stefano Spezia, Dec 27 2019
a(n) = ((2*n+1)^4 - 1)/16. - Jianing Song, Jan 03 2023
Sum_{n>=1} 1/a(n) = 6 - 2*Pi*tanh(Pi/2). - Amiram Eldar, Jan 08 2023

A004831 Numbers that are the sum of at most 2 nonzero 4th powers.

Original entry on oeis.org

0, 1, 2, 16, 17, 32, 81, 82, 97, 162, 256, 257, 272, 337, 512, 625, 626, 641, 706, 881, 1250, 1296, 1297, 1312, 1377, 1552, 1921, 2401, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4096, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6561, 6562, 6577, 6642

Offset: 1

N. J. A. Sloane

nonn

Apart from 0, 1, 2, there are no three consecutive terms up to 10^16. The first two consecutive terms not of the form n^4, n^4+1 are 3502321 = 25^4 + 42^4, 3502322 = 17^4 + 43^4. - Charles R Greathouse IV, Oct 17 2017

T. D. Noe, Table of n, a(n) for n = 1..1000

Subsequences include A003336, A000583 and A002645.

a004831 n = a004831_list !! (n-1)
a004831_list = [x ^ 4 + y ^ 4 | x <- [0..], y <- [0..x]]
-- Reinhard Zumkeller, Jul 15 2013

Reap[For[n = 0, n < 10000, n++, If[MatchQ[ PowersRepresentations[n, 2, 4], {{, },_}], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 30 2017 *)

is(n)=#thue(thueinit(z^4+1),n) \\ Ralf Stephan, Oct 18 2013

list(lim)=my(v=List(),t); for(m=0,sqrtnint(lim\=1,4), for(n=0, min(sqrtnint(lim-m^4,4),m), listput(v,n^4+m^4))); Set(v) \\ Charles R Greathouse IV, Sep 28 2015

Call f(x) the number of terms if this sequence up to x. Then x^(7/16) << f(x) << x^(1/2); in other words, n^2 << a(n) << n^(16/7). The upper bound becomes O(n^2) if A230562 is finite. - Charles R Greathouse IV, Jul 12 2024

A007204 Crystal ball sequence for D_4 lattice.

Original entry on oeis.org

1, 25, 169, 625, 1681, 3721, 7225, 12769, 21025, 32761, 48841, 70225, 97969, 133225, 177241, 231361, 297025, 375769, 469225, 579121, 707281, 855625, 1026169, 1221025, 1442401, 1692601, 1974025, 2289169, 2640625, 3031081, 3463321, 3940225, 4464769, 5040025

Offset: 0

N. J. A. Sloane and J. H. Conway, Apr 28 1994

Equals binomial transform of [1, 24, 120, 192, 96, 0, 0, 0, ...]. - Gary W. Adamson, Aug 13 2009
Hypotenuse of Pythagorean triangles with hypotenuse a square: A057769(n)^2 + A069074(n-1)^2 = a(n)^2. - Martin Renner, Nov 12 2011
Numbers n such that n*x^4 + x^2 + 1 is reducible. - Arkadiusz Wesolowski, Nov 04 2013

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for crystal ball sequences
Index entries for sequences related to D_4 lattice
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016754, A057769, A060300, A069074.

Cf. A000290, A000583, A001844, A005563, A099761.

[(2*n^2+2*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Nov 18 2016

A007204:=n->(2*n^2+2*n+1)^2; seq(A007204(n), n=0..30);

Table[(2n^2+2n+1)^2,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,25,169,625,1681},40] (* Harvey P. Dale, Mar 03 2013 *)

a(n)=(2*n^2+2*n+1)^2 \\ Charles R Greathouse IV, Feb 08 2017

G.f.: (1 + 54*x^2 + 20*x + 20*x^3 + x^4)/(1-x)^5.
a(0)=1, a(1)=25, a(2)=169, a(3)=625, a(4)=1681, a(n)=5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Mar 03 2013
Sum_{n>=0} 1/a(n) = Pi*(sinh(Pi) - Pi)/(2*(cosh(Pi) + 1)) = 1.0487582722070177... - Ilya Gutkovskiy, Nov 18 2016
a(n) = A016754(n) + A060300(n). - Bruce J. Nicholson, Apr 14 2017
a(n) = A001844(n)^2 = (2*n^2+2*n+1)^2. - Bruce J. Nicholson, May 15 2017
a(n) = A000583(n+1) + A099761(n) + 2*A005563(n-1)*A000290(n). - Charlie Marion, Jan 14 2021
E.g.f.: exp(x)*(1 + 24*x + 60*x^2 + 32*x^3 + 4*x^4). - Stefano Spezia, Jun 06 2021

More terms from Harvey P. Dale, Mar 03 2013

A016780 a(n) = (3*n+1)^4.

Original entry on oeis.org

1, 256, 2401, 10000, 28561, 65536, 130321, 234256, 390625, 614656, 923521, 1336336, 1874161, 2560000, 3418801, 4477456, 5764801, 7311616, 9150625, 11316496, 13845841, 16777216, 20151121, 24010000, 28398241, 33362176, 38950081, 45212176, 52200625, 59969536, 68574961

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. A000583 (n^4), A016777 (3n+1), A016778, A016779, A016781.

[(3*n+1)^4: n in [0..30]]; // Vincenzo Librandi, Sep 21 2011

(3*Range[0,30]+1)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,256,2401,10000,28561},30] (* Harvey P. Dale, Oct 21 2015 *)

From Harvey P. Dale, Oct 21 2015: (Start)
a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5).
G.f.: -((1+251*x+1131*x^2+545*x^3+16*x^4)/(-1+x)^5). (End)
a(n) = A000583(A016777(n)). - Michel Marcus, Nov 06 2015
E.g.f.: exp(x)*(1+255*x+945*x^2+594*x^3+81*x^4). - Wolfdieter Lang, Apr 02 2017
Sum_{n>=0} 1/a(n) = PolyGamma(3, 1/3)/486. - Amiram Eldar, Mar 29 2022

A085537 a(n) = n^4 - n^3.

Original entry on oeis.org

0, 0, 8, 54, 192, 500, 1080, 2058, 3584, 5832, 9000, 13310, 19008, 26364, 35672, 47250, 61440, 78608, 99144, 123462, 152000, 185220, 223608, 267674, 317952, 375000, 439400, 511758, 592704, 682892, 783000, 893730, 1015808, 1149984, 1297032, 1457750, 1632960

Offset: 0

N. J. A. Sloane, Jul 05 2003

For n>=1, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for a fixed x in {1,2,3,4} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Let K_n denote the complete graph on n (n>1) vertices. The sequence corresponds to the Wiener index of K_n X K_n (Cartesian product of K_n with itself). - K.V.Iyer, Mar 12 2009
Lewis proved that the order of a solvable nonabelian finite group |G| is less than or equal to e^4 - e^3, where when d is an irreducible character degree of G, then there is a positive integer e such that |G| = d(d+e). - Jonathan Vos Post, Jun 21 2012

Michael B. Porter, Table of n, a(n) for n = 0..100000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Mark L. Lewis, Bounding group orders by large character degrees: A question of Snyder, arXiv:1206.4334 [math.GR], Jun 19 2012.
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A diagonal of A228273.
Cf. A000578, A000583, A092364, A146489.
Cf. A085540 (same sequence with initial 0 dropped).

Table[(n - 1) n^3, {n, 0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 54, 192, 500}, {0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[2 x^2 (4 + 7 x + x^2)/(1 - x)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

PARI
```
A085537(n) = n^4-n^3
```

From R. J. Mathar, Sep 12 2008: (Start)
a(n) = A085540(n-1).
G.f.: 2*x^2*(4 + 7*x + x^2)/(1-x)^5. (End)
a(n) = A000583(n) - A000578(n). - Omar E. Pol, Jun 23 2012
Sum_{n>=2} 1/a(n) = 3 - zeta(2) - zeta(3) = A152419. - Daniel Suteu, Feb 06 2017
a(n) = 2*A092364(n+1). - Bruno Berselli, Sep 08 2017
Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. - Amiram Eldar, Jul 05 2020
E.g.f.: exp(x)*x^2*(4 + 5*x + x^2). - Stefano Spezia, Jul 06 2021
Product_{n>=2} (1 - 1/a(n)) = A146489. - Amiram Eldar, Nov 22 2022

A100019 a(n) = n^4 + n^3 + n^2.

Original entry on oeis.org

0, 3, 28, 117, 336, 775, 1548, 2793, 4672, 7371, 11100, 16093, 22608, 30927, 41356, 54225, 69888, 88723, 111132, 137541, 168400, 204183, 245388, 292537, 346176, 406875, 475228, 551853, 637392, 732511, 837900, 954273, 1082368, 1222947

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 19 2004

a(n) are the numbers m such that: j^2 = j + m + sqrt(j*m) with corresponding numbers j given by A002061(n+1), and with sqrt(j*m) = A027444(n) = n* A002061(n+1). - Richard R. Forberg, Sep 03 2013.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000583, A002523, A091940, A071253, A059826, A027445, A053699.

[n^4+n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

A100019:=n->n^4+n^3+n^2: seq(A100019(n), n=0..50); # Wesley Ivan Hurt, Apr 14 2017

f[n_]:=n^4+n^3+n^2;Array[f,50,0] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)

a(n) = n^4 + n^3 + n^2 \\ Indranil Ghosh, Apr 15 2017

From Indranil Ghosh, Apr 15 2017: (Start)
G.f.: -x(3 + 13x + 7x^2 + x^3)/(x - 1)^5
E.g.f.: exp(x)*x*(3 + 11x + 7x^2 + x^3)
(End)

A141046 a(n) = 4*n^4.

Original entry on oeis.org

0, 4, 64, 324, 1024, 2500, 5184, 9604, 16384, 26244, 40000, 58564, 82944, 114244, 153664, 202500, 262144, 334084, 419904, 521284, 640000, 777924, 937024, 1119364, 1327104, 1562500, 1827904, 2125764, 2458624, 2829124, 3240000, 3694084, 4194304, 4743684, 5345344

Offset: 0

Fredrik Johansson, Jul 31 2008

Nonnegative integers a(n) such that (-a(n))^(1/4) is a Gaussian integer, since (n + n*i)^4 = -4*n^4
For n > 1, a(n) + k^4 is not prime for any k. - Derek Orr, May 31 2014
Suppose the vertices of a triangle are (T(n), T(n+j)), (T(n+2*j), T(n+3*j)) and (T(n+4*j), T(n+5*j)) where T(n) is the n-th triangular number. Then the area of this triangle will be a(j). - Charlie Marion, Mar 06 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A000583, A001105, A005843, A008586.

a141046 = (* 4) . (^ 4)  -- Reinhard Zumkeller, Jan 25 2012

Mathematica
```
Table[4 n^4, {n, 0, 20}]
```

a(n)=4*n^4 \\ Charles R Greathouse IV, Jan 26 2012

a(n) = 4*n^4.
a(n) = A008586(A000583(n)) = A000290(A005843(A000290(n))). - Reinhard Zumkeller, Jan 25 2012
G.f.: 4*x*(1 + x)*(1 + 10*x + x^2)/(1 - x)^5. - Chai Wah Wu, Jun 22 2016
From G. C. Greubel, Jun 22 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
E.g.f.: 4*x*(1 + 7*x + 6*x^2 + x^3)*exp(x). (End)
a(n) = A001105(n)^2. - Bruce J. Nicholson, Apr 03 2017
From Amiram Eldar, Jan 29 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^4/360.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/2880.
Product_{n>=1} (1 + 1/a(n)) = 2*cosh(Pi/2)^2/Pi^2.
Product_{n>=1} (1 - 1/a(n)) = 2*sin(Pi/sqrt(2))*sinh(Pi/sqrt(2))/Pi^2. (End)

cp's OEIS Frontend

A262955 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is a pentagonal number (A000326) or twice a pentagonal number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A266153 Least positive integer y such that -n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A038992 Number of sublattices of index n in generic 5-dimensional lattice.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A219086 a(n) = floor((n + 1/2)^4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A004831 Numbers that are the sum of at most 2 nonzero 4th powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

A007204 Crystal ball sequence for D_4 lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A016780 a(n) = (3*n+1)^4.

Views

Author

Keywords

Links

Crossrefs