A008514 -id:A008514 - cp's OEIS frontend

A002817 Doubly triangular numbers: a(n) = n(n+1)(n^2+n+2)/8.

0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310, 198765, 222111, 247456, 274911, 304590

Offset: 0

N. J. A. Sloane

Number of inequivalent ways to color vertices of a square using <= n colors, allowing rotations and reflections. Group is dihedral group D_8 of order 8 with cycle index (1/8)*(x1^4 + 2*x4 + 3*x2^2 + 2*x1^2*x2); setting all x_i = n gives the formula a(n) = (1/8)*(n^4 + 2*n + 3*n^2 + 2*n^3).
Number of semi-magic 3 X 3 squares with a line sum of n-1. That is, 3 X 3 matrices of nonnegative integers such that row sums and column sums are all equal to n-1. - [Gupta, 1968, page 653; Bell, 1970, page 279]. - Peter Bertok (peter(AT)bertok.com), Jan 12 2002. See A005045 for another version.
Also the coefficient h_2 of x^{n-3} in the shelling polynomial h(x)=h_0*x^n-1 + h_1*x^n-2 + h_2*x^n-3 + ... + h_n-1 for the independence complex of the cycle matroid of the complete graph K_n on n vertices (n>=2) - Woong Kook (andrewk(AT)math.uri.edu), Nov 01 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-4) is equal to the number of 5-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Starting with offset 1 = binomial transform of [1, 5, 10, 9, 3, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009
Starting with "1" = row sums of triangle A178238. - Gary W. Adamson, May 23 2010
The equation n*(n+1)*(n^2 + n + 2)/8 may be arrived at by solving for x in the following equality: (n^2+n)/2 = (sqrt(8x+1)-1)/2. - William A. Tedeschi, Aug 18 2010
Partial sums of A006003. - Jeremy Gardiner, Jun 23 2013
Doubly triangular numbers are revealed in the sums of row sums of Floyd's triangle.
1, 1+5, 1+5+15, ...
             1
          2     3
       4     5     6
    7     8     9     10
11    12    13    14    15
- Tony Foster III, Nov 14 2015
From Jaroslav Krizek, Mar 04 2017: (Start)
For n>=1; a(n) = sum of the different sums of elements of all the nonempty subsets of the sets of numbers from 1 to n.
Example: for n = 6; nonempty subsets of the set of numbers from 1 to 3: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}; sums of elements of these subsets: 1, 2, 3, 3, 4, 5, 6; different sums of elements of these subsets: 1, 2, 3, 4, 5, 6; a(3) = (1+2+3+4+5+6) = 21, ... (End)
a(n) is also the number of 4-cycles in the (n+4)-path complement graph. - Eric W. Weisstein, Apr 11 2018

			G.f. = x + 6*x^2 + 21*x^3 + 55*x^4 + 120*x^5 + 231*x^6 + 406*x^7 + 666*x^8 + ...

A. Björner, The homology and shellability of matroids and geometric lattices, in Matroid Applications (ed. N. White), Encyclopedia of Mathematics and Its Applications, 40, Cambridge Univ. Press 1992.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(3,r).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, p. 292.

T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 0..10000 (first 1000 terms computed by T. D. Noe)
G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions and magic squares, p. 37.
Weymar Astaiza, Alexander J. Barrios, Henry Chimal-Dzul, Stephan Ramon Garcia, Jaaziel de la Luz, Victor H. Moll, Yunied Puig, and Diego Villamizar, Symmetric tensor powers of graphs, arXiv:2309.13741 [math.CO], 2023. See p. 12.
Matthias Beck, The number of "magic" squares and hypercubes, arXiv:math/0201013 [math.CO], 2002-2005.
A. G. Bell, Partitioning integers in n dimensions, The Computer Journal, 13 (1970), 278-283.
Miklos Bona, A New Proof of the Formula for the Number of 3 X 3 Magic Squares, Mathematics Magazine, Vol. 70, No. 3 (Jun., 1997), pp. 201-203.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33(4) (1966), pp. 771-782.
Brian Conrey and Alex Gamburd, Pseudomoments of the Riemann zeta-function and pseudomagic squares, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278. See H4 on p. 269.
P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R2.
Robert W. Donley, Partitions for semi-magic squares of size three, arXiv:1911.00977 [math.CO], 2019.
I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
Hansraj Gupta, Enumeration of symmetric matrices, Duke Math. J. 35 (3), 653-659, (September 1968).
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Milan Janjic, Two Enumerative Functions
Neven Juric, Illustration of the 55 3 X 3 matrices
Michal Opler, Pavel Valtr, and Tung Anh Vu, On the Arrangement of Hyperplanes Determined by n Points, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Henry Warburton, On Self-Repeating Series, Transactions of the Cambridge Philosophical Society, Vol. 9, 471-486, 1856.
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Path Complement Graph
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A006003 (first differences), A165211 (mod 2).
Multiple triangular: A000217, A064322, A066370.
Cf. A001496, A001621, A178238, A005045, A002721.
Cf. A006528 (square colorings).
Cf. A236770 (see crossrefs).
Cf. A016754, A007204.
Row n=3 of A257493 and row n=2 of A331436 and A343097.
Cf. A000332.
Cf. A000292 (3-cycle count of \bar P_{n+4}), A060446 (5-cycle count of \bar P_{n+3}), A302695 (6-cycle count of \bar P_{n+5}).

Maple
```
A002817 := n->n*(n+1)*(n^2+n+2)/8;
```

a[ n_] := n (n + 1) (n^2 + n + 2) / 8; (* Michael Somos, Jul 24 2002 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,1,6,21,55},40] (* Harvey P. Dale, Jul 18 2011 *)
nn=50;Join[{0},With[{c=(n(n+1))/2},Flatten[Table[Take[Accumulate[Range[ (nn(nn+1))/2]], {c,c}],{n,nn}]]]] (* Harvey P. Dale, Mar 19 2013 *)

{a(n) = n * (n+1) * (n^2 + n + 2) / 8}; /* Michael Somos, Jul 24 2002 */

concat(0, Vec(x*(1+x+x^2)/(1-x)^5 + O(x^50))) \\ Altug Alkan, Nov 15 2015

def A002817(n): return (m:=n*(n+1))*(m+2)>>3 # Chai Wah Wu, Aug 30 2024

a(n) = 3*binomial(n+2, 4) + binomial(n+1, 2).
G.f.: x*(1 + x + x^2)/(1-x)^5. - Simon Plouffe (in his 1992 dissertation); edited by N. J. A. Sloane, May 13 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - Warut Roonguthai, Dec 13 1999
a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + a(n-5) = A000217(A000217(n)). - Ant King, Nov 18 2010
a(n) = Sum(Sum(1 + Sum(3*n))). - Xavier Acloque, Jan 21 2003
a(n) = A000332(n+1) + A000332(n+2) + A000332(n+3), with A000332(n) = binomial(n, 4). - Mitch Harris, Oct 17 2006 and Bruce J. Nicholson, Oct 22 2017
a(n) = Sum_{i=1..C(n,2)} i = C(C(n,2) + 1, 2) = A000217(A000217(n+1)). - Enrique Pérez Herrero, Jun 11 2012
Euler transform of length 3 sequence [6, 0, -1]. - Michael Somos, Nov 19 2015
E.g.f.: x*(8 + 16*x + 8*x^2 + x^3)*exp(x)/8. - Ilya Gutkovskiy, Apr 26 2016
Sum_{n>=1} 1/a(n) = 6 - 4*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = 1.25269064911978447... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n)*A000124(n)/2.
a(n) = ((n-1)^4 + 3*(n-1)^3 + 2*(n-1)^2 + 2*n))/8. - Bruce J. Nicholson, Apr 05 2017
a(n) = (A016754(n)+ A007204(n)- 2) / 32. - Bruce J. Nicholson, Apr 14 2017
a(n) = a(-1-n) for all n in Z. - Michael Somos, Apr 17 2017
a(n) = T(T(n)) where T are the triangular numbers A000217. - Albert Renshaw, Jan 05 2020
a(n) = 2*n^2 - n + 6*binomial(n, 3) + 3*binomial(n, 4). - Ryan Jean, Mar 20 2021
a(n) = (A008514(n) - 1)/16. - Charlie Marion, Dec 20 2024

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

A152913 Primes of the form n^4 + (n+1)^4.

Original entry on oeis.org

17, 97, 337, 881, 3697, 10657, 16561, 49297, 66977, 89041, 149057, 847601, 988417, 1146097, 1972097, 2522257, 2836961, 3553777, 3959297, 4398577, 5385761, 7166897, 11073217, 17653681, 32530177, 41532497, 44048497, 55272097, 61627201

Offset: 1

Vincenzo Librandi, Dec 15 2008

Also primes in A008514.

Sequence is disjoint to A005385: If n^4 + (n+1)^4 is a prime p, then (p-1)/2 = n^4 + 2*n^3 + 3*n^2 + 2*n. (p-1)/2 = 8 for n = 1 and (p-1)/2 is divisible by n for n > 1. In each case, (p-1)/2 is not prime.

			For n=3, n^4 + (n+1)^4 = 337 is prime and (337-1)/2 = 168 = 3*56 is not prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A155211, A008514.

[ a: n in [1..80] | IsPrime(a) where a is n^4+(n+1)^4 ];

f[n_]:=n^4+(n+1)^4;lst={};Do[a=f[n];If[PrimeQ[a],AppendTo[lst,a]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 30 2009 *)
Select[Table[n^4+(n+1)^4,{n,0,700}],PrimeQ]
Select[Total/@Partition[Range[100]^4,2,1],PrimeQ] (* Harvey P. Dale, Sep 29 2023 *)

Edited and extended by Klaus Brockhaus, Dec 21 2008

A160827 a(n) = 3n^4 + 12n^3 + 30n^2 + 36n + 17.

Original entry on oeis.org

17, 98, 353, 962, 2177, 4322, 7793, 13058, 20657, 31202, 45377, 63938, 87713, 117602, 154577, 199682, 254033, 318818, 395297, 484802, 588737, 708578, 845873, 1002242, 1179377, 1379042, 1603073, 1853378, 2131937, 2440802, 2782097, 3158018, 3570833

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), May 27 2009

Sums of 3 consecutive fourth powers.

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. A008514.

[3*n^4+12*n^3+30*n^2+36*n+17: n in [0..40]]; // Vincenzo Librandi, Aug 27 2011

A000583 := proc(n) n^4 ; end: A160827 := proc(n) add(A000583(i),i=n..n+2) ; end: seq(A160827(n),n=0..40) ; # R. J. Mathar, May 29 2009

Total/@Partition[Range[0,40]^4,3,1] (* or *) LinearRecurrence[{5,-10,10,-5,1},{17,98,353,962,2177},40] (* Harvey P. Dale, Nov 16 2014 *)
CoefficientList[Series[(2*x^4+7*x^3+33*x^2+13*x+17)/(1-x)^5, {x, 0, 50}], x] (* G. C. Greubel, Apr 30 2018 *)

a(n)=3*n^4+12*n^3+30*n^2+36*n+17 \\ Charles R Greathouse IV, Oct 16 2015

A160827_list, m = [], [72, -36, 30, 15, 17]
for _ in range(10**2):
    A160827_list.append(m[-1])
    for i in range(4):
        m[i+1] += m[i] # Chai Wah Wu, Jan 23 2016

a(n) = Sum_{i=0..2} A000583(n+i) = Sum_{j=n..n+2} j^4.
G.f.: (2*x^4+7*x^3+33*x^2+13*x+17)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009
E.g.f.: (17 + 81*x + 87*x^2 + 30*x^3 + 3*x^4)*exp(x). - G. C. Greubel, Apr 30 2018

Edited and corrected by R. J. Mathar, May 29 2009

A237516 Pyramidal centered square numbers.

Original entry on oeis.org

1, 15, 91, 325, 861, 1891, 3655, 6441, 10585, 16471, 24531, 35245, 49141, 66795, 88831, 115921, 148785, 188191, 234955, 289941, 354061, 428275, 513591, 611065, 721801, 846951, 987715, 1145341, 1321125, 1516411, 1732591, 1971105, 2233441, 2521135, 2835771, 3178981, 3552445, 3957891, 4397095, 4871881

Offset: 1

Kival Ngaokrajang, Feb 08 2014

a(n) is sum of natural numbers filled in order-n diamond.
First differences give A173962.
The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A002378(n). - Miguel-Ángel Pérez García-Ortega, Jun 05 2025

Colin Barker, Table of n, a(n) for n = 1..1000
José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.
Kival Ngaokrajang, Illustration for n = 1..6.
Eric Weisstein's World of Mathematics, Diamond.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A001844, A002061, A173962.

Cf. A002378, A008514, A384288, A384566.

Table[Sum[i, {i, 2n(n + 1) + 1}], {n, 0, 29}] (* Alonso del Arte, Feb 09 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{1,15,91,325,861},60] (* Harvey P. Dale, Apr 21 2018 *)
a=Table[(n(n+1)),{n,0,29}];Apply[Join,Map[{(#+1)(2#+1)}&,a]] (* Miguel-Ángel Pérez García-Ortega, Jun 05 2025 *)

Vec(-x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jan 17 2015

a(n) = 2*n^4 - 4*n^3 + 5*n^2 - 3*n + 1.
a(n) = Sum_{i = 1..(2*n*(n + 1) + 1)} i.
From Colin Barker, Jan 17 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.: -x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5. (End)
a(n) = A000217(A001844(n-1)). - Ivan N. Ianakiev, Jun 14 2015
a(n) = A002061(n)*A001844(n-1). - Bruce J. Nicholson, May 14 2017
a(n) = (A002378(n)+1)*(2*A002378(n)+1). - Miguel-Ángel Pérez García-Ortega, Jun 05 2025
E.g.f.: -1 + exp(x)*(1 + 7*x^2 + 8*x^3 + 2*x^4). - Elmo R. Oliveira, Aug 22 2025

A384288 Length of the long leg in the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

12, 84, 312, 840, 1860, 3612, 6384, 10512, 16380, 24420, 35112, 48984, 66612, 88620, 115680, 148512, 187884, 234612, 289560, 353640, 427812, 513084, 610512, 721200, 846300, 987012, 1144584, 1320312, 1515540, 1731660, 1970112, 2232384, 2520012, 2834580

Offset: 1

Miguel-Ángel Pérez García-Ortega, May 31 2025

			Triangles begin:
  n=1:      5,   12,   13;
  n=2:     13,   84,   85;
  n=3:     25,  312,  313;
  ...
This sequence gives the middle column.

José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A002378 (inradius), A001844 (short leg), A008514 (sum of the legs), A237516 (semiperimeter), A384566 (area).

a(n) = 2 * A002378(n) * (A002378(n) + 1).

A384566 Area of the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

0, 30, 546, 3900, 17220, 56730, 153510, 360696, 762120, 1482390, 2698410, 4652340, 7665996, 12156690, 18654510, 27821040, 40469520, 57586446, 80354610, 110177580, 148705620, 197863050, 259877046, 337307880, 433080600, 550518150, 693375930, 865877796, 1072753500, 1319277570, 1611309630

Offset: 0

Miguel-Ángel Pérez García-Ortega, Jun 03 2025

a(n) is multiple of 6 for all n.

			For n=1, the short leg is A384288(1,1) = 5 and the long leg is A384288(1,2) = 12 so the area is then a(1) = (5 * 12 )/2 = 30.

José Miguel Blanco Casado and Miguel Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A002378, A384288, A008514, A237516.

a=Table[(n(n+1)),{n,0,30}];Apply[Join,Map[{#(#+1)(2#+1)}&,a]]

a(n) = (A384288(n,1) * A384288(n,2))/2.

a(n) = A002378(n)*(A002378(n) + 1)*(2*A002378(n) + 1).

A210694 T(n,k)=Number of (n+1)X(n+1) -k..k symmetric matrices with every 2X2 subblock having sum zero.

Original entry on oeis.org

5, 13, 9, 25, 35, 17, 41, 91, 97, 33, 61, 189, 337, 275, 65, 85, 341, 881, 1267, 793, 129, 113, 559, 1921, 4149, 4825, 2315, 257, 145, 855, 3697, 10901, 19721, 18571, 6817, 513, 181, 1241, 6497, 24583, 62281, 94509, 72097, 20195, 1025, 221, 1729, 10657, 49575

Offset: 1

R. H. Hardin, with R. J. Mathar in the Sequence Fans Mailing List, Mar 30 2012

Table starts
...5....13.....25......41.......61.......85.......113.......145........181
...9....35.....91.....189......341......559.......855......1241.......1729
..17....97....337.....881.....1921.....3697......6497.....10657......16561
..33...275...1267....4149....10901....24583.....49575.....91817.....159049
..65...793...4825...19721....62281...164305....379793....793585....1531441
.129..2315..18571...94509...358061..1103479...2920695...6880121...14782969
.257..6817..72097..456161..2070241..7444417..22542017..59823937..143046721
.513.20195.281827.2215269.12030821.50431303.174571335.521638217.1387420489
Solutions are determined by the diagonal, extended with x(i,j) = (x(i,i)+x(j,j))/2 * (-1)^(i-j)

			Some solutions for n=3 k=4
.-2..1.-3..0....0.-1..0..1....4..0..1.-1....2.-1.-1.-2....3.-2..1..0
..1..0..2..1...-1..2.-1..0....0.-4..3.-3...-1..0..2..1...-2..1..0.-1
.-3..2.-4..1....0.-1..0..1....1..3.-2..2...-1..2.-4..1....1..0.-1..2
..0..1..1..2....1..0..1.-2...-1.-3..2.-2...-2..1..1..2....0.-1..2.-3

R. H. Hardin, Table of n, a(n) for n = 1..10000

cf. A179927
Column 1 is A000051(n+1)
Column 2 is A007689(n+1)
Column 3 is A074605(n+1)
Column 4 is A074611(n+1)
Column 5 is A074615(n+1)
Column 6 is A074619(n+1)
Column 7 is A074622(n+1)
Column 8 is A074624(n+1)
Row 1 is A001844
Row 2 is A005898
Row 3 is A008514
Row 4 is A008515
Row 5 is A008516
Row 6 is A036085
Row 7 is A036086
Row 8 is A036087

T(n,k)=k^(n+1)+(k+1)^(n+1)

A267691 a(n) = (n + 1)(6n^4 - 21n^3 + 31n^2 - 31*n + 30)/30.

Original entry on oeis.org

1, 1, 2, 18, 99, 355, 980, 2276, 4677, 8773, 15334, 25334, 39975, 60711, 89272, 127688, 178313, 243849, 327370, 432346, 562667, 722667, 917148, 1151404, 1431245, 1763021, 2153646, 2610622, 3142063, 3756719, 4464000, 5274000, 6197521, 7246097, 8432018, 9768354

Offset: 0

Ilya Gutkovskiy, Jan 19 2016

			a(0) = 1,
a(1) = 1 + 0^4 = 1,
a(2) = 1 + 1^4 = 2,
a(3) = 2 + 2^4 = 18,
a(4) = 18+ 3^4 = 99, etc.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Essentially the same as A000538.
Cf. A000124, A000583, A008514, A056520, A154323, A263689.
Cf. A013662 (zeta(4)).

[(n+1)*(6*n^4-21*n^3+31*n^2-31*n+30)/30: n in [0..35]]; // Vincenzo Librandi, Jan 20 2016

Table[(n + 1) (6 n^4 - 21 n^3 + 31 n^2 - 31 n + 30)/30, {n, 0, 30}]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 1, 2, 18, 99, 355}, 40] (* Vincenzo Librandi, Jan 20 2016 *)

a(n)=(n+1)*(6*n^4-21*n^3+31*n^2-31*n+30)/30 \\ Charles R Greathouse IV, Jan 19 2016

Vec((1-5*x+11*x^2+x^3+16*x^4)/(x-1)^6 + O(x^100)) \\ Altug Alkan, Jan 19 2016

G.f.: (1 - 5*x + 11*x^2 + x^3 + 16*x^4)/(1 - x)^6.
a(n + 1) = a(n) + n^4.
a(n + 1) = A000538(n) + 1.
a(n + 2) - a(n) = A008514(n).
Sum_{n>=0} 1/a(n) = 2.570450909491318975...
Sum_{n>=1} 1/(a(n + 1) - a(n)) = zeta(4) = Pi^4/90.

cp's OEIS Frontend

A334121 a(n) is the multiplicative inverse of A008514(n) modulo A008514(n+1).

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A334137 a(n) is the multiplicative inverse of A008514(n+1) modulo A008514(n).

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A002817 Doubly triangular numbers: a(n) = n*(n+1)*(n^2+n+2)/8.

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A152913 Primes of the form n^4 + (n+1)^4.

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A160827 a(n) = 3*n^4 + 12*n^3 + 30*n^2 + 36*n + 17.

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A237516 Pyramidal centered square numbers.

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A384288 Length of the long leg in the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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A002817 Doubly triangular numbers: a(n) = n(n+1)(n^2+n+2)/8.

A160827 a(n) = 3n^4 + 12n^3 + 30n^2 + 36n + 17.

A267691 a(n) = (n + 1)(6n^4 - 21n^3 + 31n^2 - 31*n + 30)/30.