A061316 -id:A061316 - cp's OEIS frontend

A006007 4-dimensional analog of centered polygonal numbers: a(n) = n(n+1)*(n^2+n+4)/12.

0, 1, 5, 16, 40, 85, 161, 280, 456, 705, 1045, 1496, 2080, 2821, 3745, 4880, 6256, 7905, 9861, 12160, 14840, 17941, 21505, 25576, 30200, 35425, 41301, 47880, 55216, 63365, 72385, 82336, 93280, 105281, 118405, 132720, 148296, 165205, 183521

Offset: 0

N. J. A. Sloane, Simon Plouffe

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..710
Per Alexandersson, Sam Hopkins, and Gjergji Zaimi, Restricted Birkhoff polytopes and Ehrhart period collapse, arXiv:2206.02276 [math.CO], 2022.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 04 2021]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A000537, A000578, A005898, A027602. - Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Cf. A000217, A061315, A061316, A061318.

[n*(n+1)*(n^2+n+4)/12: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=n^3;lst={};s=0;Do[s+=(f[n]+f[n+1]+f[n+2]);AppendTo[lst,s/9],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
Table[2Binomial[n+2,4]+Binomial[n+1,2],{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,5,16,40},40] (* Harvey P. Dale, Sep 30 2011 *)

a(n)=n*(n+1)*(n^2+n+4)/12 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1+x^2)/(1-x)^5.
a(n) = 2*binomial(n + 2, 4) + binomial(n + 1, 2).
a(n) = A061316(n)/3 = A061315(n, 3) = sqrt(A061318(n)-A061316(n)).
a(0)=0, a(1)=1, a(2)=5, a(3)=16, a(4)=40, a(n) = 5*a(n-1) -  10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 30 2011
For n>0, a(n) = (A000217(n-1)^2 + A000217(n)^2 + A000217(n+1)^2 - 1)/9. - Richard R. Forberg, Dec 25 2013
Sum_{n>=1} 1/a(n) = 15/4 - tanh(sqrt(15)*Pi/2)*Pi*sqrt(3/5). - Amiram Eldar, Aug 23 2022
E.g.f.: exp(x)*(12 + 48*x + 42*x^2 + 12*x^3 + x^4)/12. - Stefano Spezia, Aug 31 2023

More terms from Henry Bottomley, Apr 24 2001

A086601 Triangular numbers + 1 squared.

Original entry on oeis.org

1, 4, 16, 49, 121, 256, 484, 841, 1369, 2116, 3136, 4489, 6241, 8464, 11236, 14641, 18769, 23716, 29584, 36481, 44521, 53824, 64516, 76729, 90601, 106276, 123904, 143641, 165649, 190096, 217156, 247009, 279841, 315844, 355216, 398161

Offset: 0

Jon Perry, Jul 23 2003

Also number of n X 2 0..1 arrays with rows and columns unimodal (cf. A223620, column 2). - Georg Fischer, Nov 03 2021

			a(5) = (t(5)+1)^2 = 16^2 = 256.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or column, (2014) Table 2 column 2.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000124, A000217, A061316, A223620.

A086601 := proc(n)
    (n+2+n^2)^2 /4 ;
end proc:
seq(A086601(n),n=0..20) ; # R. J. Mathar, May 14 2014

(Accumulate[Range[0,40]]+1)^2 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,16,49,121},40] (* Harvey P. Dale, Jan 14 2020 *)

PARI
```
w=vector(40,i,(t(i)+1)^2)
```

a(n) = (A000217(n) + 1)^2.
a(n) = (binomial(2+n,2) - binomial(n,1))^2. - Zerinvary Lajos, May 30 2006, corrected by R. J. Mathar, May 14 2014
a(n) = A000124(n)^2. - Omar E. Pol, Oct 30 2007
a(n) = 1 + A061316(n). Zerinvary Lajos, Apr 25 2008
G.f.: ( -1+x-6*x^2+x^3-x^4 ) / (x-1)^5. - R. J. Mathar, May 14 2014

A061314 Table read by descending antidiagonals where T(n,k) = T(n,k-1) + T(n,k-1)^2/k^2 and T(n,0)=n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 6, 3, 0, 4, 15, 12, 4, 0, 5, 40, 48, 20, 5, 0, 6, 140, 304, 120, 30, 6, 0, 7, 924, 6080, 1720, 255, 42, 7, 0, 8, 24640, 1484736, 186620, 7480, 483, 56, 8, 0, 9, 12415040, 61235956672, 1393267596, 3504380, 26404, 840, 72, 9, 0, 10

Offset: 0

Henry Bottomley, Apr 24 2001

Not always an integer.

			The table begins:
    0,       0,       0,       0,          0,...
    1,       2,       3,       4,          5,...
    2,       6,      15,      40,        140,...
    3,      12,      48,     304,       6080,...
    4,      20,     120,    1720,     186620,...
    5,      30,     255,    7480,    3504380,...
    6,      42,     483,   26404,   43599605,...
    7,      56,     840,   79240,  392515340,...
    8,      72,    1368,  209304, 2738219580,...
    ...

Rows include A000004, A000027 and A061322. Columns include A001477, A002378, A061316, A061318 and A061320.

T(n, k) = T(n, k-1) + A061315(n, k)^2.

A061318 Column 3 of A061314.

Original entry on oeis.org

0, 4, 40, 304, 1720, 7480, 26404, 79240, 209304, 499140, 1095160, 2242504, 4332640, 7966504, 14036260, 23829040, 39156304, 62512740, 97268904, 147902080, 220270120, 321933304, 462529540, 654208504, 912130600, 1255036900, 1705896504, 2292638040, 3048972304, 4015313320

Offset: 0

Henry Bottomley, Apr 24 2001

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1)

Cf. A006007, A061314, A061316, A061319.

CoefficientList[Series[-4x (1+x+22x^2+22x^3+22x^4+x^5+x^6)/(x-1)^9,{x,0,30}],x] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,4,40,304,1720,7480,26404,79240,209304},30] (* Harvey P. Dale, Jul 04 2023 *)

a(n) = A061316(n) + A006007(n)^2 = 4*A061319(n).

G.f. -4*x*(1+x+22*x^2+22*x^3+22*x^4+x^5+x^6)/(x-1)^9. - R. J. Mathar, Aug 11 2012

a(25)-a(29) from Stefano Spezia, Aug 31 2023

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A006007 4-dimensional analog of centered polygonal numbers: a(n) = n(n+1)*(n^2+n+4)/12.

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A086601 Triangular numbers + 1 squared.

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A061314 Table read by descending antidiagonals where T(n,k) = T(n,k-1) + T(n,k-1)^2/k^2 and T(n,0)=n.

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A061318 Column 3 of A061314.

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