A027441 -id:A027441 - cp's OEIS frontend

A034959 Divide even numbers into groups with prime(n) elements and add together.

2, 18, 70, 182, 484, 884, 1666, 2546, 4048, 6612, 8928, 13172, 17794, 22274, 28576, 37524, 48380, 57340, 71556, 85626, 98550, 118658, 138112, 163404, 196134, 224220, 249672, 281838, 310650, 347136, 420624, 467670, 525806, 571846, 655898

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			{0,2} #2 S=2;
{4,6,8} #3 S=18;
{10,12,14,16,18} #5 S=70;
{20,22,24,26,28,30,32} #7 S=182.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A006003, A027441, A034960.

Cf. A046992, A034956-A034958.

from itertools import islice
from sympy import nextprime
def A034959_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p-1)
        a, p = a+p, nextprime(p)
A034959_list = list(islice(A034959_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = 2*Sum_{k=(A007504(n-1)+1)..A007504(n)} (k-1), n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) - 1), n > 1.
a(n) = 2*(A000217(A007504(n) - 1) - A000217(A007504(n-1) - 1)), n > 1.
If we define A007504(0):=0, then the formulas above are also true for n=1.
a(n) = 2*A034957(n).
a(n) = A034960(n) - A000040(n).
(End)

A179268 Product of numbers between and including n and n^2.

Original entry on oeis.org

1, 24, 181440, 3487131648000, 646300418472124416000000, 3099944389915843478899995401256960000000, 844835922269816056767016893501799134566045599137792000000000

Offset: 1

Reinhard Zumkeller, Jul 06 2010

nonn

a(n) = Product_{k=n..n^2} k;

a(n) = A088020(n)/A000142(n-1).

			a(2) = 2*3*4 = 24;
a(3) = 3*4*5*6*7*8*9 = 181440.

Vincenzo Librandi, Table of n, a(n) for n = 1..23

Cf. A126804, A027441.

[Factorial(n^2) / Factorial(n-1): n in [1..10]]; // Vincenzo Librandi, May 31 2011

Table[Times@@Range[n,n^2],{n,10}] (* Harvey P. Dale, Sep 16 2020 *)

a(n) = (n^2)! / (n-1)!.

Definition clarified by Harvey P. Dale, Sep 16 2020

A034957 Divide natural numbers in groups with prime(n) elements and add together.

Original entry on oeis.org

1, 9, 35, 91, 242, 442, 833, 1273, 2024, 3306, 4464, 6586, 8897, 11137, 14288, 18762, 24190, 28670, 35778, 42813, 49275, 59329, 69056, 81702, 98067, 112110, 124836, 140919, 155325, 173568, 210312, 233835, 262903, 285923, 327949, 355001, 393285

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

Natural numbers starting from 0,1,2,3,...

			{0,1} #2 S=1;
{2,3,4} #3 S=9;
{5,6,7,8,9} #5 S=35;
{10,11,12,13,14,15,16} #7 S=91.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A006003, A027441, A034956.

Cf. A046992, A034958, A034959, A034960.

{1}~Join~Map[Abs@ Apply[Subtract, Map[PolygonalNumber, #]] &, Partition[Accumulate@ Prime@ Range@ 37 - 1, 2, 1]] (* Michael De Vlieger, Oct 06 2019 *)
Module[{nn=40,tprs},tprs=Total[Prime[Range[nn]]];Total/@TakeList[Range[0,tprs],Prime[Range[nn]]]] (* Harvey P. Dale, Apr 18 2025 *)

from itertools import islice
from sympy import nextprime
def A034957_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p-1)>>1
        a, p = a+p, nextprime(p)
A034957_list = list(islice(A034957_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (k-1), n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) - 1)/2, n > 1.
a(n) = (A000217(A007504(n) - 1) - A000217(A007504(n-1) - 1)), n > 1.
If we define A007504(0):=0, then the formulas above are also true for n=1.
a(n) = A034959(n)/2.
a(n) = A034956(n) - A000040(n).
(End)

A068253 1/3 of the number of colorings of an n X n square array with 3 colors.

Original entry on oeis.org

1, 6, 82, 2604, 193662, 33865632, 13956665236, 13574876544396, 31191658416342674, 169426507164530254380, 2176592549084872196370724, 66158464020552857153017287240, 4759146677426447759184119036493676, 810410082813497381147177065840601910384

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..24

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.
See A047938 for number of improper colorings.
Main diagonal of A078099.
Twice A207993 for n>1.

M[1] = {{1}}; M[m_] := M[m] = {{M[m - 1], Transpose[M[m - 1]]}, {Array[0 &, {2^(m - 2), 2^(m - 2)}], M[m - 1]}} // ArrayFlatten; W[m_] := M[m] + Transpose[M[m]]; T[m_, 1] := 2^(m - 1); T[1, n_] := 2^(n - 1); T[m_, n_] := MatrixPower[W[m], n - 1] // Flatten // Total; a[n_] := T[n, n]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 01 2017, after code from A078099 *)

For formula see A078099.

More terms from Vladeta Jovovic, Jul 22 2004
a(11)-a(12) from Alois P. Heinz, Mar 25 2009
a(13)-a(14) from Andrew Howroyd, Jun 26 2017

A068271 1/4 the number of colorings of an n X n rhombic hexagonal array with 4 colors.

Original entry on oeis.org

1, 12, 264, 11424, 1008576, 184910592, 71033971200, 57469424744448, 98237339264864256, 355574469749489123328, 2729407814499050197254144, 44482040254775494064841818112, 1540473331004371306422199656382464, 113440401780206156918876627438624833536

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Andrew Howroyd, Table of n, a(n) for n = 1..16

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a(9) from Alois P. Heinz, May 02 2012

a(10)-a(14) from Andrew Howroyd, Jun 25 2017

A108396 Triangle read by rows: T(n,k) = n*(1+n^k)/2, 0<=k<=n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 3, 6, 15, 42, 4, 10, 34, 130, 514, 5, 15, 65, 315, 1565, 7815, 6, 21, 111, 651, 3891, 23331, 139971, 7, 28, 175, 1204, 8407, 58828, 411775, 2882404, 8, 36, 260, 2052, 16388, 131076, 1048580, 8388612, 67108868, 9, 45, 369, 3285, 29529, 265725

Offset: 0

Reinhard Zumkeller, Jun 02 2005

Row sums give A108397;
T(n,0) = A001477(n);
T(n,1) = A000217(n) for n>0;
T(n,2) = A006003(n) for n>1;
T(n,3) = A027441(n) for n>2;
T(n,4) = A021003(n) for n>3;
T(n,n) = A108398(n).

			.  0:  0
.  1:  1  1
.  2:  2  3   5
.  3:  3  6  15   42
.  4:  4 10  34  130   514
.  5:  5 15  65  315  1565   7815
.  6:  6 21 111  651  3891  23331  139971
.  7:  7 28 175 1204  8407  58828  411775  2882404
.  8:  8 36 260 2052 16388 131076 1048580  8388612  67108868
.  9:  9 45 369 3285 29529 265725 2391489 21523365 193710249 1743392205 .

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened

Cf. A079901, A000312, A033918, A001477, A000217, A006003, A027441, A021003, A108398, A108397 (row sums), A256512 (central terms).

a108396 n k = a108396_tabl !! n !! k
a108396_row n = a108396_tabl !! n
a108396_tabl = zipWith (\v ws -> map (flip div 2 . (* v) . (+ 1)) ws)
                       [0..] a079901_tabl
-- Reinhard Zumkeller, Mar 31 2015

Join[{0},Flatten[Table[n (1+n^k)/2,{n,10},{k,0,n}]]] (* Harvey P. Dale, Mar 19 2015 *)

Offset changed by Reinhard Zumkeller, Mar 31 2015

A168029 a(n) = n*(n^6 + 1)/2.

Original entry on oeis.org

0, 1, 65, 1095, 8194, 39065, 139971, 411775, 1048580, 2391489, 5000005, 9743591, 17915910, 31374265, 52706759, 85429695, 134217736, 205169345, 306110025, 446935879, 640000010, 900544281, 1247178955, 1702412735, 2293235724, 3051757825, 4015905101

Offset: 0

N. J. A. Sloane, Dec 11 2009

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Sequences of the form n*(n^m + 1)/2: A001477 (m=0), A000217 (m=1), A006003 (m=2), A027441 (m=3), A021003 (m=4), A167963 (m=5), this sequence (m=6), A168067 (m=7), A168116 (m=8), A168118 (m=9), A168119 (m=10).

[n*(n^6+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 10 2014

CoefficientList[Series[x(1 +57x +603x^2 +1198x^3 +603x^4 +57x^5 +x^6)/ (1-x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 10 2014 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1}, {0,1,65,1095,8194,39065, 139971,411775}, 41] (* Harvey P. Dale, Jan 24 2019 *)

[n*(n^6+1)/2 for n in range(41)] # G. C. Greubel, Jan 12 2023

G.f.: x*(1+57*x+603*x^2+1198*x^3+603*x^4+57*x^5+x^6)/(1-x)^8. - Vincenzo Librandi, Dec 10 2014

E.g.f.: (x/2)*(2 +63*x +301*x^2 +350*x^3 +140*x^4 +21*x^5 +x^6)*exp(x). - G. C. Greubel, Jan 12 2023

More terms from Vincenzo Librandi, Dec 10 2014

A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 176, 5490, 65600, 455875, 2239776, 8647716, 27962880, 78920325, 200002000, 464447126, 1003294656, 2039332295, 3935444800, 7261533000, 12884914176, 22089914121, 36733221360, 59442494650, 93866696000, 144987663051, 219503536736, 326295822700, 476993088000

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a:= n-> (248 +(-1012 +(1786 +(-1791 +(1120 +(-448 +(112 +(-16+n)*n) *n) *n) *n) *n) *n) *n) *n/6:
seq(a(n), n=3..40);  #  Alois P. Heinz, May 02 2012

From Alois P. Heinz, May 02 2012: (Start)
G.f.: (1089*x^6+10934*x^5+26015*x^4+18500*x^3+3775*x^2+166*x+1) / (x-1)^10*x^3.
a(n) = (n^9 -16*n^8 +112*n^7 -448*n^6 +1120*n^5 -1791*n^4 +1786*n^3 -1012*n^2 +248*n)/6.  (End)

A068245 1/6 the number of colorings of a 4 X 4 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 7616, 5141250, 552093440, 20631905875, 395001645696, 4771909547076, 41190314035200, 275192443300005, 1502690499112000, 6971521964029766, 28275884687022336, 102456840191225975, 337289521082456320, 1022310183284613000, 2883605488481550336, 7636012822945480521

Offset: 3

R. H. Hardin, Feb 24 2002

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

[(n^11 -26*n^10 +310*n^9 -2240*n^8 +10915*n^7 -37726*n^6 +94576*n^5 -172395*n^4 +224588*n^3 -199854*n^2 +109788*n -28340)*n *(n-1)*(n-2)^3/6: n in [3..19]]; // Bruno Berselli, May 03 2012

a:= n-> (-226720+ (1445104+ (-4304712+ (7968348+ (-10265148+ (9755858+ (-7068408+ (3975561+ (-1749715+ (602408+ (-160859+ (32703+ (-4898+ (510+ (-33+n)*n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n/6:
seq(a(n), n=3..40); #  Alois P. Heinz, May 02 2012

From Alois P. Heinz, May 02 2012: (Start)
G.f.: -(7926831*x^13 +710120929*x^12 +16477733814*x^11 +144915014346*x^10 +569769493505*x^9 +1086745824783*x^8 +1040642122932*x^7 +499586289612*x^6 +115866023553*x^5 +11940350895*x^4 +465727286*x^3 +5011914*x^2 +7599*x+1) *x^3 / (x-1)^17.
a(n) = (n^16 -33*n^15 +510*n^14 -4898*n^13 +32703*n^12 -160859*n^11 +602408*n^10 -1749715*n^9  +3975561*n^8 -7068408*n^7 +9755858*n^6 -10265148*n^5 +7968348*n^4 -4304712*n^3 +1445104*n^2 -226720*n)/6. (End)

Extended beyond a(15) by Alois P. Heinz, May 02 2012

A068254 1/4 the number of colorings of an n X n square array with 4 colors.

Original entry on oeis.org

1, 21, 2403, 1500183, 5110723191, 95013316876491, 9639473169171326643, 5336900216006709884938623, 16124704040675904181778734982451, 265865038636937159336134567410478299051

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..16

Main diagonal of A222444.

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

A222444 = Cases[Import["https://oeis.org/A222444/b222444.txt", "Table"], {, }][[All, 2]];
a[n_] := A222444[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Sep 23 2019 *)

a(9)-a(10) from Alois P. Heinz, Apr 27 2012

cp's OEIS Frontend

A034959 Divide even numbers into groups with prime(n) elements and add together.

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A179268 Product of numbers between and including n and n^2.

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A034957 Divide natural numbers in groups with prime(n) elements and add together.

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A068253 1/3 of the number of colorings of an n X n square array with 3 colors.

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A068271 1/4 the number of colorings of an n X n rhombic hexagonal array with 4 colors.

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A108396 Triangle read by rows: T(n,k) = n*(1+n^k)/2, 0<=k<=n.

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A168029 a(n) = n*(n^6 + 1)/2.

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A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

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