cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A111492 Triangle read by rows: a(n,k) = (k-1)! * C(n,k).

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 6, 8, 6, 5, 10, 20, 30, 24, 6, 15, 40, 90, 144, 120, 7, 21, 70, 210, 504, 840, 720, 8, 28, 112, 420, 1344, 3360, 5760, 5040, 9, 36, 168, 756, 3024, 10080, 25920, 45360, 40320, 10, 45, 240, 1260, 6048, 25200, 86400, 226800, 403200, 362880
Offset: 1

Views

Author

Ross La Haye, Nov 15 2005

Keywords

Comments

For k > 1, a(n,k) = the number of permutations of the symmetric group S_n that are pure k-cycles.
Reverse signed array is A238363. For a relation to (Cauchy-Euler) derivatives of the Vandermonde determinant, see Chervov link. - Tom Copeland, Apr 10 2014
Dividing the k-th column of T by (k-1)! for each column generates A135278 (the f-vectors, or face-vectors for the n-simplices). Then ignoring the first column gives A104712, so T acting on the column vector (-0,d,-d^2/2!,d^3/3!,...) gives the Euler classes for hypersurfaces of degree d in CP^n. Cf. A104712 and Dugger link therein. - Tom Copeland, Apr 11 2014
With initial i,j,n=1, given the n X n Vandermonde matrix V_n(x_1,...,x_n) with elements a(i=row,j=column)=(x_j)^(i-1), its determinant |V_n|, and the column vector of n ones C=(1,1,...,1), the n-th row of the lower triangular matrix T is given by the column vector determined by (1/|V_n|) * V_n(:x_1*d/dx_1:,...,:x_n*d/dx_n:)|V_n| * C, where :x_j*d/dx_j:^n = (x_j)^n*(d/dx_j)^n. - Tom Copeland, May 20 2014
For some other combinatorial interpretations of the first three columns of T, see A208535 and the link to necklace polynomials therein. Because of the simple relation of the array to the Pascal triangle, it can easily be related to many other arrays, e.g., T(p,k)/(p*(k-1)!) with p prime gives the prime rows of A185158 and A051168 when the non-integers are rounded to 0. - Tom Copeland, Oct 23 2014

Examples

			a(3,3) = 2 because (3-1)!C(3,3) = 2.
1;
2 1;
3 3 2;
4 6 8 6;
5 10 20 30 24;
6 15 40 90 144 120;
7 21 70 210 504 840 720;
8 28 112 420 1344 3360 5760 5040;
9 36 168 756 3024 10080 25920 45360 40320;

Links

Programs

  • Magma
    /* As triangle: */ [[Factorial(k-1)*Binomial(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 21 2014
  • Mathematica
    Flatten[Table[(k - 1)!Binomial[n, k], {n, 10}, {k, n}]]

Formula

a(n, k) = (k-1)!C(n, k) = P(n, k)/k.
E.g.f. (by columns) = exp(x)((x^k)/k).
a(n, 1) = A000027(n);
a(n, 2) = A000217(n-1);
a(n, 3) = A007290(n);
a(n, 4) = A033487(n-3).
a(n, n) = A000142(n-1);
a(n, n-1) = A001048(n-1) for n > 1.
Sum[a(n, k), {k, 1, n}] = A002104(n);
Sum[a(n, k), {k, 2, n}] = A006231(n).
a(n,k) = sum(j=k..n-1, j!/(j-k)!) (cf. Chervov link). - Tom Copeland, Apr 10 2014
From Tom Copeland, Apr 28 2014: (Start)
E.g.f. by row: [(1+t)^n-1]/t.
E.g.f. of row e.g.f.s: {exp[(1+t)*x]-exp(x)}/t.
O.g.f. of row e.g.f.s: {1/[1-(1+t)*x] - 1/(1-x)}/t.
E.g.f. of row o.g.f.s: -exp(x) * log(1-t*x). (End)

A167875 One third of product plus sum of three consecutive nonnegative integers; a(n)=(n+1)(n^2+2n+3)/3.

Original entry on oeis.org

1, 4, 11, 24, 45, 76, 119, 176, 249, 340, 451, 584, 741, 924, 1135, 1376, 1649, 1956, 2299, 2680, 3101, 3564, 4071, 4624, 5225, 5876, 6579, 7336, 8149, 9020, 9951, 10944, 12001, 13124, 14315, 15576, 16909, 18316, 19799, 21360, 23001, 24724, 26531
Offset: 0

Views

Author

Klaus Brockhaus, Nov 14 2009

Keywords

Comments

a(n) = ((n*(n+1)*(n+2))+(n+(n+1)+(n+2)))/3, n >= 0.
Equals A006527 without initial term 0: a(n) = A006527(n+1).
Binomial transform of A167876.
Inverse binomial transform of A080930.
a(n) = A007290(n+2)+n+1.
a(n) = A014820(n)/(n+1) for n > 0.
a(n) = A116731(n+2)-1.
a(n) = A033547(n+1)-n.
a(n) = A054602(n)/3.
a(n) = A086514(n+3)-2.
a(n) = A002061(n+1)+a(n-1) for n > 0.
a(n) = A005894(n)-a(n-1) for n > 0.
First bisection is A057813.
Second differences are in A004277.
a(n) = A177342(n)*(-1)+a(n-1)*5 with n>0. For n=8, a(8)=-A177342(8)+a(7)*5=-631+176*5=249. - Bruno Berselli, May 18 2010

Examples

			a(0) = (0*1*2+0+1+2)/3 = (0+3)/3 = 1.
a(1) = (1*2*3+1+2+3)/3 = (6+6)/3 = 4.
a(6)-4*a(5)+6*a(4)-4*a(3)+a(2) = 119-4*76+6*45-4*24+11 = 0. - _Bruno Berselli_, May 26 2010

Links

Crossrefs

Cf. A001477 (nonnegative integers),
A006527 ((n^3+2*n)/3),
A167876 (1, 3, 4, 2, 0, 0, 0, 0, ...),
A080930,
A007290 (2*C(n, 3)),
A014820 ((1/3)*(n^2+2*n+3)*(n+1)^2),
A116731,
A033547 (n*(n^2+5)/3),
A054602 (Sum_{d|3} phi(d)*n^(3/d)),
A086514 ((n^3-6*n^2+14*n-6)/3),
A002061 (n^2-n+1),
A005894 (centered tetrahedral numbers),
A057813 ((2*n+1)*(4*n^2+4*n+3)/3),
A004277 (1 and the positive even numbers),
A028387 (n+(n+1)^2),
A166941, A166942, A166943.

Programs

  • Magma
    [ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ];
  • Mathematica
    Select[Table[(n*(n+1)*(n+2)+n+(n+1)+(n+2))/3,{n,0,5!}],IntegerQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)
(Times@@#+Total[#])/3&/@Partition[Range[0,65],3,1]  (* Harvey P. Dale, Mar 14 2011 *)
  • PARI
    a(n)=(n+1)*(n^2+2*n+3)/3 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (n^3+3*n^2+5*n+3)/3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+2 for n > 3; a(0)=1, a(1)=4, a(2)=11, a(3)=24.
G.f.: (1+x^2)/(1-x)^4.
a(n) = SUM(A109613(k)*A005408(n-k): 0<=k<=n). - Reinhard Zumkeller, Dec 05 2009
a(n)-4*a(n-1)+6*a(n-2)-4*a(n-3)+a(n-4)=0 for n>3. - Bruno Berselli, May 26 2010

A175676 a(n) = binomial(n,3) mod n.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, 0, 7, 0, 0, 8, 0, 0, 9, 0, 0, 10, 0, 0, 11, 0, 0, 12, 0, 0, 13, 0, 0, 14, 0, 0, 15, 0, 0, 16, 0, 0, 17, 0, 0, 18, 0, 0, 19, 0, 0, 20, 0, 0, 21, 0, 0, 22, 0, 0, 23, 0, 0, 24, 0, 0, 25, 0, 0, 26, 0, 0, 27, 0, 0, 28, 0, 0, 29, 0, 0, 30, 0, 0, 31, 0
Offset: 1

Views

Author

Zak Seidov, Aug 07 2010

Keywords

Comments

Number of partitions of n+3 into 3 parts that are in arithmetic progression. - Wesley Ivan Hurt, Dec 07 2020

Links

Crossrefs

Cf. A007290.

Programs

Formula

a(n) = n/3 if n==0 (mod 3) else a(n) = 0.
G.f.: x^3 / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Mar 11 2011
a(n) = A008620(n-1)*A079978(n). - Bruno Berselli, Jun 22 2012
a(n) = (n + 2*n*cos((2*n*Pi)/3))/9. - Kritsada Moomuang, Apr 02 2018

A206817 Sum_{0

Original entry on oeis.org

1, 10, 73, 520, 3967, 33334, 309661, 3166468, 35416555, 430546642, 5655609529, 79856902816, 1206424711303, 19419937594990, 331860183278677, 6000534640290364, 114462875817046051, 2297294297649673738, 48394006967070653425
Offset: 2

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

In the following guide to related sequences,
c(n) = Sum_{0
t(n) = Sum_{0
s(k).................c(n)........t(n)
k....................A000217.....A000292
k^2..................A016061.....A004320
k^3..................A206808.....A206809
k^4..................A206810.....A206811
k!...................A206816.....A206817
prime(k).............A152535.....A062020
prime(k+1)...........A185382.....A206803
2^(k-1)..............A000337.....A045618
k(k+1)/2.............A007290.....A034827
k-th quarter-square..A049774.....A206806

Examples

			a(3) = (2-1) + (6-1) + (6-2) = 10.

Links

Crossrefs

Cf. A000142, A206816.

Programs

  • Mathematica
    s[k_] := k!; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1];
Table[c[n], {n, 2, 32}]          (* A206816 *)
Flatten[Table[t[n], {n, 2, 20}]] (* A206817 *)
  • PARI
    a(n)=sum(j=1,n,j!*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015
  • PARI
    a(n)=my(t=1); sum(j=1,n,t*=j; t*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015
  • Sage
    [sum([sum([factorial(k)-factorial(j) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,21)] # Danny Rorabaugh, Apr 18 2015

Formula

a(n) = a(n-1)+(n-1)s(n)-p(n-1), where s(n) = n! and p(k) = 1!+2!+...+k!.
a(n) = Sum_{k=2..n} A206816(k).

A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 4, 8, 20, 44, 10, 20, 43, 84, 152, 20, 40, 78, 140, 240, 372, 35, 70, 130, 224, 369, 558, 824, 56, 112, 200, 332, 528, 780, 1132, 1544, 84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448
Offset: 1

Views

Author

N. J. A. Sloane, Jun 13 2020

Keywords

Comments

It follows from the definitions that T(n,k) + A334705(n,k) = A334703(n,k) for 1 <= k <= n.

Examples

			Triangle begins:
0,
0, 0,
1, 2, 8,
4, 8, 20, 44,
10, 20, 43, 84, 152,
20, 40, 78, 140, 240, 372,
35, 70, 130, 224, 369, 558, 824,
56, 112, 200, 332, 528, 780, 1132, 1544,
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712,
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448,
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992,
...
This is the lower half of a symmetric array. The full symmetric array begins:
0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...
0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, ...
1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, ...
4, 8, 20, 44, 84, 140, 224, 332, 472, 648, 864, 1120, ...
10, 20, 43, 84, 152, 240, 369, 528, 734, 988, 1295, 1652, ...
20, 40, 78, 140, 240, 372, 558, 780, 1064, 1408, 1826, 2304, ...
35, 70, 130, 224, 369, 558, 824, 1132, 1519, 1982, 2542, 3172, ...
56, 112, 200, 332, 528, 780, 1132, 1544, 2052, 2652, 3372, 4172, ...
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 3480, 4393, 5396, ...
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448, 5586, 6824, ...
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992, 8508, ...
220, 440, 732, 1120, 1652, 2304, 3172, 4172, 5396, 6824, 8508, 10332, ...
...

Links

Crossrefs

This is a companion to the triangles A334703 and A334705.
Rows (or columns) 1,2,3,4 of the full array are A000292, A007290, A057566, A334706. The main diagonal is A000938.

Extensions

Rows 6 onwards from Tom Duff. - N. J. A. Sloane, Jun 19 2020

A052149 Number of nonsquare rectangles on an n X n board.

Original entry on oeis.org

0, 4, 22, 70, 170, 350, 644, 1092, 1740, 2640, 3850, 5434, 7462, 10010, 13160, 17000, 21624, 27132, 33630, 41230, 50050, 60214, 71852, 85100, 100100, 117000, 135954, 157122, 180670, 206770, 235600, 267344, 302192, 340340, 381990, 427350, 476634
Offset: 1

Views

Author

Ronald Arms (ron.arms(AT)stanfordalumni.org), Jan 23 2000

Keywords

Comments

Partial sums of A045991 (n^3-n^2). - Jeremy Gardiner, Jun 30 2013

Examples

			a(10) = 10 * 9 * 11 * 32 / 12 = 2640.
a(5) = 170 and the sum from 1 to 5 is 15, giving 1*(15-1)=14, 2*(15-2)=26, 2*(15-3)=36, 4*(15-4)=44 and 5*(15-5)=50; adding 14+26+36+44+50=170. Do the same for each n and get a(n). - _J. M. Bergot_, Oct 31 2014

Links

Crossrefs

Cf. A035291, A045991, A033487, A007290, A000537, A000330, A000914.

Programs

  • Magma
    I:=[0, 4, 22, 70, 170]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, Apr 28 2012
  • Maple
    a:=n->sum(j^3-j^2, j=0..n): seq(a(n), n=1..37); # Zerinvary Lajos, May 08 2008
  • Mathematica
    CoefficientList[Series[2*x*(2+x)/(1-5*x+10*x^2-10*x^3+ 5*x^4-x^5), {x,0,50}], x] (* Vincenzo Librandi, Apr 28 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,4,22,70,170},40] (* Harvey P. Dale, Jul 30 2019 *)
  • PARI
    a(n) = sum(k=1,n,(k-1)*k^2) \\ Michel Marcus, Nov 09 2012

Formula

a(n) = n*(n-1)*(n+1)*(3*n+2)/12.
G.f.: 2*x^2*(2+x)/(1-5*x+10*x^2-10*x^3+5*x^4-x^5). - Colin Barker, Jan 04 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Apr 28 2012
a(n) = A033487(n-1) - A007290(n+1) starting at n=1. - J. M. Bergot, Jun 04 2012
a(n) = Sum_{k=1..n} (k-1)*k^2. - Michel Marcus, Nov 09 2012
a(n) = A000537(n) - A000330(n) = 2*A000914(n-1). - Luciano Ancora, Mar 16 2015
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=2} 1/a(n) = 81*log(3)/5 - 9*sqrt(3)*Pi/5 - 192/25.
Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 - 48*log(2)/5 - 318/25. (End)

A064999 Partial sums of sequence (essentially A002378): 1, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...

Original entry on oeis.org

1, 3, 9, 21, 41, 71, 113, 169, 241, 331, 441, 573, 729, 911, 1121, 1361, 1633, 1939, 2281, 2661, 3081, 3543, 4049, 4601, 5201, 5851, 6553, 7309, 8121, 8991, 9921, 10913, 11969, 13091, 14281, 15541, 16873, 18279, 19761, 21321, 22961, 24683
Offset: 0

Views

Author

Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Oct 31 2001

Keywords

Comments

Equals triangle A144328 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 18 2008
a(n) is the number of parking functions of size n+1 avoiding the patterns 123 and 312. - Lara Pudwell, Apr 10 2023

Links

Crossrefs

Cf. A002378, A007290.
Cf. A144328. - Gary W. Adamson, Sep 18 2008

Programs

  • Magma
    [(n^3+3*n^2+2*n+3)/3: n in [0..50]]; // Vincenzo Librandi, Feb 28 2016
  • Maple
    a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^2-n od: seq(a[n], n=0..42); # Zerinvary Lajos, Jun 05 2008
  • Mathematica
    Table[(x^3 - x + 3)/3, {x, 1, 100}] (* Artur Jasinski, Feb 14 2007 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 3, 9, 21}, 50] (* Vincenzo Librandi, Feb 28 2016 *)
  • PARI
    { for (n=0, 1000, if (n, a+=n*(n + 1), a=1); write("b064999.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 03 2009
  • PARI
    a(n) = (n^3+3*n^2+2*n+3)/3; \\ Altug Alkan, May 16 2018

Formula

a(n) = A007290(n+2) + 1 = (n^3 + 3*n^2 + 2*n + 3)/3.
a(0) = 1, a(n) = n*(n+1) + a(n-1) for n > 1. - Gerald McGarvey, Sep 26 2004
O.g.f.: (1 - x + 3x^2 - x^3)/(1 - x)^4.

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Nov 12 2001

A092271 Triangle read by rows. First in a series of triangular arrays counting permutations of partitions.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 8, 6, 1, 24, 30, 20, 10, 1, 120, 144, 90, 40, 15, 1, 720, 840, 504, 210, 70, 21, 1, 5040, 5760, 3360, 1344, 420, 112, 28, 1, 40320, 45360, 25920, 10080, 3024, 756, 168, 36, 1, 362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45, 1, 3628800, 3991680, 2217600, 831600, 237600, 55440, 11088, 1980, 330, 55, 1
Offset: 1

Views

Author

Alford Arnold, Feb 14 2004

Keywords

Comments

Generate signatures in accordance with A086141. Map to partitions in accordance with A025487. Calculate the number of permutations in accordance with Abramowitz and Stegun, p. 831 (reference M2). Display the results as illustrated by A090774. The second array is:
3
20 15
90 120 45
504 630 420 105
...
Apart from the main diagonal, appears to be the same as A211603 (see also A238363) and is related to the infinitesimal generator of A008290; if so, the off-diagonal triangle entries are given by binomial(n,k)*(n-k-1)! for n >= 2 and 0 <= k <= n-2. - Peter Bala, Feb 13 2017
Let aut(p) denote the size of the centralizer of the partition p (see A339016 for the definition). Then row(n) = [n!/aut(p) for p in P], where P are the partitions of n with largest part k and length n + 1 - k. Row sums are A121726. - Peter Luschny, Nov 19 2020

Examples

			The triangle begins:
1:    1
2:    1   1
3:    2   3  1
4:    6   8  6  1
5:   24  30 20 10  1
6:  120 144 90 40 15 1
  ...
From _Peter Luschny_, Nov 19 2020: (Start):
The combinatorial interpretation is illustrated by this computation of row 6:
6! / aut([6])                = 720 / A339033(6, 1) = 720/6   = 120 = T(6, 1)
6! / aut([5, 1])             = 720 / A339033(6, 2) = 720/5   = 144 = T(6, 2)
6! / aut([4, 1, 1])          = 720 / A339033(6, 3) = 720/8   =  90 = T(6, 3)
6! / aut([3, 1, 1, 1])       = 720 / A339033(6, 4) = 720/18  =  40 = T(6, 4)
6! / aut([2, 1, 1, 1, 1])    = 720 / A339033(6, 5) = 720/48  =  15 = T(6, 5)
6! / aut([1, 1, 1, 1, 1, 1]) = 720 / A339033(6, 6) = 720/720 =   1 = T(6, 6)
-------------------------------------------------------------------------------
                                                         Sum:  410 = A121726(6)
(End)

References

  • Abramowitz and Stegun, p. 831.

Links

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A007290, A025487, A086141, A090774, A008290, A111492, A211603, A238363, A121726, A339016, A339033.

Programs

  • Mathematica
    f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!];Table[Append[Map[f, Select[Partitions[n], Count[#, Except[1]] == 1 &]], 1], {n,1, 10}] // Grid (* Geoffrey Critzer, Nov 07 2015 *)
  • SageMath
    def A092271(n, k):
    if n == k: return 1
    return factorial(n) // ((n + 1 - k)*factorial(k - 1))
for n in (1..9): print(n, [A092271(n, k) for k in (1..n)])
def A092271Row(n):
    if n == 0: return [1]
    f = factorial(n); S = []
    for k in range(n,0,-1):
        for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
            S.append(f // p.aut())
    return S
for n in (1..9): print(A092271Row(n)) # Peter Luschny, Nov 20 2020

Extensions

More terms from Geoffrey Critzer, Nov 10 2015

A212014 Total number of states of the first n subshells of the nuclear shell model in which the subshells are ordered by energy level in increasing order.

Original entry on oeis.org

2, 6, 8, 14, 18, 20, 28, 34, 38, 40, 50, 58, 64, 68, 70, 82, 92, 100, 106, 110, 112, 126, 138, 148, 156, 162, 166, 168, 184, 198, 210, 220, 228, 234, 238, 240, 258, 274, 288, 300, 310, 318, 324, 328, 330, 350, 368, 384, 398, 410, 420, 428, 434, 438, 440, 462, 482, 500, 516, 530, 542, 552, 560, 566, 570, 572
Offset: 1

Views

Author

Omar E. Pol, Jul 15 2012

Keywords

Examples

			Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus. Triangle begins:
    2;
    6,   8;
   14,  18,  20;
   28,  34,  38,  40;
   50,  58,  64,  68,  70;
   82,  92, 100, 106, 110, 112;
  126, 138, 148, 156, 162, 166, 168;
  ...
Column 1 gives positive terms of A033547. Right border gives positive terms of A007290.
Example 2: written as an irregular triangle in which row j is related to the j-th shell of nucleus. In this case note that row 4 has only one term. Triangle begins:
    2;
    6,   8;
   14,  18,  20;
   28;
   34,  38,  40,  50;
   58,  64,  68,  70,  82;
   92, 100, 106, 110, 112, 126;
  138, 148, 156, 162, 166, 168, 184;
  ...
First seven terms of right border give the "magic numbers" A018226.

References

  • M. Goeppert Mayer, Nuclear configurations in the spin-orbit coupling model. I. Empirical evidence, Phys. Rev. 78: 16 (1950). II. Theoretical considerations, Phys. Rev. 78: 22 (1950).

Links

Crossrefs

Partial sums of A212012. Other versions are A210984, A212124, A213364, A213374.
Cf. A004736, A007290, A033547, A018226, A212013.

Programs

  • Mathematica
    2*Accumulate[Flatten[Range[Range[15], 1, -1]]] (* Paolo Xausa, Mar 14 2025 *)

Formula

a(n) = 2*A212013(n).

A115262 Correlation triangle for n+1.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 14, 11, 5, 6, 14, 20, 20, 14, 6, 7, 17, 26, 30, 26, 17, 7, 8, 20, 32, 40, 40, 32, 20, 8, 9, 23, 38, 50, 55, 50, 38, 23, 9, 10, 26, 44, 60, 70, 70, 60, 44, 26, 10, 11, 29, 50, 70, 85, 91, 85, 70, 50, 29, 11
Offset: 0

Views

Author

Paul Barry, Jan 18 2006

Keywords

Comments

This sequence (formatted as a square array) gives the counts of all possible squares in an m X n rectangle. For example, 11 = 8 (1 X 1 squares) + 3 (2 X 2 square) in 4 X 2 rectangle. - Philippe Deléham, Nov 26 2009
From Clark Kimberling, Feb 07 2011: (Start)
Also the accumulation array of min{n,k}, when formatted as a rectangle.
This is the accumulation array of the array M=A003783 given by M(n,k)=min{n,k}; see A144112 for the definition of accumulation array.
The accumulation array of A115262 is A185957. (End)
From Clark Kimberling, Dec 22 2011: (Start)
As a square matrix, A115262 is the self-fusion matrix of A000027 (1,2,3,4,...). See A193722 for the definition of fusion and A202673 for characteristic polynomials associated with A115622. (End)

Examples

			Triangle begins
  1;
  2,  2;
  3,  5,  3;
  4,  8,  8,  4;
  5, 11, 14, 11,  5;
  6, 14, 20, 20, 14,  6;
  ...
When formatted as a square matrix:
  1,  2,  3,  4,  5, ...
  2,  5,  8, 11, 14, ...
  3,  8, 14, 20, 26, ...
  4, 11, 20, 30, 40, ...
  5, 14, 26, 40, 55, ...
  ...

Crossrefs

Cf. A000027, A202673, A271916.
For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).
Diagonals (1,5,...): A000330 (square pyramidal numbers),
diagonals (2,8,...): A007290,
diagonals (3,11,...): A051925,
diagonals (4,14,...): A159920,
antidiagonal sums: A001752.

Programs

  • Mathematica
    U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
(* Clark Kimberling, Dec 22 2011 *)

Formula

Let f(m,n) = m*(m-1)*(3*n-m-1)/6. This array is (with a different offset) the infinite square array read by antidiagonals U(m,n) = f(n,m) if m < n, U(m,n) = f(m,n) if m <= n. See A271916. - N. J. A. Sloane, Apr 26 2016
G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).
Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).
T(2n,n) - T(2n,n+1) = n+1.
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