A052928 -id:A052928 - cp's OEIS frontend

A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 4, 4, 5, 0, 1, 4, 8, 7, 7, 0, 1, 6, 13, 17, 12, 11, 0, 1, 6, 18, 28, 30, 19, 15, 0, 1, 8, 24, 50, 58, 53, 30, 22

Offset: 0

Gus Wiseman, Aug 20 2023

A way of writing n as a positive linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i > 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(2,2)) are a way of writing 8 as a positive linear combination of (1,1,2), namely 8 = 3*1 + 1*1 + 2*2.

			Triangle begins:
  1
  0  1
  0  1  2
  0  1  2  3
  0  1  4  4  5
  0  1  4  8  7  7
  0  1  6 13 17 12 11
  0  1  6 18 28 30 19 15
  0  1  8 24 50 58 53 30 22
Row n = 4 counts the following linear combinations:
  .  1*4  2*2      2*1+1*2      4*1
          1*1+1*3  1*1+1*1+1*2  3*1+1*1
          1*2+1*2  1*1+1*2+1*1  2*1+2*1
          1*3+1*1  1*2+1*1+1*1  2*1+1*1+1*1
                                1*1+1*1+1*1+1*1
Row n = 5 counts the following linear combinations:
  .  1*5  1*1+1*4  2*1+1*3      3*1+1*2          5*1
          1*2+1*3  2*2+1*1      2*1+1*1+1*2      4*1+1*1
          1*3+1*2  1*1+1*1+1*3  2*1+1*2+1*1      3*1+2*1
          1*4+1*1  1*1+1*2+1*2  1*1+1*1+1*1+1*2  3*1+1*1+1*1
                   1*1+1*3+1*1  1*1+1*1+1*2+1*1  2*1+2*1+1*1
                   1*2+1*1+1*2  1*1+1*2+1*1+1*1  2*1+1*1+1*1+1*1
                   1*2+1*2+1*1  1*2+1*1+1*1+1*1  1*1+1*1+1*1+1*1+1*1
                   1*3+1*1+1*1
Array begins:
  1   0   0   0    0    0    0     0
  1   1   1   1    1    1    1     1
  2   2   4   4    6    6    8     8
  3   4   8   13   18   24   33    40
  5   7   17  28   50   70   107   143
  7   12  30  58   108  179  286   428
  11  19  53  109  223  394  696   1108
  15  30  86  194  420  812  1512  2619

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Row k = 0 is A000007.
Row k = 1 is A000012.
Column n = 0 is A000041.
Column n = 1 is A000070.
Row sums are A006951.
Row k = 2 is A052928 except initial terms.
The case of strict integer partitions is A116861.
Central column is T(2n,n) = A(n,n) = A364907(n).
With rows reversed we have the nonnegative version A365004.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A237113, A364272, A364910, A364911, A364914, A364915, A365002.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combp[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]
- or -
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n-k,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]

As an array, also the number of ways to write n-k as a nonnegative linear combination of an integer partition of k (see programs).

A123684 Alternate A016777(n) with A000027(n).

Original entry on oeis.org

1, 1, 4, 2, 7, 3, 10, 4, 13, 5, 16, 6, 19, 7, 22, 8, 25, 9, 28, 10, 31, 11, 34, 12, 37, 13, 40, 14, 43, 15, 46, 16, 49, 17, 52, 18, 55, 19, 58, 20, 61, 21, 64, 22, 67, 23, 70, 24, 73, 25, 76, 26, 79, 27, 82, 28, 85, 29, 88, 30, 91, 31, 94, 32, 97, 33, 100, 34, 103, 35, 106, 36

Offset: 1

Alford Arnold, Oct 11 2006

a(n) is a diagonal of Table A123685.
The arithmetic average of the first n terms gives the positive integers repeated (A008619). - Philippe Deléham, Nov 20 2013
Images under the modified '3x-1' map: a(n) = n/2 if n is even, (3n-1)/2 if n is odd. (In this sequence, the numbers at even indices n are n/2 [A000027], and the numbers at odd indices n are 3((n-1)/2) + 1 [A016777] = (3n-1)/2.) The latter correspondence interestingly mirrors an insight in David Bařina's 2020 paper (see below), namely that 3(n+1)/2 - 1 = (3n+1)/2. - Kevin Ge, Oct 30 2024

			The natural numbers begin 1, 2, 3, ... (A000027), the sequence 3*n + 1 begins 1, 4, 7, 10, ... (A016777), therefore A123684 begins 1, 1, 4, 2, 7, 3, 10, ...
1/1 = 1, (1+1)/2 = 1, (1+1+4)/3 = 2, (1+1+4+2)/4 = 2, ... - _Philippe Deléham_, Nov 20 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
David Bařina, Convergence verification of the Collatz problem

Cf. A000027, A008619, A016777, A016825, A031878, A052928, A064455.
Cf. A071045, A092530, A093005, A105638, A123685, A137501, A225126.
Cf. A014682 (modified Collatz map)

import Data.List (transpose)
a123684 n = a123684_list !! (n-1)
a123684_list = concat $ transpose [a016777_list, a000027_list]
-- Reinhard Zumkeller, Apr 29 2013

&cat[ [ 3*n-2, n ]: n in [1..36] ]; // Klaus Brockhaus, May 12 2007

/* From the fourteenth formula: */ [&+[1+k*(-1)^k: k in [0..n]]: n in [0..80]]; // Bruno Berselli, Jul 16 2013

A123684:=n->n-1/4-(1/2*n-1/4)*(-1)^n: seq(A123684(n), n=1..70); # Wesley Ivan Hurt, Jul 26 2014

CoefficientList[Series[(1 +x +2*x^2)/((1-x)^2*(1+x)^2), {x,0,70}], x] (* Wesley Ivan Hurt, Jul 26 2014 *)
LinearRecurrence[{0,2,0,-1},{1,1,4,2},80] (* Harvey P. Dale, Apr 14 2025 *)

print(vector(72, n, if(n%2==0, n/2, (3*n-1)/2))) \\ Klaus Brockhaus, May 12 2007

print(vector(72, n, n-1/4-(1/2*n-1/4)*(-1)^n)); \\ Klaus Brockhaus, May 12 2007

[(n + (2*n-1)*(n%2))//2 for n in range(1,71)] # G. C. Greubel, Mar 15 2024

From Klaus Brockhaus, May 12 2007: (Start)
G.f.: x*(1+x+2*x^2)/((1-x)^2*(1+x)^2).
a(n) = (1/4)*(4*n - 1 - (2*n - 1)*(-1)^n).
a(2n-1) = A016777(n-1) = 3(n-1) + 1.
a(2n) = A000027(n) = n.
a(n) = A071045(n-1) + 1.
a(n) = A093005(n) - A093005(n-1) for n > 1.
a(n) = A105638(n+2) - A105638(n+1) for n > 1.
a(n) = A092530(n) - A092530(n-1) - 1.
a(n) = A031878(n+1) - A031878(n) - 1. (End)
a(2*n+1) + a(2*n+2) = A016825(n). - Paul Curtz, Mar 09 2011
a(n)= 2*a(n-2) - a(n-4). - Paul Curtz, Mar 09 2011
From Jaroslav Krizek, Mar 22 2011 (Start):
a(n) = n + a(n-1) for odd n; a(n) = n - A064455(n-1) for even n.
a(n) = A064455(n) - A137501(n).
Abs(a(n) - A064455(n)) = A052928(n). (End)
a(n) = A225126(n) for n > 1. - Reinhard Zumkeller, Apr 29 2013
a(n) = Sum_{k=1..n} (1 + (k-1)*(-1)^(k-1)). - Bruno Berselli, Jul 16 2013
a(n) = n + floor(n/2) for odd n; a(n) = n/2 for even n. - Reinhard Muehlfeld, Jul 25 2014

More terms from Klaus Brockhaus, May 12 2007

A198442 Number of sequences of n coin flips that win on the last flip, if the sequence of flips ends with (1,1,0) or (1,0,0).

Original entry on oeis.org

0, 0, 2, 3, 6, 8, 12, 15, 20, 24, 30, 35, 42, 48, 56, 63, 72, 80, 90, 99, 110, 120, 132, 143, 156, 168, 182, 195, 210, 224, 240, 255, 272, 288, 306, 323, 342, 360, 380, 399, 420, 440, 462, 483, 506, 528, 552, 575, 600, 624, 650, 675, 702, 728, 756, 783, 812

Offset: 1

Paul Weisenhorn, Oct 25 2011

If the sequence ends with (1,1,0) Abel wins; if it ends with (1,0,0) Kain wins.
Abel(n) = A002620(n-1) = (2*n*(n - 2) + 1 - (-1)^n)/8.
Kain(n) = A004526(n-1) = floor((n - 1)/2).
Win probability for Abel = sum(Abel(n)/2^n) = 2/3.
Win probability for Kain = sum(Kain(n)/2^n) = 1/3.
Mean length of the game = sum(n*a(n)/2^n) = 16/3.
Essentially the same as A035106. - R. J. Mathar, Oct 27 2011
The sequence 2*a(n) is denoted as chi(n) by McKee (1994) and is the degree of the division polynomial f_n as a polynomial in x. He notes that "If x is given weight 1, a is given weight 2, and b is given weight 3, then all the terms in f_n(a, b, x) have weight chi(n)". - Michael Somos, Jan 09 2015
In Duistermaat (2010), at the end of section 11.2 The Elliptic Billiard, on page 492 the number of k-periodic fibers counted with multiplicities of the QRT root is given by equation (11.2.8) as "1/4 k^2 + 3{k/2}(1 - {k/2}) - 1 = n^2 - 1 when k = 2n, n^2 + n when k = 2n+1, for every integer k." - Michael Somos, Mar 14 2023

			For n = 6 the a(6) = 8 solutions are (0,0,0,1,1,0), (0,1,0,1,1,0),(0,0,1,1,1,0), (1,0,1,1,1,0), (0,1,1,1,1,0),(1,1,1,1,1,0) for Abel and
  (0,0,0,1,0,0), (0,1,0,1,0,0) for Kain.
G.f. = 2*x^3 + 3*x^4 + 6*x^5 + 8*x^6 + 12*x^7 + 15*x^8 + 20*x^9 + ...

J. J. Duistermaat, Discrete Integrable Systems, 2010, Springer Science+Business Media.
A. Engel, Wahrscheinlichkeitsrechnung und Statistik, Band 2, Klett, 1978, pages 25-26.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
J. McKee, Computing division polynomials, Math. Comp. 63 (1994), 767-771. MR1248973 (95a:11110)
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000004, A002620, A004526, A004652, A005843, A008585, A008586, A023443, A028242, A047221, A047336, A052928, A242477, A265611.

Cf. A035106, A079524, A238377.

[(2*n^2-5-3*(-1)^n)/8: n in [1..60]]; // Vincenzo Librandi, Oct 28 2011

for n from 1 by 2 to 99 do
  a(n):=(n^2-1)/4:
  a(n+1):=(n+1)^2/4-1:
end do:
seq(a(n),n=1..100);

a[ n_] := Quotient[ n^2 - 1, 4]; (* Michael Somos, Jan 09 2015 *)

a(n)=([1,1,0,0,0,0;0,0,1,1,0,0;0,1,0,0,1,0;0,0,0,1,1,0;0,0,0,0,0,2;0,0,0,0,0,2]^n)[1,5] \\ Charles R Greathouse IV, Oct 26 2011

{a(n) = (n^2 - 1) \ 4}; /* Michael Somos, Jan 09 2015 */

sub a {
    my ($t, $n) = (0, shift);
    for (0..((1<<$n)-1)) {
        my $str = substr unpack("B32", pack("N", $_)), -$n;
        $t++ if ($str =~ /1.0$/ and not $str =~ /1.0./);
    }
    return $t
} # Charles R Greathouse IV, Oct 26 2011

def A198442():
    yield 0
    x, y = 0, 2
    while True:
       yield x
       x, y = x + y, x//y + 1
a = A198442(); print([next(a) for i in range(57)]) # Peter Luschny, Dec 22 2015

a(n) = (2*n^2 - 5 - 3*(-1)^n)/8.
a(2*n) = n^2 - 1; a(2*n+1) = n*(n + 1).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) with n>=4.
G.f.: x^3*(2 - x)/((1 + x)*(1 - x)^3). - R. J. Mathar, Oct 27 2011
a(n) = a(-n) for all n in Z. a(0) = -1. - Michael Somos, Jan 09 2015
0 = a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(-1 - a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jan 09 2015
1 = a(n) - a(n+1) - a(n+2) + a(n+3), 2 = a(n) - 2*a(n+2) + a(n+4) for all n in Z. - Michael Somos, Jan 09 2015
a(n) = A002620(n+2) - A052928(n+2) for n >= 1. (Note A265611(n) = A002620(n+1) + A052928(n+1) for n >= 1.) - Peter Luschny, Dec 22 2015
a(n+1) = A110654(n)^2 + A110654(n)*(2 - (n mod 2)), n >= 0. - Fred Daniel Kline, Jun 08 2016
a(n) = A004526(n)*A004526(n+3). - Fred Daniel Kline, Aug 04 2016
a(n) = floor((n^2 - 1)/4). - Bruno Berselli, Mar 15 2021

a(12) inserted by Charles R Greathouse IV, Oct 26 2011

A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 4, 4, 1, 0, 7, 7, 8, 4, 1, 0, 11, 12, 17, 13, 6, 1, 0, 15, 19, 30, 28, 18, 6, 1, 0, 22, 30, 53, 58, 50, 24, 8, 1, 0, 30, 45, 86, 109, 108, 70, 33, 8, 1, 0, 42, 67, 139, 194, 223, 179, 107, 40, 10, 1, 0, 56, 97, 213, 328, 420, 394, 286, 143, 50, 10, 1, 0

Offset: 0

Gus Wiseman, Aug 23 2023

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			Array begins:
  1  1  2   3   5    7     11
  0  1  2   4   7    12    19
  0  1  4   8   17   30    53
  0  1  4   13  28   58    109
  0  1  6   18  50   108   223
  0  1  6   24  70   179   394
  0  1  8   33  107  286   696
  0  1  8   40  143  428   1108
  0  1  10  50  199  628   1754
  0  1  10  61  254  882   2622
  0  1  12  72  332  1215  3857
  0  1  12  84  410  1624  5457
  0  1  14  99  517  2142  7637
The A(4,2) = 6 ways:
  2*2
  0*1+4*1
  1*1+3*1
  2*1+2*1
  3*1+1*1
  4*1+0*1

Alois P. Heinz, Rows n = 0..200, flattened

Row n = 0 is A000041, strict A000009.
Row n = 1 is A000070.
Column k = 0 is A000007.
Column k = 1 is A000012.
Column k = 2 is A052928 except initial terms.
Antidiagonal sums are A006951.
The case of strict integer partitions is A116861.
Main diagonal is A364907.
The transpose is A364912, also the positive version.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A066328, A108917, A237113, A364272, A364910, A364911, A364915, A365002.

b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, m)+add(b(n-i, min(i, n-i), m-i*j), j=0..m/i)))
    end:
A:= (n, k)-> b(k$2, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jan 28 2024

nn=5;
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
tabv=Table[Length[Join@@Table[combs[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,nn},{k,0,nn}]
Table[tabv[[k+1,n-k+1]],{n,0,nn},{k,0,n}]

Also the number of ways to write n-k as a *positive* linear combination of an integer partition of k.

Antidiagonals 8-11 from Alois P. Heinz, Jan 28 2024

A005582 a(n) = n(n+1)(n+2)*(n+7)/24.

Original entry on oeis.org

0, 2, 9, 25, 55, 105, 182, 294, 450, 660, 935, 1287, 1729, 2275, 2940, 3740, 4692, 5814, 7125, 8645, 10395, 12397, 14674, 17250, 20150, 23400, 27027, 31059, 35525, 40455, 45880, 51832, 58344, 65450, 73185, 81585, 90687, 100529, 111150, 122590, 134890

Offset: 0

N. J. A. Sloane

a(n) = number of Dyck (n+2)-paths with exactly 2 rows of peaks. A row of peaks is a maximal sequence of peaks all at the same height and 2 units apart. For example, UDUDUD ( = /\/\/\ ) contains exactly one row of peaks, as does UUUDDD, but UDUUDDUD has three and a(1)=2 counts UDUUDD, UUDDUD. - David Callan, Mar 02 2005
If X is an n-set and Y a fixed 2-subset of X then a(n-4) is equal to the number of (n-4)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n>=7, a(n-7) is the number of (0,1) n X n matrices A<=P^(-1)+I+P having exactly two 1's in every row and column with perA=16. - Vladimir Shevelev, Apr 12 2010
Row 2 of the convolution array A213550. - Clark Kimberling, Jun 20 2012
a(n-1) = risefac(n, 4)/4! - risefac(n, 2)/2! is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 4 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015
Consider the array formed by the second polygonal numbers of increasing rank:
  A000217(-1-n): 0,  1,  3,  6, 10, 15, ...
  A000270(-1-n): 1,  4,  9, 16, 25, 36, ...
  A000326(-1-n): 2,  7, 15, 26, 40, 57, ...
  A000384(-1-n): 3, 10, 21, 36, 55, 78, ...
Then the antidiagonal sums yield this sequence. - Michael Somos, Nov 23 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
Vladimir S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=4) (First published: San Francisco: Holden-Day, Inc., 1964)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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].
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
Milan Janjic, Two Enumerative Functions
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [See p. 301]
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
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A005581.

Cf. A000211, A052928, A128209, A176222.

[seq(binomial(n,4)+2*binomial(n,3), n=2..43)]; # Zerinvary Lajos, Jul 26 2006
seq((n+4)*binomial(n,4)/n, n=3..43); # Zerinvary Lajos, Feb 28 2007
A005582:=(-2+z)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n(n+1)(n+2)(n+7)/24,{n,0,40}] (* Harvey P. Dale, Jun 01 2012 *)

concat(0, Vec(x*(2-x)/(1-x)^5 + O(x^100))) \\ Altug Alkan, Dec 10 2015

a(n) = binomial(n+3, n-1) + binomial(n+2, n-1).
a(n) = binomial(n,4) + 2*binomial(n,3), n>=2. - Zerinvary Lajos, Jul 26 2006
From Colin Barker, Jan 28 2012: (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*(2-x)/(1-x)^5. (End)
a(n) = Sum_{k=1..n} ( Sum_{i=1..k} i(n-k+2) ). - Wesley Ivan Hurt, Sep 26 2013
a(n+1) = A127672(8+n, n), n >= 0, with the Chebyshev C-polynomial coefficients A127672(n, k). See the Abramowitz-Stegun reference.  - Wolfdieter Lang, Dec 10 2015
E.g.f.: (1/24)*x*(48 + 60*x + 16*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017
Sum_{n>=1} 1/a(n) = 853/1225. - Amiram Eldar, Jan 02 2021
a(n) = A005587(-7-n) for all n in Z. - Michael Somos, Nov 23 2021

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000

A176222 a(n) = (n^2 - 3*n + 1 + (-1)^n)/2.

Original entry on oeis.org

0, 3, 5, 10, 14, 21, 27, 36, 44, 55, 65, 78, 90, 105, 119, 136, 152, 171, 189, 210, 230, 253, 275, 300, 324, 351, 377, 406, 434, 465, 495, 528, 560, 595, 629, 666, 702, 741, 779, 820, 860, 903, 945, 990, 1034, 1081, 1127, 1176, 1224, 1275, 1325, 1378, 1430

Offset: 3

Vladimir Shevelev, Apr 12 2010

Let I = I_n be the n X n identity matrix and P = P_n be the incidence matrix of the cycle (1,2,3,...,n).
Let T = P^(-1)+I+P.
  11000...01
  11100....0
  01110.....
  00111.....
  ..........
  00.....111
  10.....011
Then a(n) is the number of (0,1) n X n matrices A <= T (i.e., an element of A can be 1 only if T has a 1 at this place) having exactly two 1's in every row and column with per(A) = 4.
a(n) is the maximum number m such that m white kings and m black kings can coexist on an n+1 X n+1 chessboard without attacking each other. - Aaron Khan, Jul 05 2022

			For n=5 the reference matrix is:
  11001
  11100
  01110
  00111
  10011
There are 2^(3*n) = 32768 0-1 matrices obtained from removing one or more 1's in it.
There are 305 such matrices with permanent 4 and there are 13 such matrices with exactly two 1's in every column and every row.
There are 5 matrices having both properties. One of them is:
  10001
  01100
  01100
  00011
  10010
From _Aaron Khan_, Jul 05 2022: (Start)
Examples of the sequence when used for kings on a chessboard:
.
A solution illustrating a(2)=3:
  +-------+
  | B B B |
  | . . . |
  | W W W |
  +-------+
.
A solution illustrating a(3)=5:
  +---------+
  | B B B B |
  | B . . . |
  | . . . W |
  | W W W W |
  +---------+
(End)

V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3 (1992), 15-19.

G. C. Greubel, Table of n, a(n) for n = 3..1000
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000211, A052928, A128209, A250000 (queens on a chessboard), A002620 (rooks on a chessboard), A355509 (knights on a chessboard).

[(n^2-3*n+1+(-1)^n)/2: n in [3..100]]; // Vincenzo Librandi, Mar 24 2011

A176222:=n->(n^2-3*n+1+(-1)^n)/2: seq(A176222(n), n=3..100); # Wesley Ivan Hurt, May 25 2015

Table[(n^2 - 3*n + 1 + (-1)^n)/2, {n, 3, 100}] (* or *) CoefficientList[Series[x (x - 3)/((1 + x)*(x - 1)^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, May 25 2015 *)
LinearRecurrence[{2,0,-2,1},{0,3,5,10},90] (* Harvey P. Dale, Jan 14 2024 *)

a(n)=(n^2-3*n+1+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n-3)/2 + ((n+1)%2) for n in (3..60)] # G. C. Greubel, Mar 22 2022

a(n) = (n - t(n))*(n - 3 + t(n))/2, where t(n) = 1-(n mod 2).
G.f.: x^4*(3-x)/( (1+x)*(1-x)^3 ). - R. J. Mathar, Mar 06 2011
From Bruno Berselli, Sep 13 2011: (Start)
a(n) + a(n+1) = A005563(n-2).
a(-n) = A084265(n). (End)
a(n) = 1 -2*n +floor(n/2) +floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
From Wesley Ivan Hurt, May 25 2015: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>4.
a(n) = Sum_{i=(-1)^n..n-2} i. (End)
a(n) = A174239(n-2) * A174239(n-1). - Paul Curtz, Jul 17 2017
With offset 0, this is ceiling(n/2)*(2*floor(n/2)+3). - N. J. A. Sloane, Jan 16 2020
E.g.f.: (1/2)*((1-x)*exp(x/2) - exp(-x/2))^2. - G. C. Greubel, Mar 22 2022

Matrix class definition checked, edited and illustrated by Olivier Gérard, Mar 26 2011

A226141 Sum of the squared parts of the partitions of n into exactly two parts.

Original entry on oeis.org

0, 2, 5, 18, 30, 64, 91, 156, 204, 310, 385, 542, 650, 868, 1015, 1304, 1496, 1866, 2109, 2570, 2870, 3432, 3795, 4468, 4900, 5694, 6201, 7126, 7714, 8780, 9455, 10672, 11440, 12818, 13685, 15234, 16206, 17936, 19019, 20940, 22140, 24262, 25585, 27918, 29370, 31924, 33511

Offset: 1

Wesley Ivan Hurt, May 27 2013

			a(5) = 30; 5 has exactly 2 partitions into two parts, (4,1) and (3,2). Squaring the parts and adding, we get: 1^2 + 2^2 + 3^2 + 4^2 = 30.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), this sequence (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Cf. A000330, A308422.

[n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8 : n in [1..80]]; // Wesley Ivan Hurt, Jun 22 2024

a:=n->sum(i^2 + (n-i)^2, i=1..floor(n/2)); seq((a(k), k=1..40);

Array[Sum[i^2 + (# - i)^2, {i, Floor[#/2]}] &, 39] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,2,5,18,30,64,91},50] (* Harvey P. Dale, Jul 23 2019 *)

a(n) = Sum_{i=1..floor(n/2)} (i^2 + (n-i)^2).
a(n) = n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8. - Giovanni Resta, May 29 2013
G.f.: x^2*(2+3*x+7*x^2+3*x^3+x^4) / ( (1+x)^3*(x-1)^4 ). - R. J. Mathar, Jun 07 2013
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7). - Wesley Ivan Hurt, Jun 22 2024
a(n) = A000330(n) - A308422(n). - Wesley Ivan Hurt, Jul 16 2025

A294270 Sum of the cubes of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 9, 44, 100, 252, 441, 848, 1296, 2150, 3025, 4572, 6084, 8624, 11025, 14912, 18496, 24138, 29241, 37100, 44100, 54692, 64009, 77904, 90000, 107822, 123201, 145628, 164836, 192600, 216225, 250112, 278784, 319634, 354025, 402732, 443556, 501068, 549081

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

a(n) is a square when n is odd. In fact: a(2*k+1) = (2*k^2 + k)^2; a(2*k) = k^2*(4*k^2 - 3*k + 1), where (2*k)^2 > 4*k^2 - 3*k + 1 > (2*k - 1)^2 for k>0. - Bruno Berselli, Nov 20 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), this sequence (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

[0] cat &cat[[k^2*(4*k^2-3*k+1),k^2*(2*k+1)^2]: k in [1..20]]; // Bruno Berselli, Nov 22 2017

Table[Sum[i^3 + (n - i)^3, {i, Floor[n/2]}], {n, 80}]

concat(0, Vec(x^2*(2 + 7*x + 27*x^2 + 28*x^3 + 24*x^4 + 7*x^5 + x^6) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Nov 20 2017

a(n) = Sum_{i=1..floor(n/2)} i^3 + (n-i)^3.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 7*x + 27*x^2 + 28*x^3 + 24*x^4 + 7*x^5 + x^6) / ((1 - x)^5*(1 + x)^4).
a(n) = n^2*(4*n^2 - 7*n + 4 + n*(-1)^n)/16.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>9. (End)

A294271 Sum of the fourth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 17, 114, 354, 1060, 2275, 4932, 8772, 15958, 25333, 41270, 60710, 91672, 127687, 182408, 243848, 333930, 432345, 572666, 722666, 931788, 1151403, 1451980, 1763020, 2182206, 2610621, 3180478, 3756718, 4514624, 5273999, 6263056, 7246096, 8515538, 9768353

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), this sequence (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

[(n*(-16 + 160*n^2 + 15*(-15 + (-1)^n)*n^3 + 96*n^4))/480 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Table[Sum[i^4 + (n - i)^4, {i, Floor[n/2]}], {n, 60}]
Table[Total[Flatten[IntegerPartitions[n,{2}]]^4],{n,40}] (* Harvey P. Dale, Mar 01 2019 *)

concat(0, Vec(x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ Colin Barker, Nov 20 2017

a(n) = sum(i=1, n\2, i^4 + (n-i)^4); \\ Michel Marcus, Nov 20 2017

a(n) = Sum_{i=1..floor(n/2)} i^4 + (n-i)^4.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5).
a(n) = (1/480)*(n*(-16 + 160*n^2 + 15*(-15 + (-1)^n)*n^3 + 96*n^4)).
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11) for n>11.
(End)

A294272 Sum of the fifth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 33, 308, 1300, 4668, 12201, 30032, 61776, 123950, 220825, 389652, 630708, 1018808, 1539825, 2331968, 3347776, 4826682, 6657201, 9233300, 12333300, 16578452, 21571033, 28256208, 35970000, 46106918, 57617001, 72503732, 89176276, 110446800, 133987425

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), this sequence (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

[(n^2*(-16 + 80*n^2 + 3*(-31 + (-1)^n)*n^3 + 32*n^4))/192 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2]}], {n, 50}]
Table[Total[Flatten[IntegerPartitions[n,{2}]]^5],{n,35}] (* or *) LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{0,2,33,308,1300,4668,12201,30032,61776,123950,220825,389652,630708},40] (* Harvey P. Dale, Jun 07 2025 *)

concat(0, Vec(x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ Colin Barker, Nov 20 2017

a(n) = Sum_{i=1..floor(n/2)} i^5 + (n-i)^5.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6).
a(n) = (1/192)*(n^2*(-16 + 80*n^2 + 3*(-31 + (-1)^n)*n^3 + 32*n^4)).
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>13.
(End)

cp's OEIS Frontend

A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A123684 Alternate A016777(n) with A000027(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A198442 Number of sequences of n coin flips that win on the last flip, if the sequence of flips ends with (1,1,0) or (1,0,0).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Perl

Sage

Formula

Extensions

A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A005582 a(n) = n*(n+1)*(n+2)*(n+7)/24.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A176222 a(n) = (n^2 - 3*n + 1 + (-1)^n)/2.

Views

Author

A005582 a(n) = n(n+1)(n+2)*(n+7)/24.