A001971 -id:A001971 - cp's OEIS frontend

A261491 a(n) = ceiling(2 + sqrt(8*n-4)).

4, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 28, 29

Offset: 1

Juhani Heino, Aug 21 2015

Conjecture: a(n) = minimal number of stones needed to surround area n in the middle of a Go board (infinite if needed).
The formula was constructed this way: when the area is in a diamond shape with x^2+(x-1)^2 places, it can be surrounded by 4x stones. So, a(1)=4, a(5)=8, a(13)=12 etc.
The positive solution to the quadratic equation 2x^2 - 2x + 1 = n is x = (2 + sqrt(8n-4))/4. And since a(n)=4x, the formula a(n) = 2 + sqrt(8n-4) holds for the positions mentioned. But incredibly also the intermediate results seem to match when the ceiling function is used.
The opposite of this would be an area of 1 X n; it demands the maximal number of stones, a(n) = 2 + 2n.
Equivalently, a(n) is the minimum (cell) perimeter of any polyomino of n cells. - Sean A. Irvine, Oct 17 2020

			Start with the 5-cell area that is occupied by 0's and surrounded by stones 1..8. Add those surrounding stones to the area, one by one. At points 1, 2, 4 and 6, the number of surrounding stones is increased; elsewhere, it is not.
Next, do the same with stones A..L. At points A, C, F and I, the number of surrounding stones is increased; elsewhere, it is not.
___D___
__A5C__
_B104E_
G30007J
_F206I_
__H8K__
___L___

Kival Ngaokrajang, Illustration of initial terms

Cf. A001971.

Array[Ceiling[2 + Sqrt[8 # - 4]] &, {86}] (* Michael De Vlieger, Oct 23 2015 *)

a(n)=sqrtint(8*n-5)+3 \\ Charles R Greathouse IV, Aug 21 2015

a(n) = ceiling(2 + sqrt(8*n-4)).

For n > 2, a(n) - a(n-1) = 1 if n is of the form 2*(k^2+k+1), 2*k^2 + 1 or (k^2+k)/2 + 1, otherwise 0. - Jianing Song, Aug 10 2021

A001975 Number of partitions of floor(5n/2) into n nonnegative integers each no more than 5.

Original entry on oeis.org

1, 1, 3, 6, 12, 20, 32, 49, 73, 102, 141, 190, 252, 325, 414, 521, 649, 795, 967, 1165, 1394, 1651, 1944, 2275, 2649, 3061, 3523, 4035, 4604, 5225, 5910, 6660, 7483, 8372, 9343, 10395, 11538, 12764, 14090, 15516, 17053, 18691, 20451, 22330, 24342, 26476, 28754

Offset: 0

N. J. A. Sloane

In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(5n/2) involving the letters a, b, c, d, e, f, having weights 0, 1, 2, 3, 4, 5 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Shalosh B. Ekhad, Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,2,-2,2,-2,0,2,-2,2,-2,2,-1,0,1,-2,1).

LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 2, -2, 0, 2, -2, 2, -2, 2, -1, 0, 1, -2, 1}, {1, 1, 3, 6, 12, 20, 32, 49, 73, 102, 141, 190, 252, 325, 414, 521, 649, 795, 967, 1165}, 50] (* Jean-François Alcover, Feb 26 2020 *)

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)); n=350; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(5*d/2);print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)), where w=floor(5n/2). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: -(x^14 -x^13 +2*x^12 +x^11 +2*x^10 +3*x^9 +x^8 +5*x^7 +x^6 +3*x^5 +2*x^4 +x^3 +2*x^2 -x+1) / ((x^4+1) *(x^2+x+1) *(x^2-x+1) *(x^2+1)^2 *(x+1)^3 *(x-1)^5). - Alois P. Heinz, Jul 25 2015

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A220691 Table A(i,j) read by antidiagonals in order A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ..., where A(i,j) is the number of ways in which we can add 2 distinct integers from the range 1..i in such a way that the sum is divisible by j.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 1, 1, 6, 0, 0, 1, 2, 10, 0, 0, 1, 2, 4, 15, 0, 0, 1, 1, 4, 6, 21, 0, 0, 0, 2, 2, 5, 9, 28, 0, 0, 0, 1, 2, 3, 7, 12, 36, 0, 0, 0, 1, 2, 3, 5, 10, 16, 45, 0, 0, 0, 0, 2, 2, 4, 6, 12, 20, 55, 0, 0, 0, 0, 1, 3, 3, 6, 8, 15, 25, 66, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Feb 18 2013

			The upper left corner of this square array starts as:
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
   1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
   3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, ...
   6, 2, 2, 1, 2, 1, 1, 0, 0, 0, 0, ...
  10, 4, 4, 2, 2, 2, 2, 1, 1, 0, 0, ...
  15, 6, 5, 3, 3, 2, 3, 2, 2, 1, 1, ...
Row 1 is all zeros, because it's impossible to choose two distinct integers from range [1]. A(2,1) = 1, as there is only one possibility to choose a pair of distinct numbers from the range [1,2] such that it is divisible by 1, namely 1+2. Also A(2,3) = 1, as 1+2 is divisible by 3.
A(4,1) = 2, as from [1,2,3,4] one can choose two pairs of distinct numbers whose sum is even: {1+3} and {2+4}.

A. Karttunen, The first 150 antidiagonals of the square array, flattened
Stackexchange, Question 142323
Index entries for sequences related to subset sums modulo m

Transpose: A220692. The lower triangular region of this square array is given by A061857, which leaves out about half of the nonzero terms. A220693 is another variant giving 2n-1 terms from the beginning of each row, thus containing all the nonzero terms of this array.

The left column of the table: A000217. The following cases should be checked: the second column: A002620, the third column: A058212 (after the first two terms), the fourth column: A001971.

a[n_, 1] := n*(n-1)/2; a[n_, k_] := Module[{r}, r = Reduce[1 <= i < j <= n && Mod[i + j, k] == 0, {i, j}, Integers]; Which[Head[r] === Or, Length[r], Head[r] === And, 1, r === False, 0, True, Print[r, " not parsed"]]]; Table[a[n-k+1, k], {n, 1, 13} , {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 04 2014 *)

See Robert Israel's formula at A061857.

A001979 Number of partitions of floor(7n/2) into n nonnegative integers each no more than 7.

Original entry on oeis.org

1, 1, 4, 10, 24, 49, 94, 169, 289, 468, 734, 1117, 1656, 2385, 3370, 4672, 6375, 8550, 11322, 14800, 19138, 24460, 30982, 38882, 48417, 59779, 73316, 89291, 108108, 130053, 155646, 185258, 219489, 258735, 303748, 355034, 413442, 479500, 554256

Offset: 0

N. J. A. Sloane

Also, the dimension of the vector space of homogeneous covariants of degree n for the binary form of degree 7. To calculate the dimension one uses the Sylvester-Cayley formula. - Leonid Bedratyuk, Dec 06 2006

In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(7n/2) involving the letters a, b, c, d, e, f, g, h, having weights 0, 1, 2, 3, 4, 5, 6, 7 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
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).
Springer, T. A., Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, (1977).
Hilbert, D., Theory of algebraic invariants. Lectures. Cambridge University Press, (1993).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Shalosh B. Ekhad, Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 1, -2, 2, -2, 2, -2, 1, 0, 0, 0, -2, 4, -4, 4, -3, 2, -1, 0, 1, -2, 3, -4, 4, -4, 2, 0, 0, 0, -1, 2, -2, 2, -2, 2, -1, 0, 1, -2, 1).

Cf. A001980.

a(n+1) = subs({x=1},convert(series((product('1-x^i','i'=8..7+n)/product('1-x^k','k'=2..n)),x,trunc(7*n/2)+1),polynom)); # Leonid Bedratyuk, Dec 06 2006

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)*(1-x^7*z)); n=450; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(7*d/2);print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)(1-x^7z)), where w=floor(7n/2). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: -(x^34 -x^33 +3*x^32 +3*x^31 +7*x^30 +12*x^29 +16*x^28 +28*x^27 +33*x^26 +46*x^25 +56*x^24 +73*x^23 +83*x^22 +90*x^21 +106*x^20 +109*x^19 +121*x^18 +110*x^17 +121*x^16 +109*x^15 +106*x^14 +90*x^13 +83*x^12 +73*x^11 +56*x^10 +46*x^9 +33*x^8 +28*x^7 +16*x^6 +12*x^5 +7*x^4 +3*x^3 +3*x^2 -x+1) / ((x^4-x^2+1) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x^4+1) *(x^2+x+1)^2 *(x^2-x+1)^2 *(x^2+1)^3 *(x+1)^5 *(x-1)^7). - Alois P. Heinz, Jul 25 2015

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A173722 Partial sums of round(n^2/8).

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 12, 18, 26, 36, 49, 64, 82, 103, 128, 156, 188, 224, 265, 310, 360, 415, 476, 542, 614, 692, 777, 868, 966, 1071, 1184, 1304, 1432, 1568, 1713, 1866, 2028, 2199, 2380, 2570, 2770, 2980, 3201, 3432, 3674, 3927, 4192, 4468, 4756, 5056, 5369

Offset: 0

Mircea Merca, Nov 26 2010

Partial sums of A001971.

			a(5) = round(1/8) + round(4/8) + round(9/8) + round(16/8) + round(25/8) = 0 + 1 + 1 + 2 + 3 = 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Cf. A001971.

A057077 := proc(n) op( 1+(n mod 4),[1,1,-1,-1]) ; end proc:
A173722 := proc(n) 3*(-1)^n/32+n^2/16+n/12+n^3/24+1/32-A057077(n)/8 ; end proc:
seq(A173722(n),n=0..80) ; # R. J. Mathar, Nov 26 2010

Table[Floor[(n + 2)*(2*n^2 - n + 6)/48], {n,0,50}] (* G. C. Greubel, Nov 29 2016 *)

a(n)=(n+2)*(2*n^2-n+6)\48 \\ Charles R Greathouse IV, Oct 19 2022

a(n) = Sum_{k=0..n} round(k^2/8).
a(n) = round((2*n^3+3*n^2+4*n)/48).
a(n) = round((2*n+1)*(2*n^2+2*n+3)/96).
a(n) = floor((n+2)*(2*n^2-n+6)/48).
a(n) = ceiling((2*n^3+3*n^2+4*n-9)/48).
a(n) = a(n-4)+n*(n-3)/2+2, n>3.
G.f.: x^2*(1-x+x^2) / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Nov 26 2010
a(n) = 3*(-1)^n/32+n^2/16+n/12+n^3/24+1/32-A057077(n)/8. - R. J. Mathar, Nov 26 2010

A366443 Number of free polyominoes of site-perimeter n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 5, 5, 23, 46, 187, 552, 2145, 7818

Offset: 1

John Mason from an idea of Allan C. Wechsler, Oct 10 2023

This sequence counts free connected (via common edges) polyominoes with given site-perimeter. The site-perimeter of a polyomino is the number of cells that are adjacent to it (via common edges). This sequence allows holes of any kind.

			a(4) = a(6) = a(7) = 1 as the monomino, domino and L-shaped tromino are the only polyominoes with site perimeter 4, 6 and 7 respectively.
a(5) = 0 as no polyomino has a site-perimeter of 5.
a(8) = 5 as the straight tromino, square tetromino, T-tetromino, S-tetromino and cross pentomino are the only polyominoes with site perimeter 8. See link "Examples".

John Mason, Examples

Cf. A000105 (free polyominoes), A001971 (the maximum size of a polyomino with site-perimeter n is given by A001971(n-2)), A057730 (perimeter instead of site-perimeter), A216820 (fixed version of current sequence).

Column sums of A338211 (without the column for 0-celled polyominoes).

a(15) corrected by Sean A. Irvine, Apr 13 2025

A001980 Number of partitions of floor(7n/2)-1 into n nonnegative integers each no greater than 7.

Original entry on oeis.org

0, 1, 4, 10, 23, 48, 94, 166, 285, 464, 734, 1109, 1646, 2371, 3366, 4652, 6357, 8519, 11309, 14754, 19103, 24399, 30956, 38797, 48355, 59665, 73264, 89145, 108011, 129864, 155554, 185017, 219336, 258438, 303604, 354665, 413213, 479048, 554033

Offset: 0

N. J. A. Sloane

In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(7n/2)-1 involving the letters a, b, c, d, e, f, g, h, having weights 0, 1, 2, 3, 4, 5, 6, 7 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 86 terms from Vincenzo Librandi)
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 2, -1, -2, 0, -1, 0, 1, 2, 2, -1, -2, 1, -4, 0, 4, -1, 2, 1, -2, -2, -1, 0, 1, 0, 2, 1, -2, 1, -1, -1, 1).

Cf. A001979.

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)*(1-x^7*z)); n=400; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(7*d/2)-1;print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)(1-x^7z)), where w=floor(7n/2)-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: -(x^24 +3*x^23 +5*x^22 +10*x^21 +17*x^20 +26*x^19 +33*x^18 +45*x^17 +55*x^16 +61*x^15 +63*x^14 +68*x^13 +67*x^12 +68*x^11 +63*x^10 +61*x^9 +55*x^8 +45*x^7 +33*x^6 +26*x^5 +17*x^4 +10*x^3 +5*x^2 +3*x +1)*x / ((x^4+x^3+x^2+x+1) *(x^4-x^2+1) *(x^2+x+1)^2 *(x^2-x +1)^2 *(x^2+1)^3 *(x+1)^5 *(x-1)^7). - Alois P. Heinz, Jul 25 2015

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

a(0)=0 inserted by Alois P. Heinz, Jul 25 2015

A243883 Numerator of circle radius r(n) at constant unit length sagitta and chord length = n.

Original entry on oeis.org

5, 1, 13, 5, 29, 5, 53, 17, 85, 13, 125, 37, 173, 25, 229, 65, 293, 41, 365, 101, 445, 61, 533, 145, 629, 85, 733, 197, 845, 113, 965, 257, 1093, 145, 1229, 325, 1373, 181, 1525, 401, 1685, 221, 1853, 485, 2029, 265, 2213, 577, 2405, 313, 2605, 677, 2813, 365, 3029

Offset: 1

Kival Ngaokrajang, Jun 13 2014

Denominator of circle radius r(n) is A143025(n+2). The integral radius appearing at n = 2, 6, 10, 14, ..., = 1, 5, 13, 25, ..., respectively which is A001844(n/4 - 1/2). Floor (r(n)) = A001971(n). For the case of sagitta = n and chord length = 1, the numerator and the denominator will be A053755(n) and A008590(n) respectively. See illustration in links.

Colin Barker, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration for n = 1..5
Wikipedia, Sagitta

Cf. A143025, A001844, A001971.

PARI
```
a(n) = numerator(n^2/8+1/2);
```

a(n) = numerator(n^2/8 + 1/2).

Empirical g.f.: -x*(x^11 +5*x^10 +x^9 +13*x^8 +2*x^7 +14*x^6 +2*x^5 +14*x^4 +5*x^3 +13*x^2 +x +5) / ((x -1)^3*(x +1)^3*(x^2 +1)^3). - Colin Barker, Jan 17 2015

A001976 Number of partitions of floor(5n/2)-1 into n nonnegative integers each no more than 5.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 32, 48, 71, 101, 141, 188, 249, 322, 414, 518, 645, 791, 966, 1160, 1389, 1645, 1943, 2268, 2642, 3053, 3521, 4026, 4596, 5214, 5907, 6648, 7473, 8359, 9339, 10380, 11526, 12747, 14085, 15498, 17039, 18671, 20444, 22308, 24326, 26452, 28746

Offset: 0

N. J. A. Sloane

In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(5n/2)-1 involving the letters a, b, c, d, e, f, having weights 0, 1, 2, 3, 4, 5 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 1, -2, 2, -2, 2, -2, 0, 2, -2, 2, -2, 2, -1, 0, 1, -2, 1).

Cf. A001975.

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)); n=350; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(5*d/2)-1;print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)), where w=floor(5n/2)-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: -(x^12 +x^11 +x^10 +2*x^9 +2*x^8 +4*x^7 +x^6 +4*x^5 +2*x^4 +2*x^3 +x^2 +x+1)*x / ((x^4+1) *(x^2+x+1) *(x^2-x+1) *(x^2+1)^2 *(x+1)^3 *(x-1)^5). - Alois P. Heinz, Jul 25 2015

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

a(0)=0 inserted by Alois P. Heinz, Jul 25 2015

A001978 Number of partitions of 3n-1 into n nonnegative integers each no more than 6.

Original entry on oeis.org

0, 1, 3, 8, 16, 32, 55, 94, 147, 227, 332, 480, 668, 920, 1232, 1635, 2124, 2738, 3470, 4368, 5424, 6695, 8172, 9922, 11934, 14287, 16968, 20068, 23572, 27584, 32087, 37199, 42901, 49325, 56450, 64424, 73223, 83012, 93764, 105661, 118674, 133003, 148616

Offset: 0

N. J. A. Sloane

In Cayley's terminology, this is the number of literal terms of degree n and of weight 3n-1 involving the letters a, b, c, d, e, f, g, having weights 0, 1, 2, 3, 4, 5, 6 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (1, 3, -2, -4, 1, 3, -1, -1, 3, 1, -4, -2, 3, 1, -1).

Cf. A001977.

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)); n=400; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=3*d-1;print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)), where w=3n-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: (x^6 +2*x^5 +2*x^4 +x^3 +2*x^2 +2*x+1)*x / ((x^2+x+1) *(x^4+x^3+x^2+x+1) *(x+1)^3 *(x-1)^6). - Alois P. Heinz, Jul 25 2015

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

a(0)=0 inserted by Alois P. Heinz, Jul 25 2015

cp's OEIS Frontend

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A001975 Number of partitions of floor(5n/2) into n nonnegative integers each no more than 5.

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A220691 Table A(i,j) read by antidiagonals in order A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ..., where A(i,j) is the number of ways in which we can add 2 distinct integers from the range 1..i in such a way that the sum is divisible by j.

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A001979 Number of partitions of floor(7n/2) into n nonnegative integers each no more than 7.

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A173722 Partial sums of round(n^2/8).

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A366443 Number of free polyominoes of site-perimeter n.

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A001980 Number of partitions of floor(7n/2)-1 into n nonnegative integers each no greater than 7.

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