A008810 -id:A008810 - cp's OEIS frontend

A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

Suppose that S={-n,...,0,...,n} and that f(w,x,y,n) is a function, where w,x,y are in S. The number of ordered triples (w,x,y) satisfying f(w,x,y,n)=0, regarded as a function of n, is a sequence t of nonnegative integers. Sequences such as t/4 may also be integer sequences for all except certain initial values of n. In the following guide, such sequences are indicated in the related sequences column and may be included in the corresponding Mathematica programs.
...
sequence... f(w,x,y,n) ..... related sequences
A211415 ... w^2+x*y-1 ...... t+2, t/4, (t/4-1)/4
A211422 ... w^2+x*y ........ (t-1)/8, A120486
A211423 ... w^2+2x*y ....... (t-1)/4
A211424 ... w^2+3x*y ....... (t-1)/4
A211425 ... w^2+4x*y ....... (t-1)/4
A211426 ... 2w^2+x*y ....... (t-1)/4
A211427 ... 3w^2+x*y ....... (t-1)/4
A211428 ... 2w^2+3x*y ...... (t-1)/4
A211429 ... w^3+x*y ........ (t-1)/4
A211430 ... w^2+x+y ........ (t-1)/2
A211431 ... w^3+(x+y)^2 .... (t-1)/2
A211432 ... w^2-x^2-y^2 .... (t-1)/8
A003215 ... w+x+y .......... (t-1)/2, A045943
A202253 ... w+2x+3y ........ (t-1)/2, A143978
A211433 ... w+2x+4y ........ (t-1)/2
A211434 ... w+2x+5y ........ (t-1)/4
A211435 ... w+4x+5y ........ (t-1)/2
A211436 ... 2w+3x+4y ....... (t-1)/2
A211435 ... 2w+3x+5y ....... (t-1)/2
A211438 ... w+2x+2y ....... (t-1)/2, A118277
A001844 ... w+x+2y ......... (t-1)/4, A000217
A211439 ... w+3x+3y ........ (t-1)/2
A211440 ... 2w+3x+3y ....... (t-1)/2
A028896 ... w+x+y-1 ........ t/6, A000217
A211441 ... w+x+y-2 ........ t/3, A028387
A182074 ... w^2+x*y-n ...... t/4, A028387
A000384 ... w+x+y-n
A000217 ... w+x+y-2n
A211437 ... w*x*y-n ........ t/4, A007425
A211480 ... w+2x+3y-1
A211481 ... w+2x+3y-n
A211482 ... w*x+w*y+x*y-w*x*y
A211483 ... (n+w)^2-x-y
A182112 ... (n+w)^2-x-y-w
...
For the following sequences, S={1,...,n}, rather than
{-n,...,0,...n}. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A132188 ... w^2-x*y
A211506 ... w^2-x*y-n
A211507 ... w^2-x*y+n
A211508 ... w^2+x*y-n
A211509 ... w^2+x*y-2n
A211510 ... w^2-x*y+2n
A211511 ... w^2-2x*y ....... t/2
A211512 ... w^2-3x*y ....... t/2
A211513 ... 2w^2-x*y ....... t/2
A211514 ... 3w^2-x*y ....... t/2
A211515 ... w^3-x*y
A211516 ... w^2-x-y
A211517 ... w^3-(x+y)^2
A063468 ... w^2-x^2-y^2 .... t/2
A000217 ... w+x-y
A001399 ... w-2x-3y
A211519 ... w-2x+3y
A008810 ... w+2x-3y
A001399 ... w-2x-3y
A008642 ... w-2x-4y
A211520 ... w-2x+4y
A211521 ... w+2x-4y
A000115 ... w-2x-5y
A211522 ... w-2x+5y
A211523 ... w+2x-5y
A211524 ... w-3x-5y
A211533 ... w-3x+5y
A211523 ... w+3x-5y
A211535 ... w-4x-5y
A211536 ... w-4x+5y
A008812 ... w+4x-5y
A055998 ... w+x+y-2n
A074148 ... 2w+x+y-2n
A211538 ... 2w+2x+y-2n
A211539 ... 2w+2x-y-2n
A211540 ... 2w-3x-4y
A211541 ... 2w-3x+4y
A211542 ... 2w+3x-4y
A211543 ... 2w-3x-5y
A211544 ... 2w-3x+5y
A008812 ... 2w+3x-5y
A008805 ... w-2x-2y (repeated triangular numbers)
A001318 ... w-2x+2y
A000982 ... w+x-2y
A211534 ... w-3x-3y
A211546 ... w-3x+3y (triply repeated triangular numbers)
A211547 ... 2w-3x-3y (triply repeated squares)
A082667 ... 2w-3x+3y
A055998 ... w-x-y+2
A001399 ... w-2x-3y+1
A108579 ... w-2x-3y+n
...
Next, S={-n,...-1,1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated inequality. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A211545 ... w+x+y>0; recurrence degree: 4
A211612 ... w+x+y>=0
A211613 ... w+x+y>1
A211614 ... w+x+y>2
A211615 ... |w+x+y|<=1
A211616 ... |w+x+y|<=2
A211617 ... 2w+x+y>0; recurrence degree: 5
A211618 ... 2w+x+y>1
A211619 ... 2w+x+y>2
A211620 ... |2w+x+y|<=1
A211621 ... w+2x+3y>0
A211622 ... w+2x+3y>1
A211623 ... |w+2x+3y|<=1
A211624 ... w+2x+2y>0; recurrence degree: 6
A211625 ... w+3x+3y>0; recurrence degree: 8
A211626 ... w+4x+4y>0; recurrence degree: 10
A211627 ... w+5x+5y>0; recurrence degree: 12
A211628 ... 3w+x+y>0; recurrence degree: 6
A211629 ... 4w+x+y>0; recurrence degree: 7
A211630 ... 5w+x+y>0; recurrence degree: 8
A211631 ... w^2>x^2+y^2; all terms divisible by 8
A211632 ... 2w^2>x^2+y^2; all terms divisible by 8
A211633 ... w^2>2x^2+2y^2; all terms divisible by 8
...
Next, S={1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated relation.
A211634 ... w^2<=x^2+y^2
A211635 ... w^2A211790
A211636 ... w^2>=x^2+y^2
A211637 ... w^2>x^2+y^2
A211638 ... w^2+x^2+y^2A211639 ... w^2+x^2+y^2<=n
A211640 ... w^2+x^2+y^2>n
A211641 ... w^2+x^2+y^2>=n
A211642 ... w^2+x^2+y^2<2n
A211643 ... w^2+x^2+y^2<=2n
A211644 ... w^2+x^2+y^2>2n
A211645 ... w^2+x^2+y^2>=2n
A211646 ... w^2+x^2+y^2<3n
A211647 ... w^2+x^2+y^2<=3n
A063691 ... w^2+x^2+y^2=n
A211649 ... w^2+x^2+y^2=2n
A211648 ... w^2+x^2+y^2=3n
A211650 ... w^3A211790
A211651 ... w^3>x^3+y^3; see Comments at A211790
A211652 ... w^4A211790
A211653 ... w^4>x^4+y^4; see Comments at A211790

			a(1) counts these 9 triples: (-1,-1,1), (-1, 1,-1), (0, -1, 0), (0, 0, -1), (0,0,0), (0,0,1), (0,1,0), (1,-1,1), (1,1,-1).

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A120486.

t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}] (* A211422 *)
(t - 1)/8                   (* A120486 *)

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

Original entry on oeis.org

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267

Offset: 0

Clark Kimberling, Jun 01 2012

In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
  A: 2,  0, -2,  1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
  B: 3, -2, -2,  3, -1;
  C: 4, -6,  4, -1;
  D: 1,  2, -2, -1,  1;
  E: 2,  1, -4,  1,  2, -1;
  F: 2, -1,  1, -2,  1;
  G: 2, -1,  0,  1, -2,  1;
  H: 2, -1,  2, -4,  2, -1,  2, -1;
  I: 3, -3,  2, -3,  3, -1;
  J: 4, -7,  8, -7,  4, -1.
  ...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212966 ... 2*w = R
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A055998 ... w = x + y - 1
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212978 ... R = 2*n - w - x
A212979 ... R = average{w,x,y}
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A001840 ... w = 3*x + y
A212984 ... 3*w = x + y
A212985 ... 3*w = 3*x + y
A001399 ... w = 2*x + 3*y
A212986 ... 2*w = 3*x + y
A008810 ... 3*x = 2*x + y .................... F
A212987 ... 3*w = 2*x + 2*y
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A212989 ... 4*w = 4*x + y
A008812 ... 5*w = 2*x + 3*y
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A212977 ... n/2 < w + x + y <= n
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A213400 ... w < R < 2*w
A069894 ... min(|w-x|,|x-y|) = 1
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213395 ... max(|w-x|,|x-y|) = w
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213496 ... x != max(|w-x|,|x-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
A213497 ... w = min(|w-x|,|x-y|)
A213499 ... w != min(|w-x|,|x-y|)
A213501 ... w != max(|w-x|,|x-y|)
A213502 ... x != min(|w-x|,|x-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties.  Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - Philippe Deléham, Mar 16 2014

			a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A047241, A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A212959 *)

a(n)=(6*n^2+8*n+3)\/4 \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) + A212960(n) = (n+1)^3.
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - Ayoub Saber Rguez, Aug 31 2021

A056109 Fifth spoke of a hexagonal spiral.

Original entry on oeis.org

1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321, 386, 457, 534, 617, 706, 801, 902, 1009, 1122, 1241, 1366, 1497, 1634, 1777, 1926, 2081, 2242, 2409, 2582, 2761, 2946, 3137, 3334, 3537, 3746, 3961, 4182, 4409, 4642, 4881, 5126, 5377, 5634, 5897, 6166, 6441

Offset: 0

Henry Bottomley, Jun 09 2000

Squared distance from (0,0,-1) to (n,n,n) in R^3. - James R. Buddenhagen, Jun 15 2013

			Illustration of initial terms:
.
.                                                o
.                           o                 o o o o
.            o           o o o o           o o o o o o o
.   o     o o o o     o o o o o o o     o o o o o o o o o o
.            o           o o o o           o o o o o o o
.                           o                 o o o o
.                                                o
.
.   1        6              17                   34
- _Aaron David Fairbanks_, Feb 16 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Henry Bottomley, Illustration of initial terms
Tanya Khovanova, Recursive Sequences
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Leo Tavares, Triple Diamond Illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008810, A122430 (prime terms).
Other spokes: A003215, A056105, A056106, A056107, A056108.
Other spirals: A054552.
Cf. A000290.

List([0..50],n->3*n^2+2*n+1); # Muniru A Asiru, Oct 07 2018

[3*n^2 + 2*n + 1: n in [0..50]]; // Vincenzo Librandi, Mar 15 2013

seq(coeff(series(n!*(exp(x)*(3*x^2+5*x+1)),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 07 2018

Table[3 n^2 + 2 n + 1, {n, 0, 100}] (* Vincenzo Librandi, Mar 15 2013 *)
LinearRecurrence[{3,-3,1},{1,6,17},60] (* Harvey P. Dale, Mar 28 2019 *)

{a(n) = 3*n^2 + 2*n + 1}; /* Michael Somos, Aug 03 2006 */

Vec((1+3*x+2*x^2)/(1-3*x+3*x^2-x^3)+O(x^100)) \\ Stefano Spezia, Oct 17 2018

a(n) = 3n^2+2n+1 = a(n-1)+6n-1 = 2a(n-1)-a(n-2)+6 = 3a(n-1)-3a(n-2)+a(n-3) = A056105(n)+4n = A056106(n)+3n = A056107(n)+2n = A056108(n)+n = A003215(n)-n.
G.f.: (1+3*x+2*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
G.f.: (1 + x) * (1 + 2*x) / (1 - x)^3. - Michael Somos, Feb 04 2012
a(n) = A008810(3*n + 1) = A056105(-n). - Michael Somos, Aug 03 2006
E.g.f.: exp(x)*(1 + 5*x + 3*x^2). - Stefano Spezia, Oct 06 2018
a(n) = A000290(n+1) + 2*A000290(n). - Leo Tavares, May 29 2023
a(n) = A069894(n) - A000290(n+1). - Jarrod G. Sage, Jul 19 2024

A056105 First spoke of a hexagonal spiral.

Original entry on oeis.org

1, 2, 9, 22, 41, 66, 97, 134, 177, 226, 281, 342, 409, 482, 561, 646, 737, 834, 937, 1046, 1161, 1282, 1409, 1542, 1681, 1826, 1977, 2134, 2297, 2466, 2641, 2822, 3009, 3202, 3401, 3606, 3817, 4034, 4257, 4486, 4721, 4962, 5209, 5462, 5721, 5986, 6257

Offset: 0

Henry Bottomley, Jun 09 2000

Also the number of (not necessarily maximal) cliques in the n X n grid graph. - Eric W. Weisstein, Nov 29 2017

			The spiral begins:
                   49--48--47--46--45
                   /                 \
                 50  28--27--26--25  44
                 /   /             \   \
               51  29  13--12--11  24  43
               /   /   /         \   \   \
             52  30  14   4---3  10  23  42  67
             /   /   /   /     \   \   \   \   \
           53  31  15   5   1===2===9==22==41==66==>
             \   \   \   \         /   /   /   /
             54  32  16   6---7---8  21  40  65
               \   \   \             /   /   /
               55  33   17--18--19--20  39  64
                 \   \                 /   /
                 56  34--35--36--37--38  63
                   \                     /
                   57--58--59--60--61--62

G. C. Greubel, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Grid Graph
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A285792 (prime terms), A113519 (semiprime terms).
Other spokes: A003215, A056106, A056107, A056108, A056109.
Other spirals: A054552.
Cf. A000290, A000567, A008810, A033577.

List([0..50], n -> 3*n^2-2*n+1); # G. C. Greubel, Dec 02 2018

[3*n^2-2*n+1: n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014

A056105:=n->3*n^2 - 2*n + 1: seq(A056105(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014

LinearRecurrence[{3, -3, 1}, {1, 2, 9}, 50] (* Harvey P. Dale, Nov 02 2011 *)
Table[3 n^2 - 2 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(-1 + x - 6 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=3*n^2-2*n+1 /* Michael Somos, Aug 03 2006 */

[3*n^2-2*n+1 for n in range(50)] # G. C. Greubel, Dec 02 2018

a(n) = 3*n^2 - 2*n + 1.
a(n) = a(n-1) + 6*n - 5.
a(n) = 2*a(n-1) - a(n-2) + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A056106(n) - n = A056107(n) - 2*n.
a(n) = A056108(n) - 3*n = A056109(n) - 4*n = A003215(n) - 5*n.
A008810(3*n-1) = A056109(-n) = a(n). - Michael Somos, Aug 03 2006
G.f.: (1-x+6*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
From Robert G. Wilson v, Jul 05 2014: (Start)
Each of the 6 primary spokes or rays has a generating formula as stated here:
1st:  90 degrees A056105 3n^2 - 2n + 1
2nd:  30 degrees A056106 3n^2 -  n + 1
3rd: 330 degrees A056107 3n^2      + 1
4th: 270 degrees A056108 3n^2 +  n + 1
5th: 210 degrees A056109 3n^2 + 2n + 1
6th: 150 degrees A003215 3n^2 + 3n + 1
Each of the 6 secondary spokes or rays has a generating formula as stated here:
1st:  60 degrees 12n^2 - 27n + 16
2nd: 360 degrees 12n^2 - 25n + 14
3rd: 300 degrees 12n^2 - 23n + 12
4th: 240 degrees 12n^2 - 21n + 10
5th: 180 degrees 12n^2 - 19n +  8
6th: 120 degrees 12n^2 - 17n +  6 = A033577(n+1)
(End)
a(n) = 1 + A000567(n). - Omar E. Pol, Apr 26 2017
a(n) = A000290(n-1) + 2*A000290(n), n >= 1. - J. M. Bergot, Mar 03 2018
E.g.f.: (1 + x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018

A184703 T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum ix(i) equal to kn.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 3, 6, 7, 3, 1, 3, 9, 14, 12, 4, 1, 4, 12, 28, 34, 21, 5, 1, 4, 17, 46, 78, 74, 35, 6, 1, 5, 22, 74, 156, 207, 154, 58, 8, 1, 5, 27, 107, 282, 476, 504, 304, 91, 10, 1, 6, 34, 154, 471, 985, 1349, 1169, 580, 142, 12, 1, 6, 41, 208, 744, 1842, 3142, 3571, 2574, 1066, 215, 15

Offset: 1

R. H. Hardin, Jan 20 2011

T(n,k) is the number of integer lattice points in k*P_n, where P_n is the polytope in R^n defined by the constraints 0 <= x_i <= 1 and Sum_{i=1..n} i x_i = n. Thus for each n, T(n,k) is an Ehrhart quasi-polynomial. - Robert Israel, Dec 21 2022

			Table starts:
   1   1    1    1     1     1      1      1       1       1       1       1
   1   2    2    3     3     4      4      5       5       6       6       7
   2   3    6    9    12    17     22     27      34      41      48      57
   2   7   14   28    46    74    107    154     208     278     357     456
   3  12   34   78   156   282    471    744    1119    1623    2279    3118
   4  21   74  207   476   985   1842   3226    5325    8414   12766   18789
   5  35  154  504  1349  3142   6575  12688   22923   39266   64315  101460
   6  58  304 1169  3571  9353  21713  46037   90595  167917  295811  499442
   8  91  580 2574  8939 26146  67105 155645  332729  665317 1257898 2268061
  10 142 1066 5439 21310 69331 195760 495251 1146377 2467215 4994696 9599863
Some solutions for n=5, k=4:
  4   4   2   0   1   2   1   0   2   4   0   0   4   0   2   2
  1   2   3   1   2   0   2   0   1   1   0   0   0   2   2   1
  2   1   4   0   1   2   2   1   4   0   4   0   4   0   2   2
  2   1   0   2   3   3   1   3   1   1   2   0   1   4   2   0
  0   1   0   2   0   0   1   1   0   2   0   4   0   0   0   2

R. H. Hardin, Table of n, a(n) for n = 1..2113

Column 1 is A000009.

Row 3 is A008810(n+1).

S:= proc(n,k,s) option remember; local j;
  if n = 1 then
    if s <= k then return 1 else return 0 fi
  fi;
  add(procname(n-1,k,s-j*n), j=0..min(s/n,k))
end proc:
[seq(seq(S(n,m-n,(m-n)*n),n=1..m-1),m=1..20)]; # Robert Israel, Dec 21 2022

S[n_, k_, s_] := S[n, k, s] = Module[{}, If[n == 1, If[s <= k, Return@1, Return@0]]; Sum[S[n - 1, k, s - j*n], {j, 0, Min[s/n, k]}]];
Table[Table[S[n, m - n, (m - n)*n], {n, 1, m - 1}], {m, 1, 20}] // Flatten (* Jean-François Alcover, Aug 21 2023, after Robert Israel *)

A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 3, 6, 7, 0, 0, 1, 0, 9, 0, 15, 0, 1, 0, 3, 12, 31, 0, 33, 8, 0, 1, 0, 17, 0, 107, 0, 77, 0, 1, 0, 5, 22, 81, 0, 395, 410, 181, 0, 0, 1, 0, 27, 0, 397, 0, 1525, 0, 443, 0, 1, 0, 5, 34, 171, 0, 2073, 4508, 6095, 0, 1113, 58, 0, 1, 0, 41, 0, 1081, 0

Offset: 1

R. H. Hardin, May 05 2013

Table starts
.0...1...0.....1....0......1.....0.......1.....0........1......0........1
.0...1...0.....1....0......1.....0.......1.....0........1......0........1
.0...1...0.....3....0......3.....0.......5.....0........5......0........7
.2...3...6.....9...12.....17....22......27....34.......41.....48.......57
.0...7...0....31....0.....81.....0.....171.....0......309......0......509
.0..15...0...107....0....397.....0....1081.....0.....2399......0.....4675
.0..33...0...395....0...2073.....0....7261.....0....19709......0....45385
.8..77.410..1525.4508..11291.25056...50659.95130...168289.283338...457627
.0.181...0..6095....0..63121.....0..364051.....0..1478059......0..4749875
.0.443...0.24893....0.360909.....0.2676331.....0.13280209......0.50435657

			Some solutions for n=4, k=4
.-1.-1.-1..1...-1.-1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1
.-1..1..1..1...-1..1..1..1...-1.-1.-1..1....1..1..1..1....1..1..1..1
.-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1....1..1..1..1...-1.-1.-1..1
.-1.-1.-1..1...-1..1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..3132

Column 1 is A063074(n/4).
Row 3 is A063196(n/2+1).
Row 4 is A008810(n+1).
Row 5 is A202254(n/2).

Empirical for row n:
n=1: a(n) = a(n-2);
n=2: a(n) = a(n-2);
n=3: a(n) = a(n-2) +a(n-4) -a(n-6);
n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5);
n=5: a(n) = 3*a(n-2) -2*a(n-4) -2*a(n-6) +3*a(n-8) -a(n-10);
n=6: [order 26, even n];
n=7: [order 42, even n];
n=8: [order 28];
n=9: [order 58, even n];
n=10: [order 90, even n];
n=11: [order 102, even n];
n=12: [order 66].

A229093 The clubs patterns appearing in n X n coins.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046

Offset: 0

Kival Ngaokrajang, Sep 13 2013

On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.

a(n) is the number of clubs patterns appearing in n X n coins. It is also A008810(n-1), except for the third term. The inverse patterns (stars or voids between clubs) is A030511 (except the second term). See illustration in links.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration for initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A008810, A030511, A074148 (heart patterns), A227906, A229154.

CoefficientList[Series[(x^7 - 2 x^6 + x^5 - x^4 + x^3 - x^2 - 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 08 2013 *)
LinearRecurrence[{2,-1,1,-2,1},{0,0,1,2,4,6,9,12,17,22},70] (* Harvey P. Dale, Feb 05 2020 *)

Vec(x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013

a(n) = ceil((n-1)^2/3) \\ Charles R Greathouse IV, Jan 06 2016

a(n) = ceiling((n-1)^2/3), a(0) = 0, a(4) = 4.

G.f.: x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Oct 07 2013

More terms from Colin Barker, Oct 08 2013

A008812 Expansion of (1+x^5)/((1-x)^2*(1-x^5)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 14, 17, 20, 25, 30, 35, 40, 45, 52, 59, 66, 73, 80, 89, 98, 107, 116, 125, 136, 147, 158, 169, 180, 193, 206, 219, 232, 245, 260, 275, 290, 305, 320, 337, 354, 371, 388, 405, 424, 443, 462, 481, 500, 521, 542, 563, 584, 605, 628, 651, 674

Offset: 0

N. J. A. Sloane

Number of 0..n arrays of six elements with zero second differences. - R. H. Hardin, Nov 16 2011
Also number of ordered triples (w,x,y) with all terms in {1,...,n+1} and w + 4*x = 5*y. Also the number of 3-tuples (w,x,y) with all terms in {1,...,n+1} and 5*w = 2*x +3*y. - Clark Kimberling, Apr 15 2012 [Corrected by Pontus von Brömssen, Jan 26 2020]
a(n) is also the number of 5 boxes polyomino (zig-zag patterns) packing into (n+3) X (n+3) square. See illustration in links. - Kival Ngaokrajang, Nov 10 2013
Also, number of ordered pairs (x,y) with both terms in {1,...,n+1} and x+4*y divisible by 5; or number of ordered pairs (x,y) with both terms in {1,...,n+1} and 2*x+3*y divisible by 5. - Pontus von Brömssen, Jan 26 2020

			For n = 5 there are 8 0..5 arrays of six elements with zero second differences: [0,0,0,0,0,0], [0,1,2,3,4,5], [1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3,3,3], [4,4,4,4,4,4], [5,4,3,2,1,0], [5,5,5,5,5,5].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A130497 (first differences).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), this sequence (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,8,11];; for n in [8..65] do a[n]:=2*a[n-1]-a[n-2] +a[n-5]-2*a[n-6]+a[n-7]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1+x^5)/((1-x)^2*(1-x^5)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^5)/((1-x)^2*(1-x^5)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^5)/(1-x)^2/(1-x^5),{x,0,65}],x] (* or *) LinearRecurrence[{2,-1,0,0,1,-2,1}, {1,2,3,4,5,8,11}, 65] (* Harvey P. Dale, Apr 17 2015 *)

Vec((1+x^5)/(1-x)^2/(1-x^5)+O(x^65)) \\ Charles R Greathouse IV, Sep 25 2012

def A008812_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^5)/((1-x)^2*(1-x^5))).list()
A008812_list(65) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^5)/((1-x)^2*(1-x^5)).

a(n) = 2*a(n-1) -a(n-2) +a(n-5) -2*a(n-6) +a(n-7). - R. H. Hardin, Nov 16 2011

More terms added by G. C. Greubel, Sep 12 2019

A008811 Expansion of x(1+x^4)/((1-x)^2(1-x^4)).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 10, 13, 16, 21, 26, 31, 36, 43, 50, 57, 64, 73, 82, 91, 100, 111, 122, 133, 144, 157, 170, 183, 196, 211, 226, 241, 256, 273, 290, 307, 324, 343, 362, 381, 400, 421, 442, 463, 484, 507, 530, 553, 576, 601, 626, 651, 676, 703, 730, 757, 784, 813

Offset: 0

N. J. A. Sloane

Number of 0..n-1 arrays of 5 elements with zero 2nd differences. - R. H. Hardin, Nov 15 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Daniel Gabric and Joe Sawada, Investigating the discrepancy property of de Bruijn sequences, University of Guelph (Canada, 2020).
János Pach and Pankaj K. Agarwal, Combinatorial Geometry, p. 220, 1995, Problem 13.10.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A129756 (first differences).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), this sequence (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[0,1,2,3,4,7];; for n in [7..60] do a[n]:=2*a[n-1]-a[n-2] +a[n-4]-2*a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1+x^4)/((1-x)^2*(1-x^4)) )); // G. C. Greubel, Sep 12 2019

f := n->n^2/4+3*n/2+g(n);
g := n->if n mod 2 = 0 then 3 elif n mod 4 = 1 then 9/4 else 13/4; fi;
seq(f(n), n=-3..50);

CoefficientList[Series[x*(1+x^4)/((1-x)^2*(1-x^4)), {x,0,60}], x] (* G. C. Greubel, Sep 12 2019 *)

concat([0], Vec(x*(1+x^4)/((1-x)^2*(1-x^4))+O(x^60))) \\ Charles R Greathouse IV, Sep 26 2012, modified by G. C. Greubel, Sep 12 2019

def A008811_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+x^4)/((1-x)^2*(1-x^4))).list()
A008811_list(60) # G. C. Greubel, Sep 12 2019

G.f.: x*(1+x^4)/((1-x)^2*(1-x^4)).
a(n) = 2*a(n-1) -a(n-2) +a(n-4) -2*a(n-5) +a(n-6). - R. H. Hardin, Nov 15 2011
a(n) = (-2*(1+(-1)^n)*(-1)^floor(n/2) + 2*n^2 + 5 - (-1)^n)/8. - Tani Akinari, Jul 24 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x) - 2*cos(x))/4. - Stefano Spezia, May 26 2021
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/2)*Pi/4 + tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - Amiram Eldar, Aug 25 2022
a(n) = 2*floor((n^2 + 4)/8) + (n mod 2). - Ridouane Oudra, Sep 08 2023

A225310 T(n,k)=Number of nXk -1,1 arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero and rows are nondecreasing (ways to put k thrusters pointing east or west at each of n positions 1..n distance from the hinge of a south-pointing gate without turning the gate).

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 1, 0, 3, 2, 0, 3, 6, 7, 0, 1, 0, 9, 16, 15, 0, 0, 3, 12, 31, 0, 35, 8, 1, 0, 17, 52, 113, 0, 87, 14, 0, 5, 22, 83, 0, 443, 474, 217, 0, 1, 0, 27, 122, 427, 0, 1787, 1576, 547, 0, 0, 5, 34, 175, 0, 2341, 5304, 7445, 0, 1417, 70, 1, 0, 41, 238, 1165, 0, 13333, 26498

Offset: 1

R. H. Hardin May 05 2013

Table starts
..0....1....0......1.....0.......1......0........1......0.........1.......0
..0....1....0......3.....0.......3......0........5......0.........5.......0
..2....3....6......9....12......17.....22.......27.....34........41......48
..2....7...16.....31....52......83....122......175....238.......317.....410
..0...15....0....113.....0.....427......0.....1165......0......2591.......0
..0...35....0....443.....0....2341......0.....8221......0.....22351.......0
..8...87..474...1787..5304...13333..29638....60007.112790....199669..336342
.14..217.1576...7445.26498...77721.197440...449693.939130...1828785.3360554
..0..547....0..31593.....0..461973......0..3437315......0..17085339.......0
..0.1417....0.136351.....0.2791167......0.26700429......0.162204059.......0

			Some solutions for n=4 k=4
..1..1..1..1...-1.-1..1..1...-1.-1.-1..1....1..1..1..1...-1..1..1..1
..1..1..1..1...-1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1...-1.-1..1..1
.-1.-1.-1.-1....1..1..1..1...-1..1..1..1....1..1..1..1...-1..1..1..1
.-1.-1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1.-1.-1..1...-1.-1.-1..1

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A063865
Column 2 is A007576
Row 3 is A008810(n+1)

Empirical for row n:
n=1: a(n) = a(n-2)
n=2: a(n) = a(n-2) +a(n-4) -a(n-6)
n=3: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5)
n=4: a(n) = a(n-1) +a(n-2) -2*a(n-5) +a(n-8) +a(n-9) -a(n-10)
n=5: [order 18]
n=6: [order 42]
n=7: [order 24]
n=8: [order 36]

Showing 1-10 of 28 results. Next

cp's OEIS Frontend

A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

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A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

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A056109 Fifth spoke of a hexagonal spiral.

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A056105 First spoke of a hexagonal spiral.

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A184703 T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum i*x(i) equal to k*n.

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A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A229093 The clubs patterns appearing in n X n coins.

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A184703 T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum ix(i) equal to kn.

A008811 Expansion of x(1+x^4)/((1-x)^2(1-x^4)).