A000982 -id:A000982 - cp's OEIS frontend

A250320 T(n,k)=Number of length n+2 0..k arrays with the sum of second differences squared multiplied by some arrangement of +-1 equal to zero.

2, 5, 8, 8, 25, 8, 13, 60, 41, 24, 18, 117, 104, 161, 42, 25, 200, 233, 652, 487, 104, 32, 321, 436, 1773, 2432, 1689, 212, 41, 480, 745, 3916, 8767, 12820, 5849, 464, 50, 681, 1152, 7969, 24126, 57833, 61092, 19981, 950, 61, 940, 1733, 14452, 57305, 197848

Offset: 1

R. H. Hardin, Nov 18 2014

Table starts
....2......5.......8.......13........18.........25.........32........41
....8.....25......60......117.......200........321........480.......681
....8.....41.....104......233.......436........745.......1152......1733
...24....161.....652.....1773......3916.......7969......14452.....24293
...42....487....2432.....8767.....24126......57305.....119004....228401
..104...1689...12820....57833....197848.....558541....1357424...2953265
..212...5849...61092...363457...1559080....5237161...14866258..37065983
..464..19981..300616..2317841..12424332...50020061..166783380.476368553
..950..67459.1423966.14305925..95711098..461868677.1809575752
.1968.221953.6523576.85334033.709795516.4110975765

			Some solutions for n=5 k=4
..2....2....4....4....0....3....1....4....2....1....4....0....4....4....2....2
..3....3....3....4....2....4....0....3....4....1....2....0....0....4....2....1
..2....0....3....1....0....0....2....0....3....0....2....2....2....3....2....1
..1....4....0....0....1....4....4....1....4....1....0....4....1....4....1....0
..4....1....1....0....3....0....2....0....3....0....2....0....3....0....3....4
..3....2....0....2....4....1....4....1....2....2....3....0....2....2....3....2
..4....3....1....0....2....2....3....4....4....4....1....4....4....0....1....3

R. H. Hardin, Table of n, a(n) for n = 1..127

Row 1 is A000982(n+1)

Empirical for column k:
k=1: a(n) = 3*a(n-1) -6*a(n-3) +3*a(n-4) +3*a(n-5) -2*a(n-6)
Empirical for row n:
n=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2
n=2: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) +2*a(n-7) -a(n-8); also a cubic polynomial plus a linear quasipolynomial with period 3

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

A208825 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 5, 5, 1, 5, 8, 16, 7, 1, 6, 13, 38, 45, 18, 1, 7, 18, 75, 155, 167, 32, 1, 8, 25, 131, 415, 828, 609, 84, 1, 9, 32, 210, 905, 2821, 4390, 2471, 185, 1, 10, 41, 316, 1755, 7582, 19657, 25202, 10143, 486, 1, 11, 50, 453, 3085, 17339, 65134, 144871

Offset: 1

R. H. Hardin, Mar 01 2012

Table starts
..1....1.....1......1......1.......1.......1........1........1........1
..2....3.....4......5......6.......7.......8........9.......10.......11
..2....5.....8.....13.....18......25......32.......41.......50.......61
..5...16....38.....75....131.....210.....316......453......625......836
..7...45...155....415....905....1755....3085.....5077.....7891....11761
.18..167...828...2821...7582...17339...35288....65769...114442...188463
.32..609..4390..19657..65134..177097..417204...883409..1720628..3135633
.84.2471.25202.144871.587682.1888153.5134796.12322101.26828152.54037203

			All solutions for n=3, k=3:
.-2....0...-1...-1...-3...-2...-3...-2
.-1....0...-1....0....1....1....0....0
..3....0....2....1....2....1....3....2

R. H. Hardin, Table of n, a(n) for n = 1..165

Row 3 is A000982(n+1).

Row 4 is A174723(n+1).

Empirical for row n:
n=2: a(k) = k + 1.
n=3: a(k) = 2*a(k-1) - 2*a(k-3) + a(k-4).
n=4: a(k) = (2/3)*k^3 + (3/2)*k^2 + (11/6)*k + 1.
n=5: a(k) = 3*a(k-1) - a(k-2) - 5*a(k-3) + 5*a(k-4) + a(k-5) - 3*a(k-6) + a(k-7).
n=6: a(k) = (22/15)*k^5 + (11/3)*k^4 + (14/3)*k^3 + (13/3)*k^2 + (43/15)*k + 1.
n=7: a(k) = 4*a(k-1) - 3*a(k-2) - 8*a(k-3) + 14*a(k-4) - 14*a(k-6) + 8*a(k-7) + 3*a(k-8) - 4*a(k-9) + a(k-10).

A208970 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first and second differences in -k..k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 4, 3, 2, 1, 2, 2, 8, 9, 8, 2, 1, 2, 5, 11, 19, 29, 15, 4, 1, 3, 5, 18, 40, 90, 87, 42, 4, 1, 3, 5, 24, 77, 221, 371, 325, 94, 7, 1, 3, 8, 35, 130, 495, 1185, 1755, 1148, 246, 7, 1, 3, 8, 45, 213, 967, 3186, 6883, 8092, 4168, 613, 14, 1

Offset: 1

R. H. Hardin, Mar 03 2012

Table starts
.1..1...1....1....1.....1.....1......1......1......1.......1.......1......1
.1..1...1....2....2.....2.....2......3......3......3.......3.......4......4
.1..1...2....2....2.....5.....5......5......8......8.......8......13.....13
.1..3...4....8...11....18....24.....35.....45.....61......76......98....119
.1..3...9...19...40....77...130....213....325....484.....687.....956...1294
.2..8..29...90..221...495...967...1801...3093...5050....7921...11994..17488
.2.15..87..371.1185..3186..7425..15658..30368..55222...95087..156612.248194
.4.42.325.1755.6883.21830.58791.140429.304536.612054.1154448.2066531

			Some solutions for n=5, k=5:
.-2...-2...-1...-3...-2...-1...-2...-2...-2...-1...-2...-1...-3...-1...-2...-2
.-2...-1....0...-2...-1...-1...-1....0....0....0....0....0...-1....0....0...-2
..0....1....1....2....2....0....2...-1....2...-1....1....0....2....0....0...-1
..2....2...-1....2....1....2....0....2...-1....0....1....0....3....1....2....3
..2....0....1....1....0....0....1....1....1....2....0....1...-1....0....0....2

R. H. Hardin, Table of n, a(n) for n = 1..184

Row 2 is A002265(n+4).

Row 3 is A000982(floor(n/3)+1).

Empirical for row n:
n=2: a(k) = a(k-1) + a(k-4) - a(k-5).
n=3: a(k) = a(k-1) + a(k-3) - a(k-4) + a(k-6) - a(k-7) - a(k-9) + a(k-10).
n=4: a(k) = 2*a(k-1) + a(k-2) - 4*a(k-3) + a(k-4) + 2*a(k-5) - a(k-6).

A250561 T(n,k)=Number of length n+2 0..k arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

2, 5, 8, 8, 25, 14, 13, 60, 83, 32, 18, 117, 302, 297, 62, 25, 200, 761, 1516, 989, 128, 32, 321, 1648, 5105, 7126, 3113, 254, 41, 480, 3125, 13732, 31525, 30780, 9611, 512, 50, 681, 5446, 31173, 106362, 177421, 127586, 29257, 1022, 61, 940, 8843, 63400, 290909

Offset: 1

R. H. Hardin, Nov 25 2014

Table starts
....2......5.......8........13.........18.........25..........32...........41
....8.....25......60.......117........200........321.........480..........681
...14.....83.....302.......761.......1648.......3125........5446.........8843
...32....297....1516......5105......13732......31173.......63400.......117749
...62....989....7126.....31525.....106362.....290909......695890......1486139
..128...3113...30780....177421.....744564....2457921.....6924692.....17094253
..254...9611..127586....937817....4808120...18934449....62245658....176612641
..512..29257..518052...4803653...29723864..137976845...522997696...1688068993
.1022..88503.2085808..24257725..180290280..980389815..4258085394..15526286669
.2048.266769.8367220.121800949.1085927844.6899647449.34261234132.140731044189

			Some solutions for n=5 k=4
..0....0....1....4....4....3....4....0....3....0....1....1....1....3....0....3
..1....2....2....0....0....0....4....1....2....4....0....1....2....0....0....3
..3....2....0....3....3....0....1....3....4....4....0....2....2....1....2....1
..3....3....2....4....0....3....2....3....3....3....4....2....0....0....4....0
..1....1....1....1....1....3....4....3....0....1....3....2....2....1....0....1
..3....0....2....1....4....4....4....4....0....1....3....1....0....3....2....4
..4....0....1....3....0....1....2....1....1....3....2....3....1....4....2....0

R. H. Hardin, Table of n, a(n) for n = 1..159

Row 1 is A000982(n+1)

Row 2 is A250321

Empirical for column k:
k=1: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3)
k=2: [order 10] for n>15
Empirical for row n:
n=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2
n=2: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) +2*a(n-7) -a(n-8); also a cubic polynomial plus a linear quasipolynomial with period 3
n=3: [order 21; also a quartic polynomial plus a linear quasipolynomial with period 60]

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

A214540 T(n,k)=Number of nXnXn triangular 0..k arrays with no element lying outside the (possibly reversed) range delimited by its sw and se neighbors, and every horizontal row having the same average value.

Original entry on oeis.org

2, 3, 2, 4, 5, 2, 5, 8, 15, 2, 6, 13, 38, 93, 2, 7, 18, 79, 344, 1007, 2, 8, 25, 152, 1181, 5360, 17213, 2, 9, 32, 263, 3198, 35567, 141470, 461465, 2, 10, 41, 418, 7801, 155102, 2280331, 6042900, 19166997, 2, 11, 50, 643, 16752, 562559, 15986168, 292689331

Offset: 1

R. H. Hardin Jul 20 2012

Table starts
.2....3....4.....5......6......7.......8.......9.......10.....11...12.13.14
.2....5....8....13.....18.....25......32......41.......50.....61...72.85
.2...15...38....79....152....263.....418.....643......942...1329.1832
.2...93..344..1181...3198...7801...16752...33605....62766.111653
.2.1007.5360.35567.155102.562559.1786114.4984385.12779966

			Some solutions for n=4 k=4
.....3........2........3........3........1........1........2........2
....2.4......2.2......2.4......3.3......2.0......0.2......2.2......2.2
...2.3.4....2.3.1....2.3.4....3.3.3....2.1.0....0.2.1....3.1.2....3.0.3
..3.2.3.4..1.3.0.4..3.2.4.3..4.3.1.4..2.1.1.0..1.0.2.1..3.1.1.3..3.0.3.2

R. H. Hardin, Table of n, a(n) for n = 1..85

Row 2 is A000982(n+1)

Empirical for row n:
n=1: a(k)=2*a(k-1)-a(k-2)
n=2: a(k)=2*a(k-1)-2*a(k-3)+a(k-4)
n=3: a(k)=2*a(k-1)-3*a(k-4)+3*a(k-6)-2*a(k-9)+a(k-10)
n=4: (symmetric, order 30)

A214595 T(n,k) = number of n X n X n triangular 0..k arrays with every horizontal row having the same average value.

Original entry on oeis.org

2, 3, 2, 4, 5, 2, 5, 8, 23, 2, 6, 13, 62, 401, 2, 7, 18, 157, 1862, 20351, 2, 8, 25, 312, 10177, 187862, 2869211, 2, 9, 32, 601, 33352, 3330677, 63120962, 1127599139, 2, 10, 41, 986, 103651, 20608352, 5495329427, 71200442882, 1248252244661, 2, 11, 50, 1619

Offset: 1

R. H. Hardin, Jul 22 2012

			Table starts
.2.....3......4.......5........6.........7.........8..........9.........10
.2.....5......8......13.......18........25........32.........41.........50
.2....23.....62.....157......312.......601.......986.......1619.......2426
.2...401...1862...10177....33352....103651....250042.....589763....1199614
.2.20351.187862.3330677.20608352.121537201.493575042.1877543213.5767190924
Some solutions for n = k = 4:
.....2........1........2........2........2........2........2........2
....3.1......0.2......2.2......3.1......2.2......1.3......4.0......4.0
...3.2.1....0.3.0....3.2.1....2.4.0....0.2.4....3.0.3....1.2.3....4.0.2
..2.2.3.1..2.1.0.1..1.2.4.1..4.2.2.0..1.4.3.0..4.0.2.2..3.2.3.0..4.0.4.0

R. H. Hardin, Table of n, a(n) for n = 1..1475

Row 2 is A000982(n+1). Other rows: A214596, A214597, A214598.

Columns: A214589, A214590, A214591, A214592, A214593, A214594.

/* helper function  mult() gives multiplicity of a composition */
mult(p, L=1, m=(#p)!)={for(k=2,#p, p[k]!=p[k-1] && m\=(-L+L=k)!); m\(#p-L+1)!}
A214595(n, k)={sum(a=1,k, prod(L=2,n, my(c=0); forpart(p=L*a, c+=mult(p), [0,k], L); c))+1} \\ M. F. Hasler, Aug 21 2025

Empirical for row n:
n=1: a(k)=2*a(k-1)-a(k-2)
n=2: a(k)=2*a(k-1)-2*a(k-3)+a(k-4)
n=3: (order 12 antisymmetric)
n=4: (order 32 symmetric)
n=5: (order 84 symmetric)
T(n, k) = Sum_{s=0..k} Product_{L=2..n} NC(s*L, L, k), where NC(s, n, k) is the number of compositions of sum s with n parts between 0 and k. - M. F. Hasler, Aug 21 2025

A008813 Expansion of (1+x^6)/((1-x)^2*(1-x^6)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 12, 15, 18, 21, 24, 29, 34, 39, 44, 49, 54, 61, 68, 75, 82, 89, 96, 105, 114, 123, 132, 141, 150, 161, 172, 183, 194, 205, 216, 229, 242, 255, 268, 281, 294, 309, 324, 339, 354, 369, 384, 401, 418, 435, 452, 469, 486, 505, 524, 543, 562

Offset: 0

N. J. A. Sloane

Number of 0..n arrays of 7 elements with zero second differences. - R. H. Hardin, Nov 16 2011

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

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), A008812 (m=5), this sequence (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,9,12];; for n in [9..70] do a[n]:=2*a[n-1]-a[n-2] +a[n-6]-2*a[n-7]+a[n-8]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^6)/((1-x)^2*(1-x^6)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^6)/((1-x)^2*(1-x^6)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^6)/(1-x)^2/(1-x^6), {x,0,70}], x] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1}, {1,2,3,4,5,6,9,12}, 70] (* Harvey P. Dale, Oct 13 2012 *)

Vec((1+x^6)/((1-x)^2*(1-x^6)) +O(x^70)) \\ Charles R Greathouse IV, Sep 26 2012

def A008813_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^6)/((1-x)^2*(1-x^6))).list()
A008813_list(70) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^6)/((1-x)^2*(1-x^6)).

a(n) = 2*a(n-1) -a(n-2) +a(n-6) -2*a(n-7) +a(n-8). - R. H. Hardin, Nov 16 2011

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

A008814 Expansion of (1+x^7)/((1-x)^2*(1-x^7)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 13, 16, 19, 22, 25, 28, 33, 38, 43, 48, 53, 58, 63, 70, 77, 84, 91, 98, 105, 112, 121, 130, 139, 148, 157, 166, 175, 186, 197, 208, 219, 230, 241, 252, 265, 278, 291, 304, 317, 330, 343, 358, 373, 388, 403, 418, 433, 448, 465, 482, 499

Offset: 0

N. J. A. Sloane

Number of 0..n arrays of 8 elements with zero second differences. - R. H. Hardin, Nov 16 2011

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

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), A008812 (m=5), A008813 (m=6), this sequence (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[1,2,3,4,5,6,7,10,13];; for n in [10..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-7]-2*a[n-8]+a[n-9]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^7)/((1-x)^2*(1-x^7)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^7)/((1-x)^2*(1-x^7)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^7)/(1-x)^2/(1-x^7), {x,0,70}], x] (* or *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,10,13}, 70] (* Harvey P. Dale, Dec 18 2012 *)

a(n)=(n*(n+2)+[7,11,13,13,11,7,1][n%7+1])/7 \\ Charles R Greathouse IV, Nov 16 2011

a(n)=(n*(n+2)+13-6*(n%7==6))\7  \\ Tani Akinari, Jul 25 2013

def A008814_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^7)/((1-x)^2*(1-x^7))).list()
A008814_list(70) # G. C. Greubel, Sep 12 2019

G.f.: (1+x^7)/((1-x)^2*(1-x^7)).

a(n) = 2*a(n-1) -a(n-2) +a(n-7) -2*a(n-8) +a(n-9). - R. H. Hardin, Nov 16 2011

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

cp's OEIS Frontend

A250320 T(n,k)=Number of length n+2 0..k arrays with the sum of second differences squared multiplied by some arrangement of +-1 equal to zero.

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

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A208825 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero.

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A208970 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first and second differences in -k..k.

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A250561 T(n,k)=Number of length n+2 0..k arrays with the sum of second differences multiplied by some arrangement of +-1 equal to zero.

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

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A214540 T(n,k)=Number of nXnXn triangular 0..k arrays with no element lying outside the (possibly reversed) range delimited by its sw and se neighbors, and every horizontal row having the same average value.

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A214595 T(n,k) = number of n X n X n triangular 0..k arrays with every horizontal row having the same average value.

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