A084570 -id:A084570 - cp's OEIS frontend

A084569 Partial sums of A084570.

1, 3, 9, 21, 44, 82, 142, 230, 355, 525, 751, 1043, 1414, 1876, 2444, 3132, 3957, 4935, 6085, 7425, 8976, 10758, 12794, 15106, 17719, 20657, 23947, 27615, 31690, 36200, 41176, 46648, 52649, 59211, 66369, 74157, 82612, 91770, 101670, 112350, 123851

Offset: 0

Paul Barry, May 31 2003

Conjecture: a(n) is the number of perimeter-magic (hollow) squares of order 3 with magic sum n+3. Order 3 means each of the 4 edges has 3 elements >=1; the square has 8 elements. The elements do not need to be distinct, and squares obtained by rotations are counted only once. The square (read ccw) for magic sum 3 has elements 1 1 1 1 1 1 1 1. The 3 squares with magic sum 4 are 1 1 2 1 1 1 2 1, 1 1 2 1 1 2 1 2 and 1 2 1 2 1 2 1 2. - R. J. Mathar, Mar 08 2025

R. J. Mathar, Generating perimeter-magic polygons, C++ (2025)
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A116701.

Accumulate[LinearRecurrence[{3,-2,-2,3,-1},{1,2,6,12,23},50]] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{1,3,9,21,44,82},50] (* Harvey P. Dale, Nov 12 2014 *)

a(n) = (-1)^n/8 + (n^4 + 6*n^3 + 17*n^2 + 30*n + 21)/24.
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (i + (-1)^i).
G.f.: ( -1+x-2*x^2 ) / ( (1+x)*(x-1)^5 ). - R. J. Mathar, Mar 08 2025
a(n)+a(n+1) = A116701(n+3)-1. - R. J. Mathar, Mar 08 2025

A035597 Number of points of L1 norm 3 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 12, 38, 88, 170, 292, 462, 688, 978, 1340, 1782, 2312, 2938, 3668, 4510, 5472, 6562, 7788, 9158, 10680, 12362, 14212, 16238, 18448, 20850, 23452, 26262, 29288, 32538, 36020, 39742, 43712, 47938, 52428, 57190, 62232, 67562

Offset: 0

N. J. A. Sloane

Sums of the first n terms > 0 of A001105 in palindromic arrangement. a(n) = Sum_{i=1 .. n} A001105(i) + Sum_{i=1 .. n-1} A001105(i), e.g. a(3) = 38 = 2 + 8 + 18 + 8 + 2; a(4) = 88 = 2 + 8 + 18 + 32 + 18 + 8 + 2. - Klaus Purath, Jun 19 2020

Apart from multiples of 3, all divisors of n are also divisors of a(n), i.e. if n is not divisible by 3, a(n) is divisible by n. All divisors d of a(n) for d !== 0 (mod) 3 are also divisors of a(abs(n-d)) and a(n+d). For all n congruent to 0,2,7 (mod 9) a(n) is divisible by 3. If n is divisible by 3^k, a(n) is divisible by 3^(k-1). - Klaus Purath, Jul 24 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII. Coordination sequences, Proc. Roy. Soc. Lond. A 458 (1996) 2369-2389.
Milzn Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
R. J. Mathar, Bivariate generating functions for non-attacking Wazirs on rectangular boards, viXra:2404.0122 (2024) page 14
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A069894.
Cf. A001105, A005900, A084570, A097869.
Cf. A001844, A001845, A005899.
Column 3 of A035607, A266213, A343599.
Row 3 of A113413, A119800, A122542.

[(4*n^3 + 2*n)/3: n in [0..40]]; // Vincenzo Librandi, Sep 19 2011

f := proc(n,m) local i; sum( 2^i*binomial(n,i)*binomial(m-1,i-1),i=1..min(n,m)); end; # n=dimension, m=norm

Table[(4n^3+2n)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,12,38},41] (* Harvey P. Dale, Sep 18 2011 *)

a(n) = (4*n^3 + 2*n)/3.
a(n) = 2*A005900(n). - R. J. Mathar, Dec 05 2009
a(0)=0, a(1)=2, a(2)=12, a(3)=38, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: (2*x*(x+1)^2)/(x-1)^4. - Harvey P. Dale, Sep 18 2011
a(n) = -a(-n), a(n+1) = A097869(4n+3) = A084570(2n+1). - Bruno Berselli, Sep 20 2011
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=3. Also, a(n) = A001845(n) - A001844(n). - Shel Kaphan, Feb 26 2023
a(n) = A005899(n)*n/3. - Shel Kaphan, Feb 26 2023
a(n) = A006331(n)+A006331(n-1). - R. J. Mathar, Aug 12 2025

A209032 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first differences in -k..k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 3, 4, 6, 2, 1, 3, 5, 12, 11, 4, 1, 4, 7, 23, 34, 33, 6, 1, 4, 10, 38, 88, 144, 86, 13, 1, 5, 12, 60, 187, 471, 576, 278, 21, 1, 5, 15, 88, 358, 1237, 2517, 2613, 873, 45, 1, 6, 19, 125, 625, 2798, 8235, 14611, 11841, 2938, 83, 1, 6, 22, 170, 1023

Offset: 1

R. H. Hardin, Mar 04 2012

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
..1...2....4.....5.....7.....10.....12......15......19......22......26.......31
..2...6...12....23....38.....60.....88.....125.....170.....226.....292......371
..2..11...34....88...187....358....625....1023....1584....2355....3374.....4700
..4..33..144...471..1237...2798...5648...10483...18174...29863...46918....71037
..6..86..576..2517..8235..22249..52208..110285..214440..390344..672932..1108883
.13.278.2613.14611.58524.186765.505857.1210780.2631514.5293759.9995616.17902216

			Some solutions for n=6, k=6:
.-4...-3...-2...-4...-2...-5...-3...-2...-5...-6...-2...-3...-3...-3...-4...-3
.-2....1...-1....2...-1...-5...-1...-1...-1...-2...-1...-1...-3...-3...-4....1
..2...-2...-1...-3....0...-1...-1....0....5....3....0....3...-3...-1...-1...-2
.-1....1....2....0...-1....5....1....3....2....5...-1...-1....1....2....4....3
..3....1....3....3....0....6....5...-1...-1....0....4....3....5....4....4....0
..2....2...-1....2....4....0...-1....1....0....0....0...-1....3....1....1....1

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

Row 2 is A004526(n+2).
Row 3 is A007997(n+5).
Row 4 is A084570.

Empirical for row n:
n=2: a(k) = a(k-1) + a(k-2) - a(k-3).
n=3: a(k) = 2*a(k-1) - a(k-2) + a(k-3) - 2*a(k-4) + a(k-5).
n=4: a(k) = 3*a(k-1) - 2*a(k-2) - 2*a(k-3) + 3*a(k-4) - a(k-5).
n=5: a(k) = 2*a(k-1) - 2*a(k-3) + 2*a(k-4) - a(k-5) - 2*a(k-6) + 2*a(k-7) + a(k-8) - 2*a(k-9) + 2*a(k-10) - 2*a(k-12) + a(k-13).

A084263 a(n) = (-1)^n/2+(n^2+n+1)/2.

Original entry on oeis.org

1, 1, 4, 6, 11, 15, 22, 28, 37, 45, 56, 66, 79, 91, 106, 120, 137, 153, 172, 190, 211, 231, 254, 276, 301, 325, 352, 378, 407, 435, 466, 496, 529, 561, 596, 630, 667, 703, 742, 780, 821, 861, 904, 946, 991, 1035, 1082, 1128, 1177, 1225, 1276, 1326, 1379, 1431

Offset: 0

Paul Barry, May 31 2003

Old name was "Modified triangular numbers".

Starting with offset 1 = row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 0, 0, 0, ...) (see example). - Gary W. Adamson, May 14 2010

			From _Gary W. Adamson_, May 14 2010: (Start)
First few rows of the triangle with row sums = A084263 =
1;
3, 1;
5, 0, 1;
7, 0, 3, 1;
9, 0, 5, 0, 1;
11, 0, 7, 0, 3, 1;
...
Example: a(4) = 11 = (7 + 0 + 3 + 1). (End)

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

Partial sums of A004442.

Cf. A000217, A002522, A059841, A084264, A084570.

[(-1)^n/2+(n^2+n+1)/2 : n in [0..80]]; // Wesley Ivan Hurt, Dec 23 2021

Table[(-1)^n/2 + (n^2 + n + 1)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 23 2021 *)
LinearRecurrence[{2,0,-2,1},{1,1,4,6},60] (* Harvey P. Dale, Jun 10 2023 *)

a(n)=n*(n+1)/2+(1-n)%2 \\ Charles R Greathouse IV, Jun 04 2013

E.g.f.: cosh(x)+exp(x)*(x+x^2/2).
a(n) = Sum_{k=0..n} k+(-1)^k.
a(n) = A000217(n)+A059841(n). Partial sums are A084570. Binomial transform is A084264.
G.f.: (1-x+2*x^2)/((1-x)^3*(1+x)). - R. J. Mathar, Apr 02 2008
a(0) = 1, a(n) = n^2 - a(n-1) + 1 for n >= 1. - Richard R. Forberg, Jun 05 2013
a(n) = 1 + floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) + a(n+1) = A002522(n+1). - R. J. Mathar, May 21 2018
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). - Wesley Ivan Hurt, Dec 23 2021

Name changed by Wesley Ivan Hurt, Dec 23 2021

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A084569 Partial sums of A084570.

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A035597 Number of points of L1 norm 3 in cubic lattice Z^n.

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A209032 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first differences in -k..k.

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A084263 a(n) = (-1)^n/2+(n^2+n+1)/2.

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