A024305 -id:A024305 - cp's OEIS frontend

A023855 a(n) = 1(n) + 2(n-1) + 3(n-2) + ... + (n+1-k)k, where k = floor((n+1)/2).

1, 2, 7, 10, 22, 28, 50, 60, 95, 110, 161, 182, 252, 280, 372, 408, 525, 570, 715, 770, 946, 1012, 1222, 1300, 1547, 1638, 1925, 2030, 2360, 2480, 2856, 2992, 3417, 3570, 4047, 4218, 4750, 4940, 5530, 5740, 6391, 6622, 7337, 7590, 8372, 8648, 9500, 9800, 10725, 11050

Offset: 1

Clark Kimberling

Given a rectangle of perimeter 2*n one can form rectangles having this perimeter for a number of different rectangles or squares depending on how large 2*n is. The sequence lists the total areas of all such rectangles for each 2*n. - J. M. Bergot, Sep 14 2011

Antidiagonal sums of triangle A075362. - L. Edson Jeffery, Jan 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1)

Cf. A000292, A023856, A023857, A024305, A024854.

a023855 n = sum $ zipWith (*) [1 .. div (n+1) 2] [n, n-1 ..]
-- Reinhard Zumkeller, Jan 23 2012

[(4*n^3 +15*n^2 +14*n +3 -3*(n+1)^2*(-1)^n)/48: n in [1..60]]; // G. C. Greubel, Jul 12 2022

seq(-(1/3)*floor((k+1)/2)^3 + (k/2)*floor((k+1)/2)^2 + ((3*k+2)/6)*floor((k+1)/2), k=1..100); # Wesley Ivan Hurt, Sep 18 2013

LinearRecurrence[{1,3,-3,-3,3,1,-1}, {1,2,7,10,22,28,50}, 60] (* Vincenzo Librandi, Jan 23 2012 *)
Table[-Ceiling[n/2] (Ceiling[n/2] + 1) (2 Ceiling[n/2] - 3 n - 2)/6, {n, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)

a(n)=if(n%2, (n+1)*(n+3)*(2*n+1)/24, n*(n+1)*(n+2)/12)

my(x='x+O('x^99)); Vec(x*(1+x+2*x^2)/((1-x)^4*(1+x)^3)) \\ Altug Alkan, Mar 03 2018

[(4*n^3 +15*n^2 +14*n +3 -3*(n+1)^2*(-1)^n)/48 for n in (1..60)] # G. C. Greubel, Jul 12 2022

a(n) = (n+1)*(n+3)*(2*n+1)/24 if n is odd, or n*(n+1)*(n+2)/12 if n is even.
G.f.: x*(1+x+2*x^2)/((1-x)^4*(1+x)^3). - Ralf Stephan, Apr 28 2004
a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+1) = -ceiling(n/2)*(ceiling(n/2)+1)*(2*ceiling(n/2)-3n-2)/6. - Wesley Ivan Hurt, Sep 19 2013
a(n) = (4*n^3 + 15*n^2 + 14*n + 3 - 3*(n+1)^2*(-1)^n)/48. - Luce ETIENNE, Oct 22 2014
a(n) = (A000292(n) + (n mod 2)*(ceiling(n/2))^2)/2. - Luc Rousseau, Feb 25 2018
E.g.f.: (1/24)*( x*(21+12*x+2*x^2)*cosh(x) + (3+12*x+15*x^2+2*x^3)*sinh(x) ). - G. C. Greubel, Jul 12 2022

Formula, program, and slight revision by Charles R Greathouse IV, Feb 23 2010

A023856 a(n) = 1(n+1-1) + 2(n+1-2) + ... + k*(n+1-k), where k = floor((n+1)/2).

Original entry on oeis.org

2, 3, 10, 13, 28, 34, 60, 70, 110, 125, 182, 203, 280, 308, 408, 444, 570, 615, 770, 825, 1012, 1078, 1300, 1378, 1638, 1729, 2030, 2135, 2480, 2600, 2992, 3128, 3570, 3723, 4218, 4389, 4940, 5130, 5740, 5950, 6622, 6853, 7590, 7843, 8648, 8924, 9800, 10100, 11050, 11375

Offset: 1

Clark Kimberling

Or, a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n-k+1), where k = floor( n/2 ) and s = (natural numbers).

Sum of the areas of the distinct rectangles with positive integer length and width such that L + W = n + 2, W < L. For example, a(5) = 28; the rectangles are 1 X 6, 2 X 5 and 3 X 4. The sum of the areas is then 1*6 + 2*5 + 3*4 = 28. - Wesley Ivan Hurt, Nov 12 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 302.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A000292, A023855, A023857, A024305, A024854.

[(n+2)*(4*n^2 + 13*n + 6 - 3*(n+2)*(-1)^n)/48 : n in [1..80]]; // Wesley Ivan Hurt, Nov 29 2017

seq(add(i*(k-i+2), i=1..ceil(k/2)), k=1..70); # Wesley Ivan Hurt, Sep 20 2013

Table[-Ceiling[n/2]*(Ceiling[n/2]+1)*(2*Ceiling[n/2]-3n-5)/6, {n, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{2,3,10,13,28,34,60},60] (* Harvey P. Dale, Jan 09 2017 *)

my(x='x+O('x^99)); Vec(x*(2+x+x^2)/((1+x)^3*(x-1)^4)) \\ Altug Alkan, Dec 17 2017

[(n+2)*(4*n^2 +13*n +6 -3*(n+2)*(-1)^n)/48 for n in (1..60)] # G. C. Greubel, Jul 12 2022

a(n) = (n+2)*(4*n^2 + 13*n + 6 - 3(n+2)(-1)^n)/48.
a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+2) = -ceiling(n/2)*(ceiling(n/2) + 1)*(2*ceiling(n/2) - 3n - 5)/6. - Wesley Ivan Hurt, Sep 20 2013
G.f.: x*(2+x+x^2) / ( (1+x)^3*(1-x)^4 ). - R. J. Mathar, Sep 25 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, Dec 01 2017
a(n - 1) = (A000292(n) - (n mod 2) * (ceiling(n / 2)) ^ 2) / 2. - Luc Rousseau, Feb 25 2018
E.g.f.: (1/24)*( x*(36 + 15*x + 2*x^2)*cosh(x) + (12 + 21*x + 18*x^2 + 2*x^3)*sinh(x) ). - G. C. Greubel, Jul 12 2022

Title simplified by Sean A. Irvine, Jun 12 2019

A023857 a(n) = 1(n+3-1) + 2(n+3-2) + .... + k*(n+3-k), where k=floor((n+1)/2).

Original entry on oeis.org

3, 4, 13, 16, 34, 40, 70, 80, 125, 140, 203, 224, 308, 336, 444, 480, 615, 660, 825, 880, 1078, 1144, 1378, 1456, 1729, 1820, 2135, 2240, 2600, 2720, 3128, 3264, 3723, 3876, 4389, 4560, 5130, 5320, 5950, 6160, 6853, 7084, 7843, 8096, 8924, 9200, 10100, 10400, 11375, 11700

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A023855, A023856, A024305, A024854.

List([1..60], n-> (4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48) # G. C. Greubel, Jun 12 2019

[(4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48: n in [1..60]]; // G. C. Greubel, Jun 12 2019

seq(sum(i*(n-i+3), i=1..ceil(n/2)), n=1..60); # Wesley Ivan Hurt, Sep 20 2013

Table[-Ceiling[n/2]*(Ceiling[n/2]+1)*(2*Ceiling[n/2]-3n-8)/6, {n,60}] (* Wesley Ivan Hurt, Sep 20 2013 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{3,4,13,16,34,40,70},60] (* Harvey P. Dale, Feb 13 2018 *)

a(n) = (4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48; \\ G. C. Greubel, Jun 12 2019

[(4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48 for n in (1..60)] # G. C. Greubel, Jun 12 2019

a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+3) = -ceiling(n/2)*(ceiling(n/2)+1)*(2*ceiling(n/2) - 3*n - 8)/6. - Wesley Ivan Hurt, Sep 20 2013
G.f. x*(3+x) / ( (1+x)^3*(1-x)^4 ). - R. J. Mathar, Sep 25 2013
a(n) = 3*A058187(n-1) + A058187(n-2). - R. J. Mathar, Sep 25 2013
a(n) = (4*n^3 + 27*n^2 + 50*n + 21 - 3*(n^2 + 6*n + 7)*(-1)^n)/48. - Luce ETIENNE, Nov 21 2014
E.g.f.: (x*(51 + 18*x + 2*x^2)*cosh(x) + (21 + 30*x + 21*x^2 + 2*x^3)*sinh(x))/24. - G. C. Greubel, Jun 12 2019

Title simplified by Sean A. Irvine, Jun 12 2019

A024854 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (natural numbers), t = (natural numbers >= 3).

Original entry on oeis.org

4, 5, 16, 19, 40, 46, 80, 90, 140, 155, 224, 245, 336, 364, 480, 516, 660, 705, 880, 935, 1144, 1210, 1456, 1534, 1820, 1911, 2240, 2345, 2720, 2840, 3264, 3400, 3876, 4029, 4560, 4731, 5320, 5510, 6160, 6370, 7084, 7315, 8096, 8349, 9200, 9476, 10400, 10700, 11700

Offset: 2

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A023855, A023856, A023857, A024305, A058187.

[(4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48: n in [2..60]]; // Vincenzo Librandi, Oct 31 2014

seq(sum(i*(k-i+3), i=1..floor(k/2)), k=2..70); # Wesley Ivan Hurt, Sep 20 2013

Table[-Floor[n/2] * (Floor[n/2] + 1) * (2 * Floor[n/2] - 3n - 8)/6, {n, 2, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)
CoefficientList[Series[- (- 4 - x + x^2)/((1 + x)^3 (x - 1)^4), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 31 2014 *)

[(4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48 for n in (2..60)] # G. C. Greubel, Jul 13 2022

a(n) = Sum_{i=1..floor(n/2)} i*(n-i+3) = -floor(n/2)*(floor(n/2)+1)*(2*floor(n/2)-3n-8)/6. - Wesley Ivan Hurt, Sep 20 2013
G.f. x^2*(4 + x - x^2) / ( (1+x)^3*(1-x)^4 ). - R. J. Mathar, Sep 25 2013
a(n) = 4*A058187(n-2) + A058187(n-3) - A058187(n-4). - R. J. Mathar, Sep 25 2013
a(n) = (4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48. - Luce ETIENNE, Nov 14 2014
E.g.f.: (1/24)*( x*(9 + 18*x + 2*x^2)*cosh(x) + (-9 + 30*x + 15*x^2 + 2*x^3)*sinh(x) ). - G. C. Greubel, Jul 13 2022

A024868 a(n) = 2(n+1) + 3n + ... + (k+1)*(n+2-k), where k = floor(n/2).

Original entry on oeis.org

6, 8, 22, 27, 52, 61, 100, 114, 170, 190, 266, 293, 392, 427, 552, 596, 750, 804, 990, 1055, 1276, 1353, 1612, 1702, 2002, 2106, 2450, 2569, 2960, 3095, 3536, 3688, 4182, 4352, 4902, 5091, 5700, 5909, 6580, 6810, 7546, 7798, 8602, 8877, 9752, 10051, 11000, 11324, 12350

Offset: 2

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A023855, A023856, A023857, A024305, A024854.

Apart from offset the same as A024306.

[19*n/24-9/16+n^3/12+11*n^2/16+(-1)^n*(3*n/8 +9/16+n^2/16): n in [2..50]]; // Vincenzo Librandi, Sep 26 2013

seq(sum((i+1)*(k-i+2), i=1..floor(k/2)), k=2..70); # Wesley Ivan Hurt, Sep 20 2013

Table[Floor[n/2] (-2Floor[n/2]^2 +3n*Floor[n/2] +9n +14)/6, {n, 2, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)
CoefficientList[Series[(6 +2x -4x^2 -x^3 +x^4)/((1+x)^3 (1-x)^4), {x, 0, 60}], x] (* Vincenzo Librandi, Sep 26 2013 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{6,8,22,27,52,61,100},50] (* Harvey P. Dale, Aug 11 2023 *)

[(1/48)*(4*n^3 +33*n^2 +38*n -27 +3*(-1)^n*(n+3)^2) for n in (2..60)] # G. C. Greubel, Jul 13 2022

a(n) = Sum_{i=1..floor(n/2)} (i+1)*(n-i+2) = floor(n/2)*(-2*floor(n/2)^2 + 3*n*floor(n/2) + 9*n + 14)/6, n>1. - Wesley Ivan Hurt, Sep 20 2013
G.f.: x^2*(6 + 2*x - 4*x^2 - x^3 + x^4) / ( (1+x)^3*(x-1)^4 ). - R. J. Mathar, Sep 25 2013
a(n) = 6*A058187(n-2) +2*A058187(n-3) -4*A058187(n-4) -A058187(n-5) +A058187(n-6). - R. J. Mathar, Sep 25 2013
a(n) = ( 4*n^3 + 33*n^2 + 38*n - 27 )/48 + (-1)^n*(n+3)^2/16. - R. J. Mathar, Sep 25 2013
E.g.f.: (1/24)*( x*(2*x^2 + 24*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 48*x - 27)*sinh(x) ). - G. C. Greubel, Jul 13 2022

cp's OEIS Frontend

A023855 a(n) = 1*(n) + 2*(n-1) + 3*(n-2) + ... + (n+1-k)*k, where k = floor((n+1)/2).

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A023856 a(n) = 1*(n+1-1) + 2*(n+1-2) + ... + k*(n+1-k), where k = floor((n+1)/2).

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A023857 a(n) = 1*(n+3-1) + 2*(n+3-2) + .... + k*(n+3-k), where k=floor((n+1)/2).

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A024854 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (natural numbers), t = (natural numbers >= 3).

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A024868 a(n) = 2*(n+1) + 3*n + ... + (k+1)*(n+2-k), where k = floor(n/2).

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A023855 a(n) = 1(n) + 2(n-1) + 3(n-2) + ... + (n+1-k)k, where k = floor((n+1)/2).

A023856 a(n) = 1(n+1-1) + 2(n+1-2) + ... + k*(n+1-k), where k = floor((n+1)/2).

A023857 a(n) = 1(n+3-1) + 2(n+3-2) + .... + k*(n+3-k), where k=floor((n+1)/2).

A024854 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (natural numbers), t = (natural numbers >= 3).

A024868 a(n) = 2(n+1) + 3n + ... + (k+1)*(n+2-k), where k = floor(n/2).