A117560 -id:A117560

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A154560

(n+3)^2*n/2 + 1.

+10
4

1, 9, 26, 55, 99, 161, 244, 351, 485, 649, 846, 1079, 1351, 1665, 2024, 2431, 2889, 3401, 3970, 4599, 5291, 6049, 6876, 7775, 8749, 9801, 10934, 12151, 13455, 14849, 16336, 17919, 19601, 21385, 23274, 25271, 27379, 29601, 31940, 34399, 36981

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

8*a(n) is the y value of a solution (x, y) to the Diophantine equation 2*x^3+12*x^2 = y^2. The corresponding x value is A152811(n+1).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: (1+5*x-4*x^2+x^3)/(1-x)^4.

a(n) = A058794(n)/2.

a(n) = A117560(n+2) - n - 1.

a(2*n) = A144129(n+1).

a(2*n-1) = A141530(n+1). a(n) = -a(-n-4). - Bruno Berselli, Sep 05 2011

EXAMPLE

a(5) = (5+3)^2*5/2+1 = 64*5/2+1 = 161.

PROG

(PARI) {for(n=0, 40, print1((n+3)^2*n/2+1, ", "))}

(Magma) [(n+3)^2*n/2 + 1: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

CROSSREFS

Cf. A058794 (row 3 of A007754), A117560 (n*(n^2-1)/2-1), A144129 (4*n^3-3*n), A141530, A152811 (2*(n^2+2*n-2)).

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Jan 12 2009

STATUS

approved

A117561

a(n) = floor(n*(n^3-n-3)/(2*(n-1))).

+10
1

3, 15, 38, 73, 124, 194, 286, 403, 548, 724, 934, 1181, 1468, 1798, 2174, 2599, 3076, 3608, 4198, 4849, 5564, 6346, 7198, 8123, 9124, 10204, 11366, 12613, 13948, 15374, 16894, 18511, 20228, 22048, 23974, 26009, 28156, 30418, 32798, 35299, 37924

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

a[n-1] is one approximation for the upper bound of the "antimagic constant" of an antimagic square of order n. The antimagic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. By analogy with the magic constant. This approximation follows from the observation that Sum[m + k, {k, 0, 2*n + 1}] <= (2*Sum[k, {k, 1, n^2}]) + (2*m) + (2*m + 1) where m is the antimagic constant for an antimagic square of order n. Stricter bounds seem likely to exist. See A117560 for the lower bounds. Note there exist no antimagic squares of order two or three, but the values are indexed here for completeness.

LINKS

Table of n, a(n) for n=2..42.

Eric Weisstein's World of Mathematics, Antimagic Square.

FORMULA

a(n) = floor(n*(n^3-n-3)/(2*(n-1))).

G.f.: x^2*(3+3*x-4*x^2-x^3+3*x^4-x^5)/(1-x)^4. - Colin Barker, Mar 29 2012

EXAMPLE

a[3] = 38 because the antimagic constant of an antimagic square of order 4 cannot exceed 38 (see comments)

MATHEMATICA

Table[Floor[n(n^3-n-3)/(2*(n-1))], {n, 2, 50}]

CROSSREFS

Cf. A117560, A050257, A049475.

KEYWORD

easy,nonn

AUTHOR

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 29 2006

STATUS

approved

A337130

a(n) is the sum of all products of pairs of numbers joined by the diagonals of an n-gon when its vertices are numbered from 1 to n in order.

+10
0

0, 0, 0, 11, 40, 99, 203, 370, 621, 980, 1474, 2133, 2990, 4081, 5445, 7124, 9163, 11610, 14516, 17935, 21924, 26543, 31855, 37926, 44825, 52624, 61398, 71225, 82186, 94365, 107849, 122728, 139095, 157046, 176680, 198099, 221408, 246715, 274131, 303770

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

For n < 4, no n-gon has a diagonal and thus a(n)=0.

LINKS

Table of n, a(n) for n=1..40.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n) = 3*binomial(n+1, 4) - n = (n-2)*(n-1)*n*(n+1)/8 - n for n>=3; a(1) = a(2) = 0.

a(n) = A000914(n-1) - A006527(n).

From Colin Barker, Aug 19 2020: (Start)

G.f.: x^4*(11 - 15*x + 9*x^2 - 2*x^3) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.

(End)

E.g.f.: x + x^2 + exp(x)*x*(-8 + 4*x^2 + x^3)/8. - Stefano Spezia, Aug 19 2020

EXAMPLE

The diagonals of 4-gon would be numbered (1,3) and (2,4). So a(4) = 1*3 + 2*4 = 11.

The diagonals of 5-gon would be numbered (1,3), (1,4), (2,4), (2,5) and (3,5). So a(5) = 1*3 + 1*4 + 2*4 + 2*5 + 3*5 = 40.

PROG

(PARI) concat([0, 0, 0], Vec(x^4*(11 - 15*x + 9*x^2 - 2*x^3) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Aug 19 2020

CROSSREFS

Partial sums of A117560. Cf. A000914 (products including sides), A007569, A007678.

KEYWORD

nonn,easy

AUTHOR

Mohammed Yaseen, Aug 17 2020

STATUS

approved

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