A014626 -id:A014626 - cp's OEIS frontend

A159355 Number of n X n arrays of squares of integers summing to 4.

5, 135, 1836, 12675, 58941, 211925, 635440, 1663821, 3921325, 8495531, 17179020, 32795295, 59626581, 103962825, 174792896, 284660665, 450710325, 695946991, 1050740300, 1554600411, 2258257485, 3226077405, 4538848176, 6296973125, 8624108701, 11671286355

Offset: 2

R. H. Hardin, Apr 11 2009

Each array either has four 1's or one 4, and all other elements 0. - Robert Israel, Jun 19 2018

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A014626, A159359.

Maple
```
seq(binomial(n^2,4)+n^2, n=2..100);
```

Vec(x^2*(5 + 90*x + 801*x^2 + 591*x^3 + 252*x^4 - 88*x^5 + 37*x^6 - 9*x^7 + x^8) / (1 - x)^9 + O(x^40)) \\ Colin Barker, Jun 19 2018

Empirical: n^2*(n^2+1)*(n^4-7*n^2+18)/24. - R. J. Mathar, Aug 11 2009
From Robert Israel, Jun 19 2018: (Start)
Empirical formula confirmed.
a(n) = binomial(n^2,4)+n^2 = A014626(n^2).
(End)
From Colin Barker, Jun 19 2018: (Start)
G.f.: x^2*(5 + 90*x + 801*x^2 + 591*x^3 + 252*x^4 - 88*x^5 + 37*x^6 - 9*x^7 + x^8) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>10.
(End)

A159255 Irregular triangle read by rows: row n gives expansion of (1-x+x^2)*(1+x)^n.

Original entry on oeis.org

1, -1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 2, 3, 3, 1, 1, 4, 6, 5, 5, 6, 4, 1, 1, 5, 10, 11, 10, 11, 10, 5, 1, 1, 6, 15, 21, 21, 21, 21, 15, 6, 1, 1, 7, 21, 36, 42, 42, 42, 36, 21, 7, 1, 1, 8, 28, 57, 78, 84, 84, 78, 57, 28, 8, 1, 1, 9, 36, 85, 135, 162, 168, 162, 135, 85, 36, 9, 1

Offset: 0

Philippe Deléham, Apr 07 2009

			Row n=0 : 1, -1, 1 ;
Row n=1 : 1, 0, 0, 1 ;
Row n=2 : 1, 1, 0, 1, 1 ;
Row n=3 : 1, 2, 1, 1, 2, 1 ;
Row n=4 : 1, 3, 3, 2, 3, 3, 1 ;
Row n=5 : 1, 5, 10, 11, 10, 11, 10, 5, 1;
Row n=6 : 1, 6, 15, 21, 21, 21, 21, 15, 6, 1;
...

Cf. A000079, A014626, A024483, A050407

row(n)=Vecrev(polcoef((1 - y + y^2)/(1 - x*(1+y)) + O(x*x^n), n))
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Mar 03 2023

G.f.: A(x,y) = (1 - y + y^2)/(1 - x*(1+y)). - Andrew Howroyd, Mar 03 2023

A014628 Number of segments (and sides) created by diagonals of an n-gon in general position.

Original entry on oeis.org

3, 8, 20, 45, 91, 168, 288, 465, 715, 1056, 1508, 2093, 2835, 3760, 4896, 6273, 7923, 9880, 12180, 14861, 17963, 21528, 25600, 30225, 35451, 41328, 47908, 55245, 63395, 72416, 82368, 93313, 105315, 118440, 132756, 148333, 165243, 183560

Offset: 3

Mohammad K. Azarian

There is a connection to A014626: number of intersection points of diagonals of n-gon, plus number of vertices, b(n) = n*(n+1)*(n^2-7*n+18)/24 and A006522: number of regions created by sides and diagonals of n-gon, c(n) = (n-1)*(n-2)*(n^2-3*n+12)/24. These are related by the Euler-formula: b(n) + c(n) - a(n) = 1. - Georg Wengler, Mar 31 2005

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

Cf. A014626, A006522, A134393.

Table[Binomial[n,2]+2Binomial[n,4],{n,3,50}] (* Harvey P. Dale, Oct 03 2020 *)

a(n) = (n^4-6*n^3+17*n^2-24*n)/12 + n; or equally n*(n-1)*(n^2-5*n+12)/12.
G.f.: x^3*(3-7*x+10*x^2-5*x^3+x^4)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = C(n,2) + 2*C(n,4). - Gary Detlefs, Jun 06 2010

G.f. proposed by Maksym Voznyy, checked and corrected by R. J. Mathar, Sep 16 2009
More terms from Erich Friedman
Offset corrected by Mohammad K. Azarian, Nov 19 2008
Offset corrected by Eric Rowland, Aug 15 2017

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A159355 Number of n X n arrays of squares of integers summing to 4.

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A159255 Irregular triangle read by rows: row n gives expansion of (1-x+x^2)*(1+x)^n.

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A014628 Number of segments (and sides) created by diagonals of an n-gon in general position.

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