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

Author

R. H. Hardin, Apr 11 2009

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A014626, A159359.

Programs

Maple

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

PARI

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

Formula

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)