A083364 Antidiagonal sums of table A083362.

1, 5, 17, 32, 71, 105, 187, 248, 389, 485, 701, 840, 1147, 1337, 1751, 2000, 2537, 2853, 3529, 3920, 4751, 5225, 6227, 6792, 7981, 8645, 10037, 10808, 12419, 13305, 15151, 16160, 18257, 19397, 21761, 23040, 25687, 27113, 30059, 31640, 34901, 36645

Offset: 0

Author

Paul D. Hanna, Apr 27 2003

Keywords

• nonn
• easy

Comments

A083362 is the square table of least distinct positive integers such that the sum of any two consecutive terms in any row form a square.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A083362, A083363.

Programs

Magma

[(4*n^3+12*n^2+18*n+9+(2*n^2+2*n-1)*(-1)^n)/8 : n in [0..40]]; // Wesley Ivan Hurt, Sep 26 2014

Maple

A083364:=n->(4*n^3+12*n^2+18*n+9+(2*n^2+2*n-1)*(-1)^n)/8: seq(A083364(n), n=0..40); # Wesley Ivan Hurt, Sep 26 2014

Mathematica

Table[(4 n^3 + 12 n^2 + 18 n + 9 + (2 n^2 + 2 n - 1) (-1)^n)/8, {n,0,50}] (* Wesley Ivan Hurt, Sep 26 2014 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,5,17,32,71,105,187},50] (* Harvey P. Dale, Aug 16 2021 *)

PARI

Vec((x^5+6*x^4+3*x^3+9*x^2+4*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 26 2014

Formula

a(2n) = n(n+1)(4n+3)+(2n+1), a(2n+1) = ((n+1)^2)(4n+3)+(2n+2), for n>=0. - Paul D. Hanna, Apr 30 2003

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). - Colin Barker, Sep 26 2014

G.f.: (x^5+6*x^4+3*x^3+9*x^2+4*x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Sep 26 2014

a(n) = (4*n^3+12*n^2+18*n+9+(2*n^2+2*n-1)*(-1)^n)/8. - Wesley Ivan Hurt, Sep 26 2014