A260955 Differences of the increasing arithmetic progression a^2+a, b^2+b, c^2+c, where b = 5*a+2, c = 7*a+3 and a >= 0.
6, 54, 150, 294, 486, 726, 1014, 1350, 1734, 2166, 2646, 3174, 3750, 4374, 5046, 5766, 6534, 7350, 8214, 9126, 10086, 11094, 12150, 13254, 14406, 15606, 16854, 18150, 19494, 20886, 22326, 23814, 25350, 26934, 28566, 30246, 31974, 33750, 35574, 37446, 39366, 41334
Offset: 0
Examples
By the definition, given a = 7 and b = 5*7+2 = 37, c = 7*7+3 = 52, it follows that a^2+a = 56, b^2+b = 1406, c^2+c = 2756, where 56, 1406, 2756 are in arithmetic progression. Therefore, 2756-1406 = 1406-56 = 1350 and 1350 is in the sequence (8th term).
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
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Magma
[24*n^2+24*n+6: n in [0..40]]; // Vincenzo Librandi, Aug 05 2015
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Mathematica
Table[24 n^2 + 24 n + 6, {n, 0, 40}] (* Bruno Berselli, Aug 05 2015 *) LinearRecurrence[{3, -3, 1}, {6, 54, 150}, 50] (* Vincenzo Librandi, Aug 05 2015 *) -
PARI
Vec(6*(1+6*x+x^2)/(1-x)^3 + O(x^100)) \\ Colin Barker, Aug 05 2015
Formula
a(n) = 24*n^2 + 24*n + 6.
From Colin Barker, Aug 05 2015: (Start)
G.f.: 6*(1 + 6*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Bruno Berselli, Aug 05 2015: (Start)
a(n) = A032528(4*n+2).
a(n)*(2*h-1)^2 = a((2*h-1)*n+h-1). For h=0, a(n) = a(-n-1); for h=7, 169*a(n) = a(13*n+6). (End)
From Elmo R. Oliveira, Dec 28 2024: (Start)
E.g.f.: 6*exp(x)*(1 + 8*x + 4*x^2).
a(n) = 6*A016754(n). (End)