A295089 a(n) = 3*n^2 + n + 3.

3, 7, 17, 33, 55, 83, 117, 157, 203, 255, 313, 377, 447, 523, 605, 693, 787, 887, 993, 1105, 1223, 1347, 1477, 1613, 1755, 1903, 2057, 2217, 2383, 2555, 2733, 2917, 3107, 3303, 3505, 3713, 3927, 4147, 4373, 4605, 4843, 5087, 5337, 5593, 5855, 6123, 6397, 6677, 6963, 7255, 7553, 7857

Offset: 0

Author

Ron Knott, Nov 14 2017

Keywords

• nonn
• easy
• changed

Comments

Numbers represented as the palindrome 313 in number base n including base n=1, base 2 (binary) and base 3 with 'illegal' digit 3: 313_1=7, 313_2=17, 313_3=33, ... 313_9=255, 313_10=313, ...

Examples

			313 in base 7 is 3*7^2 + 1*7 + 3 = 157.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A049451, A056108, A131649, A144391.

Programs

Mathematica

Array[3 #^2 + # + 3 &, 52, 0] (* Michael De Vlieger, Nov 15 2017 *)
LinearRecurrence[{3, -3, 1}, {3, 7, 17}, 52] (* or *)
CoefficientList[Series[-(5 x^2 - 2 x + 3)/(x - 1)^3, {x, 0, 51}], x] (* Robert G. Wilson v, Nov 29 2017 *)

PARI

a(n) = 3*n^2 + n + 3; \\ Michel Marcus, Dec 15 2017

Formula

a(n) = A131649(n+3) + 1, n >= 2 (conjectured).

a(n) = A056108(n) + 2 = A049451(n) + 3 = A144391(n) + 4.

From Elmo R. Oliveira, Sep 02 2025: (Start)

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

E.g.f.: (3 + 4*x + 3*x^2)*exp(x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)