A302537 a(n) = (n^2 + 13*n + 2)/2.

1, 8, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205, 1255, 1306, 1358, 1411, 1465, 1520, 1576

Offset: 0

Author

Franck Maminirina Ramaharo, Apr 09 2018

Keywords

• nonn
• easy

Comments

Binomial transform of [1, 7, 1, 0, 0, 0, ...].

Numbers m > 0 such that 8*m + 161 is a square.

Examples

			Illustration of initial terms (by the formula a(n) = A052905(n) + 3*n):
.                                                                    o
.                                                                  o o
.                                                    o           o o o
.                                                  o o         o o o o
.                                      o         o o o       o o o o o
.                                    o o       o o o o     o o o o o o
.                          o       o o o     o o o o o   o . . . . . o
.                        o o     o o o o   o . . . . o   o . . . . . o
.                o     o o o   o . . . o   o . . . . o   o . . . . . o
.              o o   o . . o   o . . . o   o . . . . o   o . . . . . o
.        o   o . o   o . . o   o . . . o   o . . . . o   o . . . . . o
.      o o   o . o   o . . o   o . . . o   o . . . . o   o . . . . . o
.  o   o o   o o o   o o o o   o o o o o   o o o o o o   o o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
----------------------------------------------------------------------
.  1     8      16        25          35            46              58
		

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Sequences whose n-th terms are of the form binomial(n, 2) + n*k + 1:

A152947 (k = 0); A000124 (k = 1); A000217 (k = 2); A034856 (k = 3);

A052905 (k = 4); A051936 (k = 5); A246172 (k = 6).

Cf. A000578, A002939, A004120, A008589, A056119, A060544, A162261.

Programs

Magma

A302537:= func< n | ((n+1)^2 +12*n +1)/2 >;
[A302537(n): n in [0..50]]; // G. C. Greubel, Jan 21 2025

Maple

a := n -> (n^2 + 13*n + 2)/2;
seq(a(n), n = 0 .. 100);

Mathematica

Table[(n^2 + 13 n + 2)/2, {n, 0, 100}]
CoefficientList[ Series[(5x^2 - 5x - 1)/(x - 1)^3, {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 16}, 51] (* Robert G. Wilson v, May 19 2018 *)

Maxima

makelist((n^2 + 13*n + 2)/2, n, 0, 100);

PARI

a(n) = (n^2 + 13*n + 2)/2; \\ Altug Alkan, Apr 12 2018

Python

def A302537(n): return (n**2 + 13*n + 2)//2
print([A302537(n) for n in range(51)]) # G. C. Greubel, Jan 21 2025

Formula

a(n) = binomial(n + 1, 2) + 6*n + 1 = binomial(n, 2) + 7*n + 1.

a(n) = a(n-1) + n + 6.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, where a(0) = 1, a(1) = 8 and a(2) = 16.

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

a(n) = A004120(n+1) for n > 1.

a(n) = A056119(n) + 1.

a(n) = A152947(n+1) + A008589(n).

a(n) = A060544(n+1) - A002939(n).

a(n) = A000578(n+1) - A162261(n) for n > 0.

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

E.g.f.: (1/2)*(2 + 14*x + x^2)*exp(x).

Sum_{n>=0} 1/a(n) = 24097/45220 + 2*Pi*tan(sqrt(161)*Pi/2) / sqrt(161) = 1.4630922534498496... - Vaclav Kotesovec, Apr 11 2018