A226450 -id:A226450

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A226449

a(n) = n*(5*n^2-8*n+5)/2.

+10
8

0, 1, 9, 39, 106, 225, 411, 679, 1044, 1521, 2125, 2871, 3774, 4849, 6111, 7575, 9256, 11169, 13329, 15751, 18450, 21441, 24739, 28359, 32316, 36625, 41301, 46359, 51814, 57681, 63975, 70711, 77904, 85569, 93721, 102375, 111546, 121249, 131499, 142311, 153700

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:

A006003(n) = A000217(n) + n*A000217(n-1) (b = triangular numbers);

A069778(n) = A000290(n+1) + (n+1)*A000290(n) (b = square numbers);

A143690(n) = A000326(n+1) + (n+1)*A000326(n) (b = pentagonal numbers);

A212133(n) = A000384(n) + n*A000384(n-1) (b = hexagonal numbers);

a(n) = A000566(n) + n*A000566(n-1) (b = heptagonal numbers);

A226450(n) = A000567(n) + n*A000567(n-1) (b = octagonal numbers);

A226451(n) = A001106(n) + n*A001106(n-1) (b = nonagonal numbers);

A204674(n) = A001107(n+1) + (n+1)*A001107(n) (b = decagonal numbers).

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: x*(1+5*x+9*x^2)/(1-x)^4.

a(n) - a(-n) = A008531(n) for n>0.

MATHEMATICA

Table[n (5 n^2 - 8 n + 5)/2, {n, 0, 40}]

CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 9, 39}, 50] (* Harvey P. Dale, May 19 2017 *)

PROG

(Magma) [n*(5*n^2-8*n+5)/2: n in [0..40]];

(Magma) I:=[0, 1, 9, 39]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

(PARI) a(n)=n*(5*n^2-8*n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. (see the comment) A000566, A006003, A069778, A143690, A204674, A212133, A226450, A226451.

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jun 07 2013

STATUS

approved

A226451

a(n) = n*(7*n^2-12*n+7)/2.

+10
3

0, 1, 11, 51, 142, 305, 561, 931, 1436, 2097, 2935, 3971, 5226, 6721, 8477, 10515, 12856, 15521, 18531, 21907, 25670, 29841, 34441, 39491, 45012, 51025, 57551, 64611, 72226, 80417, 89205, 98611, 108656, 119361, 130747, 142835, 155646, 169201, 183521

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

See the comment in A226449.

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: x*(1+7*x+13*x^2)/(1-x)^4.

a(n) = A001106(n) + n*A001106(n-1).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Oct 15 2023

MATHEMATICA

Table[n (7 n^2 - 12 n + 7)/2, {n, 0, 40}]

CoefficientList[Series[x (1 + 7 x + 13 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

PROG

(Magma) [n*(7*n^2-12*n+7)/2: n in [0..40]];

(Magma) I:=[0, 1, 11, 51]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 18 2013