A368529 -id:A368529 - cp's OEIS frontend

A368528 a(n) = Sum_{k=1..n} k^2 * 3^(n-k).

0, 1, 7, 30, 106, 343, 1065, 3244, 9796, 29469, 88507, 265642, 797070, 2391379, 7174333, 21523224, 64569928, 193710073, 581130543, 1743391990, 5230176370, 15690529551, 47071589137, 141214767940, 423644304396, 1270932913813, 3812798742115

Offset: 0

Seiichi Manyama, Dec 28 2023

Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Cf. A000290, A000330, A047520, A368529.

Cf. A052150, A368524.

PARI
```
a(n) = sum(k=1, n, k^2*3^(n-k));
```

G.f.: x * (1+x)/((1-3*x) * (1-x)^3).
a(n) = 6*a(n-1) - 12*a(n-2) + 10*a(n-3) - 3*a(n-4).
a(n) = A052150(n-1) + A052150(n-2) for n > 1.
a(n) = (3^(n+1) - (n^2 + 3*n + 3))/2.
a(0) = 0; a(n) = 3*a(n-1) + n^2.

A368530 a(n) = Sum_{k=1..n} k^3 * 4^(n-k).

Original entry on oeis.org

0, 1, 12, 75, 364, 1581, 6540, 26503, 106524, 426825, 1708300, 6834531, 27339852, 109361605, 437449164, 1749800031, 6999204220, 27996821793, 111987293004, 447949178875, 1791796723500, 7167186903261, 28668747623692, 114674990506935, 458699962041564

Offset: 0

Seiichi Manyama, Dec 29 2023

Index entries for linear recurrences with constant coefficients, signature (8,-22,28,-17,4).

Cf. A000578, A000537, A213575, A066999.
Cf. A097788, A368525.
Cf. A014825, A368529.

PARI
```
a(n) = sum(k=1, n, k^3*4^(n-k));
```

G.f.: x * (1+4*x+x^2)/((1-4*x) * (1-x)^4).
a(n) = 8*a(n-1) - 22*a(n-2) + 28*a(n-3) - 17*a(n-4) + 4*a(n-5).
a(n) = (11*4^(n+1) - (9*n^3 + 36*n^2 + 60*n + 44))/27.
a(0) = 0; a(n) = 4*a(n-1) + n^3.

cp's OEIS Frontend

A368528 a(n) = Sum_{k=1..n} k^2 * 3^(n-k).

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