A260260 a(n) = n*(16*n^2 - 21*n + 7)/2.

0, 1, 29, 132, 358, 755, 1371, 2254, 3452, 5013, 6985, 9416, 12354, 15847, 19943, 24690, 30136, 36329, 43317, 51148, 59870, 69531, 80179, 91862, 104628, 118525, 133601, 149904, 167482, 186383, 206655, 228346, 251504, 276177, 302413, 330260, 359766, 390979

Offset: 0

Author

Bruno Berselli, Jul 21 2015

Keywords

• nonn
• easy

Comments

Similar sequences, where P(s, m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number:

A000578: P(3, m)*P( 3, m) - P(3, m-1)*P( 3, m-1);

A213772: P(3, m)*P( 4, m) - P(3, m-1)*P( 4, m-1) for m>0;

A005915: P(3, m)*P( 5, m) - P(3, m-1)*P( 5, m-1) " ;

A130748: P(3, m)*P( 6, m) - P(3, m-1)*P( 6, m-1) for m>1;

A027849: P(3, m)*P( 7, m) - P(3, m-1)*P( 7, m-1) for m>0;

A214092: P(3, m)*P( 8, m) - P(3, m-1)*P( 8, m-1) " ;

A100162: P(3, m)*P( 9, m) - P(3, m-1)*P( 9, m-1) " ;

A260260: P(3, m)*P(10, m) - P(3, m-1)*P(10, m-1), this sequence;

A100165: P(3, m)*P(11, m) - P(3, m-1)*P(11, m-1) for m>0.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• OEIS Wiki, Figurate numbers.
• Wikipedia, Polygonal numbers: Table of values.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000217, A000292, A001107.

Subsequence of A047275.

Sequences of the same type (see comment): A000578, A005915, A027849, A100162, A100165, A130748, A213772, A214092.

Programs

Magma

[n*(16*n^2-21*n+7)/2: n in [0..40]];

Mathematica

Table[n (16 n^2 - 21 n + 7)/2, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{0,1,29,132},40] (* Harvey P. Dale, May 08 2025 *)

PARI

vector(40, n, n--; n*(16*n^2-21*n+7)/2)

Sage

[n*(16*n^2-21*n+7)/2 for n in (0..40)]

Formula

G.f.: x*(1 + 25*x + 22*x^2)/(1 - x)^4. [corrected by Georg Fischer, May 10 2019]

a(n) = A000217(n)*A001107(n) - A000217(n-1)*A001107(n-1), with A000217(-1) = 0.

a(n) = A000292(n) + 25*A000292(n-1) + 22*A000292(n-2), with A000292(-2) = A000292(-1) = 0.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020

E.g.f.: exp(x)*x*(2 + 27*x + 16*x^2)/2. - Elmo R. Oliveira, Aug 08 2025