A254407 a(n) = n*(n+1)*(11*n +10)/6.

0, 7, 32, 86, 180, 325, 532, 812, 1176, 1635, 2200, 2882, 3692, 4641, 5740, 7000, 8432, 10047, 11856, 13870, 16100, 18557, 21252, 24196, 27400, 30875, 34632, 38682, 43036, 47705, 52700, 58032, 63712, 69751, 76160, 82950, 90132, 97717, 105716, 114140, 123000

Offset: 0

Author

Bruno Berselli, Jan 30 2015

Keywords

• nonn
• easy

Comments

Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:

s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;

s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);

s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);

s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);

s=7: a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).

Examples

			532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.
		

Links

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

Crossrefs

Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.

Programs

Magma

[n*(n+1)*(11*n+10)/6: n in [0..40]];

Maple

A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # G. C. Greubel, Mar 31 2021

Mathematica

Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]
Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* Vincenzo Librandi, Jan 31 2015 *)

Maxima

makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);

PARI

vector(40, n, n--; n*(n+1)*(11*n+10)/6)

Sage

[n*(n+1)*(11*n+10)/6 for n in (0..40)]

Formula

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

a(-n) = -A132112(n-1).

a(n) = Sum_{k=0..n} A011875(11*k+2).

Equivalently, partial sums of A254963.

E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021