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A269342 a(n) = (n + 1)*(2*n + 1)*(4*n + 9)/3. 0
3, 26, 85, 196, 375, 638, 1001, 1480, 2091, 2850, 3773, 4876, 6175, 7686, 9425, 11408, 13651, 16170, 18981, 22100, 25543, 29326, 33465, 37976, 42875, 48178, 53901, 60060, 66671, 73750, 81313, 89376, 97955, 107066, 116725, 126948, 137751, 149150, 161161 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Table of n, a(n) for n=0..38.

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

FORMULA

G.f.: (3 + 14*x - x^2)/(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = Sum_{k=0..n} (8*k^2 + 12*k + 3).

Sum_{n>=0} 1/a(n) = 3*(80*log(2) + 5*Pi - 48)/175 = 0.397024075075621559...

EXAMPLE

a(0) = 0*2 + 1*3 = 3;

a(1) = 0*2 + 1*3 + 2*4 + 3*5 = 26;

a(2) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 = 85;

a(3) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 = 196;

a(4) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 + 8*10 + 9*11 = 375, etc.

MATHEMATICA

Table[(n + 1) (2 n + 1) (4 n + 9)/3, {n, 0, 38}]

LinearRecurrence[{4, -6, 4, -1}, {3, 26, 85, 196}, 39]

Table[Sum[8 k^2 + 12 k + 3, {k, 0, n}], {n, 0, 38}]

PROG

(PARI) Vec((3 + 14*x - x^2)/(1 - x)^4 + O(x^50)) \\ Michel Marcus, Feb 25 2016

(MAGMA) [(n+1)*(2*n+1)*(4*n+9)/3: n in [0..50]]; // Vincenzo Librandi, Feb 25 2016

CROSSREFS

Cf. A000466, A005563, A033996, A195241.

Sequence in context: A177511 A294494 A048372 * A292001 A211622 A062124

Adjacent sequences:  A269339 A269340 A269341 * A269343 A269344 A269345

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Feb 24 2016

STATUS

approved

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)