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A130423 Main diagonal of array A[k,n] = n-th sum of 3 consecutive k-gonal numbers, k>2. 2
4, 14, 39, 88, 170, 294, 469, 704, 1008, 1390, 1859, 2424, 3094, 3878, 4785, 5824, 7004, 8334, 9823, 11480, 13314, 15334, 17549, 19968, 22600, 25454, 28539, 31864, 35438, 39270, 43369, 47744, 52404, 57358, 62615, 68184, 74074, 80294, 86853 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The first row of the array is the sum of 3 consecutive triangular numbers = A000217(n) + A000217(n+1) + A000217(n+2) = Centered triangular numbers: 3*n*(n-1)/2 + 1, for n>1. The second row of the array is the sum of 3 consecutive squares = Number of points on surface of square pyramid: 3*n^2 + 2 (n>1). The first column of the array is k+1 = 4, 5, 6, 7, 8, 9, ... The second column of the array is A016825 = 4*n + 2 (for n>2). The third column of the array is A017377 = 10*n + 9 (for n>0).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Polygonal Number

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

FORMULA

a(n) = A[n+2,n] = P(k+2,n) + P(k+2,n+1) + P(k+2,n+2) where P(k,n) = k*((n-2)*k - (n-4))/2.

a(n) = n*(3*n^2-3*n+8)/2. G.f.: x*(4-2*x+7*x^2)/(1-x)^4. [Colin Barker, Apr 30 2012]

a(1)=4, a(2)=14, a(3)=39, a(4)=88, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Aug 15 2012

EXAMPLE

The array begins:

k / A[k,n]

3.|.4.10.19.31..46..64..85.109.136.166....=A005448(n+1).

4.|.5.14.29..50..77.110.149.194.245.302...=A005918(n).

5.|.6.18.39..69.108.156.213.279.354.438...=A129863(n).

6.|.7.22.49..88.139.202.277.364.463.574...

7.|.8.26.59.107.170.248.341.449.572.710...

8.|.9.30.69.126.201.294.405.534.681.846...

MAPLE

P := proc(k, n) n*((k-2)*n-k+4)/2 ; end: A := proc(k, n) add( P(k, i), i=n..n+2) ; end: A130423 := proc(n) A(n+3, n) ; end: seq(A130423(n), n=0..40) ; # R. J. Mathar, Jun 14 2007

MATHEMATICA

CoefficientList[Series[(4-2*x+7*x^2)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 28 2012 *)

Table[n (3n^2-3n+8)/2, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {4, 14, 39, 88}, 40] (* Harvey P. Dale, Aug 15 2012 *)

PROG

(MAGMA) I:=[4, 14, 39, 88]; [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, Jun 28 2012

CROSSREFS

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A005448, A005918, A016825, A017377, A129803, A129863.

Sequence in context: A114845 A141755 A064463 * A266423 A055484 A055279

Adjacent sequences:  A130420 A130421 A130422 * A130424 A130425 A130426

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, May 26 2007

EXTENSIONS

More terms from R. J. Mathar, Jun 14 2007

STATUS

approved

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Last modified September 20 10:51 EDT 2021. Contains 347584 sequences. (Running on oeis4.)