A243201 Odd octagonal numbers indexed by triangular numbers.

1, 21, 133, 481, 1281, 2821, 5461, 9633, 15841, 24661, 36741, 52801, 73633, 100101, 133141, 173761, 223041, 282133, 352261, 434721, 530881, 642181, 770133, 916321, 1082401, 1270101, 1481221, 1717633, 1981281, 2274181, 2598421, 2956161, 3349633, 3781141, 4253061, 4767841, 5328001

Offset

0,2

Links

Table of n, a(n) for n=0..36.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 3*n^4 + 6*n^3 + 7*n^2 + 4*n + 1.

a(n) = (n^2 + n + 1)*(3*n^2 + 3*n + 1).

a(n) = ((3*n^2 + 3*n + 2)^2 - 1)/3.

a(n) = A003215(n) * A002061(n + 1).

a(n) = A022522(n) / A005408(n).

a(n) = A000567(n^2 + n + 1).

a(n) = A014641((n^2 + n)/2).

a(n) = 1 + A140676(n^2 + n).

a(n) = 1 + A187156((n^2 + n + 4)/2) (empirical).

G.f.: (1 + 16*x + 38*x^2 + 16*x^3 + x^4)/(1 - x)^5. - Bruno Berselli, Jun 03 2014

E.g.f.: exp(x)*(1 + 20*x + 46*x^2 + 24*x^3 + 3*x^4). - Stefano Spezia, Apr 16 2022

Example

a(2) = 133 because the second triangular number is 3 and third odd octagonal number is 133.
a(3) = 481 because the third triangular number is 6 and the sixth odd octagonal number is 481.
a(4) = 1281 because the fourth triangular number is 10 and the tenth odd octagonal number is 1281.

Mathematica

Table[((3 n^2 + 3 n + 2)^2 - 1)/3, {n, 0, 39}] (* Alonso del Arte, Jun 01 2014 *)

Prog

(Magma) [3*n^4+6*n^3+7*n^2+4*n+1: n in [0..40]]; // Bruno Berselli, Jun 03 2014
(Sage) [3*n^4+6*n^3+7*n^2+4*n+1 for n in (0..40)] # Bruno Berselli, Jun 03 2014

Crossrefs

Row 5 of A059259 (coefficients of 1 + 4*n + 7*n^2 + 6*n^3 + 3*n^4 + 0*n^5 which is a formula for the within sequence).

Column 5 of A081297.

Column 6 of A072024.

Diagonal T(n + 1, n) of A219069, n > 0.

Cf. A000217, A000567, A002061, A003215, A005408, A014641, A022522, A140676, A187156.

Sequence in context: A308348 A089369 A328861 * A264554 A125331 A126489

Adjacent sequences: A243198 A243199 A243200 * A243202 A243203 A243204

Keyword

nonn,easy

Author

Mathew Englander, Jun 01 2014

Status

approved