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

Author

Mathew Englander, Jun 01 2014

Keywords

• nonn
• easy

Examples

			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.
		

Links

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

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.

Programs

Magma

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

Mathematica

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

Sage

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

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