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A243201 Odd octagonal numbers indexed by triangular numbers. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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.
Sequence in context: A308348 A089369 A328861 * A264554 A125331 A126489
KEYWORD
nonn,easy
AUTHOR
Mathew Englander, Jun 01 2014
STATUS
approved

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Last modified March 3 21:06 EST 2024. Contains 370517 sequences. (Running on oeis4.)