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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 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

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