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 A247792 a(n) = 9*n^2 + 1. 2
 1, 10, 37, 82, 145, 226, 325, 442, 577, 730, 901, 1090, 1297, 1522, 1765, 2026, 2305, 2602, 2917, 3250, 3601, 3970, 4357, 4762, 5185, 5626, 6085, 6562, 7057, 7570, 8101, 8650, 9217, 9802, 10405, 11026, 11665, 12322, 12997, 13690, 14401, 15130, 15877, 16642, 17425, 18226, 19045, 19882 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The odd numbers of the form 9n^2 + 1 are listed in A158591 (36n^2 + 1). The even numbers of the form 9n^2 + 1 are given by 36x^2 - 36x + 10, x > 0. Every integer n>0 give three perfect squares and consecutives from 2^2. The formulas for each value of n are: a(n)-6n, a(n)-1 and a(n)+6n. - Miquel Cerda, Sep 19 2016 These squares are, for n>0, A000290(3*n-1), 3*n and (3n+1) and the sum of them is 3*a(n) - 1. - Miquel Cerda, Sep 26 2016 LINKS Karl V. Keller, Jr., Table of n, a(n) for n = 0..10000 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = (3n)^2 + 1 = 9n^2 + 1 = A016766(n) + 1. G.f.: (1+7*x+10*x^2)/(1-x)^3. - Vincenzo Librandi, Sep 27 2014 a(n) = ((3n-1)^2 + (3n+1)^2)/2 = (A016790(n-1) + A016778(n))/2. - Miquel Cerda, Jun 25 2016 From Ilya Gutkovskiy, Jun 25 2016: (Start) E.g.f.: (1 + 9*x + 9*x^2)*exp(x). Dirichlet g.f.: 9*zeta(s-2) + zeta(s). Sum_{n>=0} 1/a(n) = (3 + Pi*coth(Pi/3))/6. (End) a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Wesley Ivan Hurt, Jun 25 2016 Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/3)*cosech(Pi/3))/2. - Amiram Eldar, Jul 15 2020 EXAMPLE a(1) = (2^2 + 4^2)/2 = 3^2 + 1 = 10, a(2) = (5^2 + 7^2)/2 = 6^2 + 1 = 37, a(3) = (8^2 + 10^2)/2 = 9^2 + 1 = 82. - Miquel Cerda, Jun 25 2016 MAPLE A247792:=n->9*n^2 + 1: seq(A247792(n), n=0..80); # Wesley Ivan Hurt, Jun 25 2016 MATHEMATICA (3Range[0, 49])^2 + 1 (* Alonso del Arte, Sep 24 2014 *) CoefficientList[Series[(1 + 7 x + 10 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 27 2014 *) PROG (Python) for n in range (0, 100): print (9*n**2+1) (PARI) a(n)=9*n^2+1 \\ Charles R Greathouse IV, Sep 26 2014 (MAGMA) [9*n^2+1: n in [0..60]]; // Vincenzo Librandi, Sep 27 2014 CROSSREFS Cf. A016766, A158591 (36n^2 + 1), A156226 (primes of the form 9n^2 + 1). Cf. also A000290. Sequence in context: A139236 A212795 A227695 * A096000 A047672 A200872 Adjacent sequences:  A247789 A247790 A247791 * A247793 A247794 A247795 KEYWORD nonn,easy AUTHOR Karl V. Keller, Jr., Sep 23 2014 STATUS approved

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Last modified August 14 07:04 EDT 2020. Contains 336477 sequences. (Running on oeis4.)