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 A316224 a(n) = n*(2*n + 1)*(4*n + 1). 3
 0, 15, 90, 273, 612, 1155, 1950, 3045, 4488, 6327, 8610, 11385, 14700, 18603, 23142, 28365, 34320, 41055, 48618, 57057, 66420, 76755, 88110, 100533, 114072, 128775, 144690, 161865, 180348, 200187, 221430, 244125, 268320, 294063, 321402, 350385, 381060, 413475, 447678, 483717 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Sums of the consecutive integers from A000384(n) to A000384(n+1)-1. This is the case s=6 of the formula n*(n*(s-2) + 1)*(n*(s-2) + 2)/2 related to s-gonal numbers. The inverse binomial transform is 0, 15, 60, 48, 0, ... (0 continued). LINKS Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA O.g.f.: 3*x*(5 + 10*x + x^2)/(1 - x)^4. E.g.f.: x*(15 + 30*x + 8*x^2)*exp(x). a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). a(n) =  3*A258582(n). a(n) = -3*A100157(-n). Sum_{n>0} 1/a(n) = 2*(3 - log(4)) - Pi. EXAMPLE Row sums of the triangle: |  0 |  ................................................................. 0 |  1 |  2  3  4  5  .................................................... 15 |  6 |  7  8  9 10 11 12 13 14  ........................................ 90 | 15 | 16 17 18 19 20 21 22 23 24 25 26 27  ........................... 273 | 28 | 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  ............... 612 | 45 | 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65  .. 1155 ... where: . first column is A000384, . second column is A130883 (without 1), . third column is A033816, . diagonal is A014106, . 0, 2, 8, 18, 32, 50, ... are in A001105. MAPLE seq(n*(2*n+1)*(4*n+1), n=0..40); # Muniru A Asiru, Jun 27 2018 MATHEMATICA Table[n (2 n + 1) (4 n + 1), {n, 0, 40}] PROG (PARI) vector(40, n, n--; n*(2*n+1)*(4*n+1)) (Sage) [n*(2*n+1)*(4*n+1) for n in (0..40)] (Maxima) makelist(n*(2*n+1)*(4*n+1), n, 0, 40); (GAP) List([0..40], n -> n*(2*n+1)*(4*n+1)); (MAGMA) [n*(2*n+1)*(4*n+1): n in [0..40]]; (Python) [n*(2*n+1)*(4*n+1) for n in range(40)] (Julia) [n*(2*n+1)*(4*n+1) for n in 0:40] |> println CROSSREFS First bisection of A059270 and subsequence of A034828, A047866, A109900, A290168. Sums of the consecutive integers from P(s,n) to P(s,n+1)-1, where P(s,k) is the k-th s-gonal number: A027480 (s=3), A055112 (s=4), A228888 (s=5). Sequence in context: A164541 A145789 A010822 * A022707 A323334 A151974 Adjacent sequences:  A316221 A316222 A316223 * A316225 A316226 A316227 KEYWORD nonn,easy AUTHOR Bruno Berselli, Jun 27 2018 STATUS approved

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Last modified February 29 09:35 EST 2020. Contains 332355 sequences. (Running on oeis4.)