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 A076454 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way. 9
 1, 21, 102, 310, 735, 1491, 2716, 4572, 7245, 10945, 15906, 22386, 30667, 41055, 53880, 69496, 88281, 110637, 136990, 167790, 203511, 244651, 291732, 345300, 405925, 474201, 550746, 636202, 731235, 836535, 952816, 1080816, 1221297, 1375045, 1542870, 1725606, 1924111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence is related to A007585 by a(n) = n*A007585(n)-sum(i=0..n-1, A007585(i)). - Vincenzo Librandi, Aug 08 2010 In fact, this is the case d=4 in the identity n*(n*(n+1)*(2*d*n-2*d+3)/6)-sum(k*(k+1)*(2*d*k-2*d+3)/6, k=0..n-1) = n*(n+1)*(3*d*n^2-d*n+4*n-2*d+2)/12. - Bruno Berselli, Mar 01 2012 Bisection of A233329 (up to an offset). - L. Edson Jeffery, Jan 23 2014 REFERENCES Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian). Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = n*(n+1)*(2*n^2-1)/2. G.f.: x*(1+16*x+7*x^2)/(1-x)^5. a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), n>=6, with a(1)=1, a(2)=21, a(3)=102, a(4)=310, a(5)=735. - L. Edson Jeffery, Dec 30 2013 MAPLE seq(1/2*n*(n+1)*(2*n^2-1), n=1..40); MATHEMATICA CoefficientList[Series[(1 + 16 x + 7 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 21, 102, 310, 735}, 40] (* Harvey P. Dale, Jun 30 2023 *) PROG (Magma) [n*(n+1)*(2*n^2-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013 CROSSREFS Cf. A002417, A007585, A076455-A076459, A233329. Sequence in context: A356738 A044272 A044653 * A208538 A303681 A304132 Adjacent sequences: A076451 A076452 A076453 * A076455 A076456 A076457 KEYWORD nonn,easy AUTHOR Floor van Lamoen, Oct 13 2002 EXTENSIONS Comments rewritten from Bruno Berselli, Mar 01 2012 More terms from Vincenzo Librandi, Dec 30 2013 STATUS approved

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Last modified August 10 18:57 EDT 2024. Contains 375058 sequences. (Running on oeis4.)