This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A061793 #47 Dec 25 2024 21:00:19 %S A061793 3,28,78,153,253,378,528,703,903,1128,1378,1653,1953,2278,2628,3003, %T A061793 3403,3828,4278,4753,5253,5778,6328,6903,7503,8128,8778,9453,10153, %U A061793 10878,11628,12403,13203,14028,14878,15753,16653,17578,18528,19503,20503,21528,22578,23653 %N A061793 a(n) = 25*n*(n + 1)/2 + 3. %C A061793 "If m is a triangular number, then so are 9*m+1, 25*m+3 and 49*m+6. (Euler, 1775)", see Burton in References. Note that A060544 is the same as 9*m+1 when m is triangular and that 9*(m*(m+1)/2)+1 is another formula for it. %C A061793 9*m+1, 25*m+3 and 49*m+6 are special cases of the identity A000290(2*r + 1)*A000217(s) + A000217(r) = A000217((2*r + 1)*s + r). - _Bruno Berselli_, Mar 01 2018 %C A061793 Complementing the previous comment, with T(n) = A000217(n), 4*T(s)+1+s = T(2*s+1), 16*T(s)+3+2s = T(4*s+2) and 36*T(s)+6+3s = T(6*s+3) are special cases of the identity A000290(2*r)*T(s) + T(r) + r*s = T(2*r*s + r). - _Charlie Marion_, Mar 28 2018 %D A061793 D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc. Boston, MA, 1976, p. 17. %H A061793 Harry J. Smith, <a href="/A061793/b061793.txt">Table of n, a(n) for n = 0..1000</a> %H A061793 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1). %F A061793 a(n) = 25*A000217(n) + 3 = A123296(n) + 3. %F A061793 From _Elmo R. Oliveira_, Oct 23 2024: (Start) %F A061793 G.f.: (3 + 19*x + 3*x^2)/(1 - x)^3. %F A061793 E.g.f.: (3 + 25*x + 25*x^2/2)*exp(x). %F A061793 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End) %p A061793 [seq(25*(n*(n+1)/2)+3,n=0..40)]; # _Muniru A Asiru_, Mar 30 2018 %t A061793 25*Accumulate[Range[0,40]]+3 (* _Harvey P. Dale_, Aug 26 2013 *) %o A061793 (PARI) a(n) = 25*n*(n + 1)/2 + 3 %o A061793 (GAP) List([0..40],n->25*(n*(n+1)/2)+3); # _Muniru A Asiru_, Mar 30 2018 %Y A061793 Cf. A000217, A000290, A060544, A123296. %K A061793 nonn,easy %O A061793 0,1 %A A061793 _Jason Earls_, Jun 22 2001 %E A061793 Corrected by _T. D. Noe_, Oct 25 2006