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 A262221 #70 Feb 16 2025 08:33:27
%S A262221 1,26,76,151,251,376,526,701,901,1126,1376,1651,1951,2276,2626,3001,
%T A262221 3401,3826,4276,4751,5251,5776,6326,6901,7501,8126,8776,9451,10151,
%U A262221 10876,11626,12401,13201,14026,14876,15751,16651,17576,18526,19501,20501,21526,22576,23651
%N A262221 a(n) = 25*n*(n + 1)/2 + 1.
%C A262221 Also centered 25-gonal (or icosipentagonal) numbers.
%C A262221 This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
%C A262221 For k=2*n, the formula shown above gives A011379.
%C A262221 Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...
%D A262221 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).
%H A262221 Bruno Berselli, <a href="/A262221/b262221.txt">Table of n, a(n) for n = 0..1000</a>
%H A262221 Bruno Berselli, <a href="/A262221/a262221_2.txt">Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2</a>.
%H A262221 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>.
%H A262221 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.
%H A262221 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A262221 G.f.: (1 + 23*x + x^2)/(1 - x)^3.
%F A262221 a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A262221 a(n) = A123296(n) + 1.
%F A262221 a(n) = A000217(5*n+2) - 2.
%F A262221 a(n) = A034856(5*n+1).
%F A262221 a(n) = A186349(10*n+1).
%F A262221 a(n) = A054254(5*n+2) with n>0, a(0)=1.
%F A262221 a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.
%F A262221 Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
%F A262221 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A262221 Sum_{n>=0} a(n)/n! = 77*e/2.
%F A262221 Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
%F A262221 E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - _Elmo R. Oliveira_, Dec 24 2024
%t A262221 Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
%t A262221 25*Accumulate[Range[0,50]]+1 (* or *) LinearRecurrence[{3,-3,1},{1,26,76},50] (* _Harvey P. Dale_, Jan 29 2023 *)
%o A262221 (PARI) vector(50, n, n--; 25*n*(n+1)/2+1)
%o A262221 (Sage) [25*n*(n+1)/2+1 for n in (0..50)]
%o A262221 (Magma) [25*n*(n+1)/2+1: n in [0..50]];
%Y A262221 Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.
%Y A262221 Cf. centered polygonal numbers listed in A069190.
%Y A262221 Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.
%K A262221 nonn,easy
%O A262221 0,2
%A A262221 _Bruno Berselli_, Sep 15 2015